Download LECTURE NOTES IN TOPOLOGICAL GROUPS 1

LECTURE NOTES IN TOPOLOGICAL GROUPS 1-12
06.02.14
MICHAEL MEGRELISHVILI
Contents
1. Lecture 1
2. Lecture 2
3. Lecture 3
4. Lecture 4
5. Lecture 5
6. Lectures 6 and 7
7. Lecture 8
8. Lecture 9
9. Lectures 10,11,12
References
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1. Lecture 1
Definition 1.1. Let (G, m) be a group and τ be a topology on G. We say that
(G, m, τ ) (or, simply, G) is a topological group if the two basic operations
m : G × G → G, (x, y) 7→ m(x, y) := xy
and the inversion
i : G → G, x 7→ x−1
are continuous.
We say also that τ is a group topology on the group G.
By TGr we denote the class of all topological groups. In most cases later we consider
Hausdorff groups. Morphisms in TGr are continuous homomorphisms. For example,
id : (R, τdiscrete ) → R is a continuous homomorphism but not id−1 . Algebraically id
is of course an isomorphism. That is, isomorphism in GR but not in TGr.
Note that every group with the discrete topology is a topological group. Hence,
Gr ⊂ T Gr. Trivial topology is also a group topology on every group. We say that
a topological group is Hausdorff, compact, metrizable, separable etc. if the given
topology on G satisfies the corresponding topological property.
Remarks 1.2.
Date: February 06, 2014.
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(1) (a short version) 1.1 is equivalent to the following condition:
G × G → G, (x, y) 7→ xy −1
is continuous.
(2) (in terms of nbds) 1.1 is equivalent to the following conditions:
(a) ∀U ∈ N (xy) ∃V ∈ N (x), W ∈ N (y) : V W ⊂ U
(b) ∀U ∈ N (x−1 ) ∃V ∈ N (x) : V −1 ⊂ U .
Note that (b) is equivalent to
(b’) ∀U ∈ N (x−1 ) U −1 ∈ N (x).
(3) (in terms of (generalized) sequences) If G is metrizable then 1.1 is equivalent
to the following condition:
xn → x, yn → y ⇒ xn yn−1 → xy −1
In general one may use the nets ( generalized sequences).
Exercise 1.3.
(1) Show that R2 and R are nonhomeomorphic topological spaces
but algebraically these groups are isomorphic.
(2) Show that the discrete spaces Z and Z × Z2 are homeomorphic but as groups
they are not isomorphic.
Definition 1.4.
(1) In terms of Definition 1.1 we say that G is: paratopological group if m is
continuous.
(2) Let (S, m) be a semigroup and τ be a topology on S. We say that the semigroup S is:
(a) topological semigroup if m : S × S → S is continuous. So, a paratopological group is a topological semigroup.
(b) semitopological if m is separately continuous. That is, if all left and right
translations
la : S → S, x 7→ ax,
ra : S → S, x 7→ xa
are continuous for every a ∈ S.
(c) right (left) topological if right (left) translations of S are continuous.
Example 1.5.
(1) (para but not topo) Let τs be the Sorgenfrey topology (standard
topological base is {[a, b)}) on the group (R, +) of all reals. Then (R, τs , +) is
a paratopological but not topological group.
Hint: [0, 1) ∈ τs but (−1, 0] ∈
/ τs .
(2) (semi but not para) For every group G the pair (G, τcof ) (with the cofinite
topology) is a semitopological group which satisfies T1 . It is a paratopological
group iff G is finite.
(3) (right but not left) For every topological space X consider the semigroup
(X X , ◦) of all selfmaps wrt product (=pointwise) topology. Then X X is right
topological. * If X ∈ T1 the teft translation lf : X X → X X is continuous iff
f ∈ C(X, X). Note that if X is compact then X X is compact by the Tychonoff
theorem.
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2. Lecture 2
Definition 2.1. A topological space X is said to be homogeneous if for every x, y ∈ X
there exists an autohomeomorphism h : X → X (notation: h ∈ H(X)) s.t. h(x) = y.
Lemma 2.2. For every semitopological group the left (right) translations are homeomorphism. For every topological group the inversion map is a homeomorphism.
Proof. Observe that la−1 = la−1 (ra−1 = ra−1 ) and i−1 = i.
Proposition 2.3. Every semitopological group (hence, also, every topological, as a
topological space) is homogeneous.
Definition 2.4. A topological space (X, τ ) is said to be of group type if there exists
a group structure m on X such that (X, τ, m) is a topological group.
Remarks 2.5.
(1) [0, 1]n for every natural n is not homogeneous hence not of
group type.
(2) The Hilbert cube [0, 1]N is homogeneous (Keller). At the same time it has the
fixed point property: every continuous map h : [0, 1]N → [0, 1]N has a fixed
point. It follows that the Hilbert cube is not of group type. Moreover, there is
no structure of a left (right) topological group on it.
(3) The Cantor set C ⊂ [0, 1] is of group type. Indeed, C is homeomorphic to the
topological space ZN2 , which is a topological group.
(4) The space of all irrational numbers R \ Q is of group type being homeomorphic
to ZN .
(5) The Sorgenfrey line as a topological space is homogeneous but not of group type
(up to a non-trivial theorem of Kakutani below: every Hausdorff topological
group with the first countable property B1 is metrizable).
Some examples of topological groups:
(1) Every group in the discrete topology.
(2) GLn (R) are locally compact metrizable topological group. It, as a metric
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space, is embedded isometrically into the Euclidean metric space Rn .
(3) The orthogonal group On (R) ⊂ GLn (R) is compact by Heine-Borel thm being
a bounded and a closed subset in the metric space GLn (R).
(4) TGr is closed under: subgroups, factor-groups, topological products, box
products.
(5) Every Euclidean space Rn and Tn the n-dimensional torus.
(6) Every normed space (more generally, any linear topological space).
(7) (Z, dp ) the integers wrt the p-adic metric. It is a precompact group (totally
bounded in its metric) and its completion is the compact topological group of
all p-adic integers.
(8) For every compact space K the group of all autohomeomorphisms H(K) endowed with the so-called compact-open topology (we define it later).
(9) For every metric space (X, d) the group of all onto isometries Iso(X, d) ⊂ X X
endowed with the pointwise topology inherited from X X .
(10) For every Banach space (V, ||·||) the group Isolin (V ) of all linear onto isometries
V → V endowed with the pointwise topology inherited from V V . For example,
if V := Rn is the Euclidean space then Isolin (V ) = On (R) the orthogonal
group.
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Note that, in contrast to the case of Rn , for infinite dimensional V the topological
group Isolin (V ) as usual is not compact. Moreover, Teleman’s theorems show that
every Hausdorff topological group G is embedded into Isolin (V ) for suitable V and
also into some H(K) for suitable compact space K. As we will see below even groups
like Z and R cannot be embedded into compact groups. We examine the question
which topological groups admit representations on good Banach spaces (like: Hilbert,
reflexive, ...). For these purposes we give a necessary basic material for topological
group theory. Including among others: first steps in uniform structures and uniformly
continuous functions on groups.
We touch also some questions from the van-Kampen Pontryagin duality theory for
locally compact abelian groups.
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2.1. First homework. Let G be a topological group.
Exercise 2.6. Prove that
(1) cl(A−1 ) = cl(A)−1 and cl(A)cl(B) ⊂ cl(AB) for every subsets A, B of G.
(2) If H ≤ G is a subgroup then cl(H) ≤ G is also a subgroup.
(3) If H G is a normal subgroup then cl(H) G is also a normal subgroup.
(4) If G, in addition, is abelian and H ≤ G then cl(H) ≤ G is also an abelian
subgroup. Give a counterexample if G is not Hausdorff.
Exercise 2.7. Prove that the function
(1) Gn → G, (x1 , x2 , · · · , xn ) 7→ xk11 xk22 · · · xknn is continuous for every given tuple
(k1 , k2 , · · · , kn ) ∈ Zn .
(2) For every nbd U ∈ N (e) of the identity e ∈ G and every given natural n ∈ N
there exists V ∈ N (e) such that V = V −1 and V n := V
· · · V} ⊂ U .
| V {z
n times
Exercise 2.8. Prove that
(1) the inversion map, left and right translations, all conjugations, are homeomorphisms G → G.
(2) G is homogeneous as a topological space.
(3) * For every pair (x, y) ∈ G × G there exists f ∈ Homeo(G, G) such that
f (x) = y and f (y) = x.
(4) Which of the following topological spaces are of the group type:
(a) (R, τs ) the Sorgenfrey line.
(b) X := {x ∈ R2 : ||x|| = 5}.
(c) X := {x ∈ R3 : ||x|| < 5}.
(d) The integers Z with the cofinite topology.
Exercise 2.9. Let A and B are subsets of G and g ∈ G. Prove that:
(1) If A is open then gA and AB are open in G.
(2) If A and B are compact then AB is also compact.
(3) If A and B are connected then AB is also connected.
(4) If A and B are closed then AB need not be closed.
(5) * If A is closed and B is compact then AB is closed.
(6) cl(A) = ∩V ∈N (e) V A = ∩V ∈N (e) V A.
Exercise 2.10. Let G be a countable topological group which is either: a) metrizable
by a complete metric; or b) locally compact and Hausdorff. Show that G is discrete.
Hint: Use the Baire Category theorem: For every Hausdorff space X which is is
either: a) metrizable by a complete metric; or b) locally compact the following holds.
For every countable cover X = ∪n∈N An where each is An is closed in X at least one of
the sets contains an interior point. That is, there exists k ∈ N such that int(Ak ) 6= ∅.
Remark: conclude that there is no complete metric on the space Q of all rationals.
Exercise 2.11.
(1) Let {Gn }n∈N be a sequence of topological groups where each
Gn is a (separable)
metrizable topological group. Show that the topological
Q
product n∈N Gn endowed with the usual Tychonoff topology is a (separable)
metrizable topological group.
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(2) ** Let {Gn }n∈N be a sequence of topological groups where each Gn is the
topological group Q of all rational numbers carrying the usual topology. Let
G := ⊕n∈N Gn = {x = (x1 , x2 , · · · ) : almost all coordinates are 0}
Q
be the direct sum endowed with the box topology τbox inherited from n∈N Gn .
Prove that (G, τbox ) is a countable non-metrizable Hausdorff topological
group.
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3. Lecture 3
Proposition 3.1. (Basic properties of nbd’s at e)
For every topological group G and every local base γ at e we have:
(1) ∀U ∈ γ ∃V ∈ γ : V 2 ⊂ U ;
(2) ∀U ∈ γ ∃V ∈ γ : V −1 ⊂ U ;
(3) ∀U ∈ γ ∀a ∈ G ∃V ∈ γ : aV a−1 ⊂ U .
Exercise 3.2. ∀G ∈ T Gr we have
(1) ∀U ∈ N (e) ∀k ∈ N ∃V ∈ N (e) : V −1 = V, V is open and V k ⊂ U.
(2) ∀U ∈ N (e) ∀compact subset K ⊂ G ∃V ∈ N (e) : xV x−1 ⊂ U ∀x ∈ K.
Lemma 3.3. (Some useful properties) Let G be a topological group. Then
(1) N (x) = xN (e) := {xU : U ∈ N (e)} and N (x) = N (e)x for every x ∈ G.
For every local base γ at e the system xγ is a local base at x ∈ G.
(2) G is discrete iff G contains an isolated point.
(3) Every conjugation is a homeomorphism.
(4) N (e)−1 = N (e).
(5) for every O ∈ τ and every A ⊂ G we have AO and OA are open in G.
(6) cl(A) ⊂ AV ∀V ∈ N (e) ∀A ⊂ G.
(7) (Homework 1) cl(A) = ∩{AV : V ∈ N (e)} = ∩{V A : V ∈ N (e)}.
(8) If G is T2 then the center Z(G) is closed in G.
Proof. We show (6) cl(A) ⊂ AV ∀V ∈ N (e) ∀A ⊂ G.
Let x ∈ cl(A). Then for every V ∈ N (e) we get
xV −1 ∩ A 6= ∅
So,
x ∈ AV.
(8) Using G ∈ T2 show that every stationary subgroup Sta := {x ∈ G : axa−1 = x}
is closed. Indeed, Sta is the closed subset fa−1 (e), where fa is the following continuous
map
fa : G → G, x 7→ axa−1 x−1 .
Now observe that Z(G) = ∩{Sta : a ∈ G}.
3.1. Separation axioms.
Theorem 3.4. Let G be a topological group. TFAE:
(1) G is T0 .
(2) G is T1 ({e} is closed in G).
(3) G is T2 (Hausdorff ).
(4) G is T3 (regular).
(5) * G is T3.5 (Tychonoff = completely regular).
Proof. Here we prove only the equivalence of (1),(2),(3) and (4).
(1) ⇒ (2): Let x 6= y ∈ G. Then by (1), without restriction of generality, say for
x, there exists
∃ U ∈ N (x) s.t. y ∈
/U
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Then ∃V ∈ N (e) : xV ⊂ U. From here
x∈
/ yV −1
but yV −1 ∈ N (y).
(2) ⇒ (4): Well known Lemma: X ∈ T3 is equivalent to the following: for every
x ∈ X and every U ∈ N (x) there exists V ∈ N (X) such that cl(V ) ⊂ U .
By the homogeneity of G it suffices to verify this for x := e ∈ G.
By Lemma 3.3 we have
cl(A) ⊂ AV ∀V ∈ N (e) ∀A ⊂ G
Let U ∈ N (e). Choose V ∈ N (e) s.t. V 2 ⊂ U . Then
cl(V ) ⊂ V 2 ⊂ U.
(4) ⇒ (1): Is trivial.
Corollary 3.5. G is Hausdorff iff {e} is closed iff for every e 6= a ∈ G there exists
U ∈ N (e) : a ∈
/ U iff ∩{U : U ∈ N (e)} = {e}.
Proposition 3.6.
(1) Every open subgroup H ≤ G is clopen in G.
(2) A subgroup H ≤ G is open iff int(H) 6= ∅.
(3) Let U ∈ N (e) be symmetric (that is, U −1 = U ). Then H := ∪n∈N U n is an
open subgroup of G.
Proof. (1) All cosets xH x ∈ G are open. So O := ∪{xH : x ∈ G, x =
6 e} is open,
too. Therefore, its complement G \ O = H is closed.
(2) If O is a nonempty subset of G and if O ⊂ H then H = ∪{hO : h ∈ H}.
(3) Observe that HH ⊂ H and H −1 = H.
Theorem 3.7. Let G ∈ T Gr ∩ T2 be a Hausdorff topological group and H ≤ G be its
topological subgroup. If H is locally compact then H is closed in G.
Proof. It is equivalent to prove in the case of cl(H) = G. So we have to show that H is
closed in cl(H). By Proposition 3.6 it suffices to show that H is open in G = Cl(H).
Since H is LC one may choose a compact nbd K of e in H.
∃U ∈ NG (e) ∩ τ : U ∩ H ⊂ K
U = U ∩ G = U ∩ cl(H) ⊂ cl(U ∩ H) ⊂ cl(K) = K
(remark1: for every open O ⊂ X and A ⊂ X we have O ∩ cl(A) ⊂ cl(O ∩ A))
(remark2: every compact subset is closed in a Hausdorff space)
So, U ⊂ K. Therefore, U ⊂ H. Hence, intG (H) 6= ∅. By Proposition 3.6 we
conclude that H is open in cl(H). Hence, also closed. So, H = cl(H).
Corollary 3.8. It is impossible to embed a locally compact noncompact group into any
Hausdorff compact group. In particular, there is no finite-dimensional topologically
faithful representation by linear isometries of a locally compact noncompact groups
(like Z, R) on finite-dimensional Euclidean spaces.
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Example 3.9. Show that every locally compact Hausdorff group G can be embedded
into a compact Hausdorff semitopological semigroup.
Hint: Use the 1-point (Alexandrov) compactification.
Definition 3.10. Let X be a topological space. A compactification of X is a continuous map f : X → Y where Y is a compact Hausdorff space and f (X) is dense in Y .
We say: proper compactification when, in addition, f is required to be a topological
embedding.
One of the standard examples of a proper compactification is the so-called 1-point
compactification ν : X ,→ X∞ := X ∪ {∞} defined for every locally compact noncompact Hausdorff space (X, τ ). Recall the topology
τ∞ := τ ∪ {X∞ \ K : K is compact in X}.
Important example of a compactification is the so-called maximal (or, Stone-Chech)
compactification β : X → βX which is proper iff X ∈ T3.5 . See for example the file
of Doron Ben Hadar downloadable from the course homepage.
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3.2. Homework 2.
Exercise 3.11. Let G1 , G2 be topological groups and f : G1 → G2 be a homomorphisms which is continuous at the point e ∈ G1 . Show that f is continuous.
Exercise 3.12. Let G be a topological group. Prove that:
(1) ∀U ∈ N (e) ∀ compact subset K ⊂ G ∃V ∈ N (e) :
xV x−1 ⊂ U ∀x ∈ K;
(2) for every compact subset K ⊂ G and a closed subset A ⊂ G with K ∩ A = ∅
there exists U ∈ N (e) s.t. U K ∩ A = ∅.
Exercise 3.13. Show that for every connected topological group G and every nbd
U ∈ N (e) we have
G = ∪n∈N U n .
Conclude that, in particular, U algebraically generates G.
A topological group G is said to be compactly generated if there exists a compact
subset K ⊂ G which algebraically generates G; that is < K >= G. For example,
every compact group, Rn , Rn × TS .
A topological space X is σ-compact if X = ∪n∈N Kn where each Kn ⊂ X is compact.
Exercise 3.14.
(1) Show that every connected LC topological group G is compactly generated and σ-compact.
(2) Give an example of a σ-compact topological group which is not compactly
generated.
Definition 3.15. Let (Y, τ ) be a topological space and X be a set. Denote by Y X
the set of all maps f : X → Y endowed with the product topology of Y X . This
topology has the standard base α which consists of all the sets:
O(x1 , · · · , xn ; U1 , · · · , Un ) := {f ∈ Y X : f (xi ) ∈ Ui }
where, F := {x1 , · · · , xn } is a finite subset of X (all xi ’s are pairwise distinct) and
Ui are nonempty open subsets in Y . Other names of this topology are: pointwise
topology, point-open topology.
Exercise 3.16.
(1) For every topological space X consider the semigroup (X X , ◦)
of all selfmaps f : X → X wrt pointwise (=product) topology. Show that X X
is a right topological semigroup.
(2) C(X, X) is a semitopological subsemigroup of X X .
* Is it true that C([0, 1], [0, 1]) is a topological semigroup ?
(3) ** Prove that the left translation lf : X X → X X is continuous if and only if
f ∈ C(X, X). Derive that if X is T1 , then the right topological semigroup
X X is semitopological iff X is discrete.
Definition 3.17. Let X be a topological space. A compactification of X is a continuous map f : X → Y where Y is a compact Hausdorff space and f (X) is dense in Y .
We say: proper compactification when, in addition, f is required to be a topological
embedding.
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One of the standard examples of a proper compactification is the so-called 1-point
compactification ν : X ,→ X∞ := X ∪ {∞} defined for every locally compact noncompact Hausdorff space (X, τ ). Recall the topology
τ∞ := τ ∪ {X∞ \ K : K is compact in X}.
Important example of a compactification is the so-called maximal (or, Stone-Chech)
compactification β : X → βX which is proper iff X ∈ T3.5 . See for example the file
of Doron Ben Hadar downloadable from the course homepage.
Exercise 3.18.
(1) Let S := R ∪ {∞} be the 1-point compactification of R. Define the ”usual”
operation + on S: x + y is already defined for x, y ∈ R. Otherwise, x + y = ∞
(that is, x+∞ = ∞+x = ∞+∞ = ∞). Show that (S, +) is a semitopological
but not topological semigroup.
(2) More generally, let (G, ·, τ ) be a locally compact non-compact Hausdorff topological group. Denote by S := G ∪ {∞} the 1-point compactification of G.
* Show that (S, ·, τ∞ ) is a semitopological but not topological semigroup.
Remark 3.19. As we know a locally compact Hausdorff group G admits an embedding
into a compact Hausdorff group iff G is compact. Exercise 3.18 shows that such G at
least admits a proper semigroup compactification ν : G ,→ S such that S is a compact
semitopological monoid.
4. Lecture 4
After Corollary 3.8 and Example 3.18 we give some remarks.
The semitopological version of Example 3.18 is false as it follows from the following.
Proposition 4.1. If S is a compact Hausdorff topological semigroup and if G is a
subgroup of S then cl(G) is a (compact) topological group (note that if S is a monoid
with the identity element eS then G is not necessarily a submonoid of S. That is, the
identity element eG of G is not necessarily eS ).
Proof. (EXERCISE)
In particular, it follows that R cannot be embedded into the compact topological
semigroup Θ(Rn ) := {f ∈ L(Rn , Rn ) : ||f || ≤ 1} of all non-expanding linear selfoperators. It can be identified with the monoid of all matrices A = (aij ) (size n × n)
such that |aij | ≤ 1.
Definition 4.2. Let G be a Hausdorff topological group. We say that f : G → S is
a semigroup compactification of G if:
(1)
(2)
(3)
(4)
f is a compactification (see Definition 3.10);
S is a compact Hausdorff right topological semigroup (see Definition 1.4);
f is a homomorphism of semigroups;
f (G) ⊂ Λ(S), where Λ(S) := {a ∈ S : la : S → S is continuous} (topological
centre of S).
Observe that then S is necessarily a monoid and f (eG ) is its neutral element.
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Remark 4.3.
(1) A Hausdorff topological group can be embedded into a compact
Hausdorff topological semigroup iff G can be embedded into a compact Hausdorff topological group.
(2) Every LC T2 topological group G admits a proper semitopological compactification (as we already know a LC T2 topological group G admits a proper
compact group compactification iff G admits a proper compactification into a
compact topological semigroup iff G is compact).
(3) * (MM, 2001) There exists a separable metrizable (complete) topological group
G such that it does not admit a proper semitopological semigroup compactification. Namely, one may take G := H+ [0, 1], the group of all orientation
preserving homeomorphisms of the unit interval [0, 1] endowed with the compact open topology. The same topology on H+ can be defined by the following
metric
d(f1 , f2 ) := sup |(f1 (t) − f2 (t)|)
0≤t≤1
The same group H+ [0, 1] cannot be embedded into Isolin V for any reflexive
(see Remark 4.4) Banach space V .
(4) * An example of an important right topological semigroup compactification
(which is not semitopological) is the maximal (Stone-Chech) compactification
β : Z ,→ βZ
of the group Z. Note that the standard semigroup structure on the semigroup
βZ is not commutative. This compactification comes from the algebra Cb (Z) =
l∞ (Z) of all (continuous) bounded functions on Z.
(5) Any T2 topological group G admits a proper right topological semigroup
compactification. Namely, one may consider the compactification βG : G ,→
βG G. This compactification comes from the algebra RUCb (G) of all bounded
right uniformly continuous functions f : G → R. The latter means that
∀ε > 0 ∀g0 ∈ G ∃U ∈ N (g0 ) : |f (gx) − f (g0 x)| < ε ∀g ∈ U ∀x ∈ G
Remark 4.4. Recall that for every Banach space V one may define the dual Banach
space V ∗ := {f : V → R} of all continuous linear functionals. Consider the canonical
bilinear map V ∗ × V → R, (f, v) 7→ f (v). Then it naturally induces the canonical
isometric inclusion map i : V → V ∗∗ into the second dual. When this map is onto
then V is said to be reflexive. For example, every Hilbert space is reflexive. An
example of a reflexive space which is not Hilbert is any lp with 1 < p < ∞, p 6= 2. A
nonreflexive separable Banach space (with separable dual) is for example c0 .
4.1. Homomorphisms and factor groups. Let q : X → Y be an onto map and
(X, τ ) ∈ T OP . Recall that the quotient topology on Y is defined by τY := {A ⊂
Y : q −1 (A) ∈ τ }. Then q is continuous and τY is the strongest topology making q
continuous. If Y carries the quotient topology wrt q : X → Y then q is said to be a
quotient map. Moreover we have the following useful lemma.
Lemma 4.5. Let f1 : X → Z, f2 : Z → Y be continuous maps. Suppose that
f2 ◦ f1 : X → Y is an onto quotient map. Then f2 is also a quotient map.
Remark 4.6. In particular, if f2 is an identity continuous map then it is a homeomorphism iff it is a quotient.
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A trivial example of a continuous onto homomorphism of topological groups which
is not a quotient is the following map
id : (R, τdiscr ) → R.
Recall also that particular examples of quotient maps are closed continuous onto
maps and open continuous onto maps.
It turns out that for continuous onto homomorphisms between topological groups
every quotient map necessarily is open (not always closed, however; take for example
any projection R2 → R).
Proposition 4.7. Let H G ∈ T Gr. Then: G/H wrt quotient topology is a topological group and the canonical onto homomorphism
q : G → G/H, x 7→ [x] = xH
is continuous and open.
Proof. The continuity of q follows from the definition of the quotient topology. We
show that q is open. Observe that
q −1 (q(U )) = U H = ∪{uH : u ∈ U } = ∪{U h : h ∈ H}
is open for every open U ⊂ G as a union of open sets. Now it is easy to see by
definitions that G/H is a topological group.
Exercise 4.8. R/Z is topologically isomorphic to T. More generally, Rn /Zn is topologically isomorphic to Tn .
Proof. Observe that f : R → T, f (t) := cis(2πt) = e2πit is open because it is ”locally
open”. Indeed, for every x0 ∈ R the image f (x0 − ε, x0 + ε) is open in T for every
0 < ε ≤ 1. Now use Lemma 4.5 and Remark 4.6.
Lemma 4.9. (Properties of the quotient groups G/H)
(1) G/H is Hausdorff iff H is closed in G.
(2) G/H is discrete iff H is open in G.
(3) Any closed subgroup H ≤ G of the finite index [G : H] < ∞ is clopen.
(4) If G is LC then G/H is also LC.
(5) (up to the Kakutani thm mentioned above)
If G is metrizable and H is closed then G/H is also metrizable (=B1 ).
(6) If f : G → Y is a continuous homomorphism and H ⊂ kerf then there exists
a unique continuous homomorphism ν : G/H → L such that f = ν ◦ π, where
π : G → G/H is the natural map.
(7) Let f : K → H be a continuous onto homomorphism of a compact group onto
the Hausdorff group H. Prove that f is open.
Proof. Exercise.
14
4.2. Homework 3 (returning date 18.11.13).
Exercise 4.10. * Prove or disprove: the topological group T2 (two-dimensional torus)
is a topological factor group of the group C r {0}.
Exercise 4.11. Prove or disprove: GLn (R)/D is a locally compact Hausdorff topological group, where D denotes the set of all invertible scalar matrices in GLn (R) for
every n ∈ N.
Exercise 4.12. Give a concrete example of a continuous onto homomorphism f : G1 →
G2 of Hausdorff separable metrizable topological groups which is not a quotient map.
Exercise 4.13. Let G be a topological group and H be its normal subgroup. Prove:
(1) G/H is discrete iff H is open in G.
(2) If G is Hausdorf and the normal subgroup H ≤ G is closed and has the finite
index [G : H] < ∞ then H is clopen.
(3) If G is LC then G/H is also LC.
(4) If G is metrizable then G/H is also metrizable (Hint: You may use the Kakutani thm mentioned above).
(5) Let f : G1 → G2 be a continuous homomorphism of topological groups. Assume that G1 is compact and G2 is Hausdorff. Show that f is an open map.
(6) If f : G → Y is a continuous homomorphism of topological groups and
H E G, H ⊂ kerf then there exists a unique continuous homomorphism
ν : G/H → L such that f = ν ◦ π, where π : G → G/H is the natural map.
Exercise 4.14. * If S is a compact Hausdorff topological semigroup and if G is a
subgroup of S then cl(G) is a (compact) topological group.
Hint: eG is an idempotent of S and also an identity of T := cl(G).
15
5. Lecture 5
5.1. Closed subgroups of R and T, Kronecker’s theorem. First of all we will
examine the following questions: what are the closed subgroups of R and T ? what
are the dense subgroups ?
We are going to prove that:
the closed subgroups of R are only:
R, aZ a ≥ 0
Every nondiscrete subgroup H ≤ R is dense in R.
the closed subgroups of T are only:
T, Ωn n ∈ N
Lemma 5.1. For every a ∈ R the set < a >= aZ is a closed discrete (and, of course,
cyclic) subgroup of R.
Theorem 5.2. Every nondiscrete subgroup H ≤ R is dense in R.
Proof. Assume that H is not discrete. Equivalently, 0 is not isolated in H. Hence,
ε ε
∀ ε > 0 ∃ gε ∈ (− , ) ∩ H 6= ∅
2 2
Clearly,
Mε :=< gε >:= {kgε : k ∈ Z} ≤ H.
This subgroup Mε is ε-dense in R. Then also H is ε-dense in R. It follows that H
is topologically dense in R (being ε-dense for every ε > 0).
Theorem 5.3. The closed subgroups of R are only:
R, aZ (a ≥ 0)
Proof. Assume that H is a closed subgroup of R and H 6= R. Then H is not dense in
R. Therefore, by Theorem 5.2 H is necessarily discrete. If H = {0} then take a = 0.
Without restriction of generality suppose that H 6= {0} is a discrete proper subgroup. Then there exists b ∈ H ∩ (0, ∞). The subset H ∩ [0, b] is compact (being
closed and bounded in R). Also, H ∩ [0, b] is discrete because H is discrete. It follows
that H ∩ [0, b] is finite. So, there exists the smallest positive element in H ∩ [0, b].
Denote it by a. We show that H =< a >. Indeed, for every x ∈ H we have:
(1) x − [ xa ]a ∈ H;
(2) 0 ≤ x − [ xa ]a < a.
Recall the following property (of the fractional part) 0 ≤ xa − [ xa ] < 1. This explains
the second fact. The first fact is easy because H is a subgroup and the integer part
[ xa ] ∈ Z.
Now by the minimality of a we necessarily have 0 = x − [ xa ]a. So, x ∈ aZ.
Theorem 5.4. The list of all (up to a topological isomorphism) Hausdorff quotients
of R is:
R, {0} and T.
16
Proof. Only the case of T is nontrivial.
By Theorem 5.3 (and the first isomorphism theorem for groups) every proper nontrivial Hausdorff quotient of R is R/aZ for some nonzero a ∈ R. Observe that R is a
topological field. Hence, for every nonzero a ∈ R the map
fa : R → R, x 7→ ax
is an automorphism of the topological group (R, +). Then it is clear that all R/aZ
(with a 6= 0) are naturally isomorphic. Finally recall that R/Z is T.
Theorem 5.5. Let a, b ∈ R. Consider the subgroup
H :=< a, b >= {na + mb : n, m ∈ Z} ≤ R
Then H is dense in R iff a, b are rationally independent.
Proof. By Theorem 5.3 we can conclude that H is not dense in R iff H is discrete
and cyclic. In this case there exists c ∈ R such that a, b ∈ cZ. So, a = k1 c, b = k2 c for
some k1 , k2 ∈ Z. Then c = ka1 = kb2 (WRG we can assume that a and b are nonzero).
So, a, b are rationally dependent, a contradiction.
Example 5.6.
(1) The subgroup
√
√
H =< 1, 2 >= {m + n 2 : m, n ∈ Z}
is dense in R.
√
(2) The subgroup A =< cis2π 2 > is dense in T.
Exercise 5.7. (Cyclic subgroups of T) Let a := cis(2πα) ∈ T. Then the order O(a) is
infinite if and only if α is irrational.
Let Ω∞ := ∪m∈N Ωm . Then:
(1) Ω∞ = {cis(2πα) : α ∈ Q}.
(2) The group Ω∞ is isomorphic to Q/Z.
(3) Ω∞ is a dense subgroup of T.
Theorem 5.8. (Kronecker’s theorem for dimension 1, ”Irrational Billiard” in the
circle group)
For every irrational α the cyclic subgroup < cis(2πα) > is dense in T.
Proof. Homework
Definition 5.9. A topological group G is said to be monothetic if it contains a dense
cyclic subgroup.
R is not monothetic because any cyclic subgroup of R is discrete.
Corollary 5.10. Every infinite cyclic subgroup of T is dense in T. In particular, T
is monothetic.
Theorem 5.11. Every closed proper subgroup H of T is the finite cyclic group
Ωm := {z ∈ C : z m = 1}
for some m ∈ Z.
17
Proof. q −1 (H) ≤ R is closed and proper. By Theorem 5.3 we know that q −1 (H) = aZ
is cyclic for some a ∈ R. Then its image H is also cyclic. By our assumption the
cyclic subgroup H is closed. Therefore, by Corollary 5.10 we conclude that H is Ωm
for some m ∈ N.
Corollary 5.12. The Hausdorff quotients of T are T and {0} (up to the topological
isomorphisms). Moreover, for every quotient onto homomorphism γ : T → T there
exists m ∈ Z such that γ(z) = z m ∀z ∈ T.
Proof. The first assertion is easy using Theorem 5.4 taking into account that the
composition of two quotients functions is a quotient. For the second assertion let
γ : R/Z → R/Z be an onto quotient homomorphism. WRG we can suppose that
Ker(γ) is a finite cyclic subgroup K :=< [ m1 ] >≤ R/Z (K is isomorphic to Ωm ).
Then we have the commutative diagram:
γ
R/Z
/ R/Z
II
O
II
II
f2
I
f1 II$
R/( m1 Z)
where
f1 : R/Z → (R/Z)/K = R/(
1
Z), f1 ([x]) = [x]
m
1
Z) → R/Z
m
is uniquely determined and satisfies f2 ([x]) = [mx].
f2 : R/(
Remark 5.13. One may reformulate Theorem 5.8 (1-dimensional Kronecker’s approximation thm) as follows. For every irrational β the sequence nβ − [nβ] (n ∈ Z) is
dense in [0, 1). Use the fact that the map q : R → R/Z is of period 1.
Equivalent formulations of Theorem 5.8:
(1) For every ε > 0 and every r ∈ R there exists an integer m ∈ Z with
|mα − r| < ε (mod 1)
(2) For every ε > 0 and every a ∈ R there exists an integer m ∈ Z with
|mα − r − n| < ε ∃n ∈ Z
Theorem 5.14. (Kronecker’s approximation theorem)
Let α1 , α2 , · · · , αn be a rationally independent finite family of real numbers. Then
for every ε > 0 and every n-tuple (r1 , r2 , · · · , rn ) of real numbers there exists an
integer m ∈ Z with
|mαk − rk | < ε (mod 1)
(equivalently, |mαk − rk − nk | < ε
∀k ∈ {1, · · · , n}
∀k ∈ {1, · · · , n} ∃nk ∈ Z)
Corollary 5.15. Every Tn is monothetic. That is, there exists a cyclic dense subgroup.
18
5.2. Homework 4 (returning date 25.11.13).
Exercise 5.16. Let G be a Hausdorff nondiscrete topological group. Show that in the
topological group G2 there exist at least 3 distinct but isomorphic proper (6= G2 )
nondiscrete closed subgroups.
Exercise 5.17. Show that there exist: an abelian topological group (G, +) and closed
subgroups H1 and H2 of G, such that the subgroup H1 + H2 is not closed. Show also
that it is impossible for G := T.
Exercise 5.18. (Cyclic subgroups of T) Let a := cis(2πα) ∈ T. Then the order O(a)
is infinite if and only if α is irrational.
Let Ω∞ := ∪m∈N Ωm . Then:
(1) Ω∞ = {cis(2πα) : α ∈ Q} ≤ T.
(2) The group Ω∞ is isomorphic to Q/Z.
(3) Ω∞ is a dense subgroup of T.
Exercise 5.19. Let G be a Hausdorff monothetic group.
(1) Show that G is abelian.
(2) * Give an example of a monothetic group G having a non-monothetic subgroup.
(3) If G is metrizable then the cardinality of the set G is not greater than the
cardinality of R.
Definition 5.20. A Hausdorff topological group G is said to be minimal if there is
no strictly coarser Hausdorff group topology on G.
Exercise 5.21. * Which of the following topological groups are minimal: Z, R, T ?
Exercise 5.22. Let S be the interval [0, 1] with the multiplication
(
t, if 0 ≤ t < 21 ;
st =
1, if 12 ≤ t ≤ 1.
Show that: S is a compact right topological semigroup with Λ(S) = ∅. The subset
T := [0, 21 ) is a subsemigroup of S and cl(T ) = [0, 12 ] is not a subsemigroup of S.
Exercise 5.23. Let S := Z∪{−∞, ∞} be the two-point compactification of Z. Extend
the usual addition by:
n + t = t + n = s + t = t n ∈ Z, s, t ∈ {−∞, ∞}
Show: (S, +) is a noncommutative compact right topological monoid having dense
commutative topological centre Λ(S) = Z. S is not semitopological.
19
6. Lectures 6 and 7
Theorem 6.1. (Kronecker’s approximation theorem)
Let α1 , β2 , · · · , αn be a rationally independent finite family of real numbers. Then
for every ε > 0 and every n-tuple (r1 , r2 , · · · , rn ) of real numbers there exists an
integer m ∈ Z with
|mαk − rk | < ε (mod 1)
(equivalently, |mαk − rk − nk | < ε
∀k ∈ {1, · · · , n}
∀k ∈ {1, · · · , n} ∃nk ∈ Z)
Corollary 6.2. Every Tn is monothetic. That is, there exists a cyclic dense subgroup.
We give an independent direct proof for the following
Theorem 6.3.
(1) The function
f : R → T2 , x 7→ ([x], [αx]) = (cis(2πx), cis(α2πx))
is a continuous homomorphism with the dense image.
(2) The subgroups H1 := {([n], [αn]) : n ∈ Z} < H := {([x], [αx]) : x ∈ R} < T2
are dense in T2 .
(3) The image of the line {(x, y) ∈ R2 : y = αx} into the torus T2 is dense.
Proof. (2) We have a flow (a group action of R on T2 ) R × T2 → T2 defined by
t([x1 ], [x2 ]) := ([x1 + t], [x2 + αt])
We show that the Z-orbit (hence, also R-orbit) of the point ([0], [0]) ∈ T2 is dense
in T2 . It suffices to check that this orbit intersects every ”meridian circle”
Sa := {([a], [x]) : x ∈ R}
in a dense subset of Sa for every given a ∈ R.
Claim: the point ([t], [αt]) of the orbit of ([0], [0]) is in Sa iff t ∈ a + Z.
Hence the set of all second coordinates of such points is
{[β(a + n)] : n ∈ Z} = {[αa + αn] : n ∈ Z} = [αa] + H0
where H0 := {[αn] : n ∈ Z}. But H0 = q(αZ) is dense in T because α is irrational
(Theorem 5.8, Kronecker’s theorem for dimension 1). Then its translation [αa] + H0
is also dense in T.
All topological groups below are assumed to be Hausdorff.
Theorem 6.4. (Hewitt and Zuckerman) Let G ∈ LCA. Then the set of all continuous characters pointwise approximates the set of all characters. Precisely, let
f : G → T be a (not necessarily continuous) character. Then for every ε > 0 and
every finite subset F ⊂ G there exists a continuous character χ : G → T such that
|f (x) − χ(x)| < ε
for every x ∈ F .
20
Theorem 6.5. Let α1 , α2 , · · · , αn be a rationally independent finite family of real
numbers. Then for every ε > 0 and every n-tuple (r1 , r2 , · · · , rn ) of real numbers
there exists an integer m ∈ Z with
|mαk − rk | < ε (mod 1)
(equivalently, |mαk − rk − nk | < ε
∀k ∈ {1, · · · , n}
∀k ∈ {1, · · · , n} ∃nk ∈ Z)
Proof. Treat (R, τdiscr ) as an (infinite dimensional, of course) vector space over the
field Q. Consider the finite dimensional Q-linear subspace H := span{α1 , · · · , αn } <
R. Then α1 , · · · , αn is its linear basis because this family is Q-independent. WRG
we can suppose that 1 ∈ H (otherwise, if 1 ∈
/ H then the family
α0 := 1, α1 , α2 , · · · , αn
is Q-independent and we can continue
with this family and add arbitrarily r0 ∈ R).
P
So we can suppose that 1 =
ci αi with some tuple (c1 , · · · , cn ) ∈ Qn .
There exists a unique Q-linear functional λH : H → R such that
fk (αk ) = rk , ∀k ∈ {1, · · · , n}.
Since R is a divisible group there exists a homomorphic extension (not necessarily
continuous wrt the standard topology of R)P
λ : (R, τdiscr ) → R. Now we define the
desired character f : R/Z → R/Z. Let a := ci ri ∈ R. Consider two cases:
(1) if a 6= 0 then consider
2π
f : T → T, f (e2πix ) = e a iλ(x)
We obtain that f is a (possibly, discontinuous) character of T. Observe that it is well
defined because λ(1) = a, hence λ(s) = sa for every s ∈ Z and we get
2π
2π
2π
2π
f (e2πi(x+s) ) = e a iλ(x+s) = e a iλ(x)+ a iλ(s) = e a iλ(x) = f (e2πix ) ∀s ∈ Z
(2) if a = 0 then define
f : T → T, f (e2πix ) = e2πiλ(x) .
Apply Hewitt-Zuckerman theorem to this function. Then for every ε > 0 and every
finite subset F ⊂ G there exists a continuous character χ : G → T such that
|f (x) − χ(x)| < ε
for every x ∈ F . On the other hand every continuous character χ ∈ T∗ has the
form um : T → T, um (t) = tm for some m ∈ Z. Define F := {q(αk )} ⊂ T. Now
substituting um (q(αk )) = um (x) instead of χ(x) and f (αk ) = rk we eventually get:
there exists an integer m ∈ Z with
|e2πimαk − e2πirk | < ε
∀k ∈ {1, · · · , n}
or
|mαk − rk | < ε (mod 1)
∀k ∈ {1, · · · , n}
21
Definition 6.6. Let G be a topological group. A (continuous) homomorphism f :
G → T is called to be a (continuous) character of G. Denote by G∗ or Hom(G, T)
the set of all continuous characters of G. It is a group wrt the following pointwise
operation
(γ1 + γ2 )(g) := γ1 (g)γ2 (g)
This group G∗ wrt compact-open topology is a LCA topological group which is said
to be the Pontryagin dual of G.
Recall that the family
[K, O] := {f : G → T : f (K) ⊂ O}
with compact K ⊂ G and open O ⊂ T is a subbase of the compact open topology on
G∗ .
(1) we have a contravariant functor LCA → LCA.
(2) The natural evaluation map w : G × G∗ → T is continuous. It induces a map
G → G∗∗ . This map is continuous.
(3) Moreover, by Pontryagin-vanCampen duality theorem this map is a topological isomorphism for every LCA group G.
An abelian group (D, ·) is said to be divisible if for every n ∈ N the canonical
endomorphism un : D → Dn , x 7→ xn is onto. Examples: T, R, C but not Z.
Lemma 6.7. Let H be a subgroup of an abelian group G and f0 : H → D be a homomorphism into a divisible group (D, ·). Then there exists a homomorphic extension
f : G → D of f0 .
By Zorn’s Lemma it is enough to show that for every x ∈
/ H there exists a homok
morphic extension on the subgroup H0 :=< x, H >= {x h : k ∈ Z, h ∈ H}.
Case (i): If xn 6= H ∀n ∈ N then define f (xn h) = f0 (h).
Case (ii): Let k ≥ 2 is the least natural number s.t. xk ∈ H. Let f0 (xk ) = d ∈ D.
Since D is divisible one may choose z ∈ D s.t. z k = d. Now define f (xn h) =
z n f0 (h).
Corollary 6.8. Let G be an abelian group. Then the characters into T separate the
points.
Proof. Let g 6= e. It is enough to show that there exists χ ∈ G∗ s.t. χ(g) 6= 1 = e ∈ T.
Case (1). Let O(g) = n. Then consider the embedding f0 : H :=< g >,→ Ωn < T.
Case (2). Let (O(g) = ∞. Then consider any homomorphism f0 : H :=< g >→<
z >< T with z = f0 (g) 6= 1.
Theorem 6.9.
(1) Z∗ = T.
For every m ∈ Z consider χm : T → T, x 7→ xm .
w:Z→T→T
Consider χt : Z → T, n 7→ tn (where t ∈ T given constant). Observe that
every homomorphism χ : Z → T has this form for some t ∈ T. Show that Z∗
is algebraically (in fact, even, topologically) isomorphic to T.
(2) T∗ = Z.
Z × T → T, (k, t) 7→ tk .
22
(3) R∗ = R.
χr : R → T, x 7→ e2πirx .
w : R → R → T, (s, t) 7→ e2πist
Zn × Tn → T and Rn × Rn → T
are well defined biadditive mappings.
In general: G × G∗ → T is a continuous biadditive mapping for every LCA group
G. The dual operator preserves the finite products. That is,
(
n
Y
Gi ) ∼
=
∗
i=1
n
Y
G∗i
i=1
LCA group G is said to be autodual if it is t. isomorphic to its dual G∗ .
Exercise 6.10. Show that every LCA group G is a subgroup of an autodual LCA
group H.
Proposition 6.11. Every character γ : T → T has the form γ = χm for some m ∈ Z.
Proof. Let K := Ker(γ). Then K is a finite cyclic group Ωm or T. In the latter case
we have the trivial character and we can take m = 0. So wrg suppose that K = Ωm
for some m ∈ N. Then T/Ωm is topologically isomorphic to T. Moreover, there exists
σ:
Theorem 6.12. Let G be a LCA group.
(1) If G is discrete then G∗ is compact.
(2) If G is compact then G∗ is discrete.
Proof. (1) If X is discrete then for C(X, Y ) we have τc−o = τp .
G∗ ⊂ Hom(G, T) ⊂ TG ∈ Comp. G∗ is pointwise closed in TG .
(2) If X is compact then for C(X, Y ) we have τc−o = τsup .
(G, U 1 ) ∈ N (e).
4
On the other hand (G, U 1 ) = 1 because
4
U 1 ⊃ χ(G) ≤ T ⇒ χ(G) = 1
4
So, the identity is an isolated point.
Remark 6.13. G∗ is a compact metrizable iff G is a countable abelian group.
Indeed, in Thm 6.12.1 we have G∗ ⊂ TG ∈ Comp for every discrete abelian G.
Hence, if G is countable then G∗ is metrizable being a subspace of the metrizable
group TG . In fact, the converse is also true.
Some categorical facts:
(1) (Pontryagin duality)
In order to study the theory of compact (metrizable) abelian groups it is
equivalent to study the theory of discrete (countable) abeiian groups. The category of compact abelian (metrizable) groups is dual (meaning: is equivalent
to the dual) of the category of abelian discrete (countable) groups.
23
(2) (Gelfand duality)
Category of all compact Hausdorff spaces is dual to the category of all
commutative Banach algebras.
(3) (Stone duality)
Category of all 0-dimensional compact Hausdorff spaces is dual to the category of all Boolean algebras.
As a nontrivial generalization of the classification of finitely generated abelian
groups Pontryagin duality provides the following
Remark 6.14. {compactly generated LCA groups} =
= {Rn ×Zm ×K}, where K runs over compact abelian groups and n, m ∈ {0, 1, · · · }.
{compactly generated connected LCA groups} = {Rn × K}
Theorem 6.15. On every abelian group G there exists a precompact Hausdorff group
topology (which is metrizable if G is countable).
Proof. There exists a set S ⊂ Hom(G, T) which separates the Qpoints of S and
|S| = card(G) for infinite G. The diagonal function f : G →
TS is an injective homomorphism into the compact group TS . So, f (G) is a precompact group.
If G is countable then we can choose a countable S. Therefore, TS (hence, f (G)) is
metrizable. Since f : G → f (G) algebraically is an isomorphism we can endow G
with the ”preimage topology” coming from f (G).
Corollary 6.16. On every infinite abelian group G there exists a nondiscrete Hausdorff group topology.
Proof. Every discrete (even every LC) subgroup of a Hausdorff group is closed. Hence,
if in Thm 6.15, f (G) would discrete in the COMPACT group TS then f (G) is finite
(being discrete and compact).
Remark 6.17. It is not true for nonabelian groups (Hesse, independently, Olshanskii
and Shelah under assumption CH). In other terms there exists an infinite discrete
group which is minimal. That is an infinite group G which is miniamal, that is, does
not admit a nondiscrete Hausdorff group topology.
Recall
Definition 6.18. A Hausdorff topological group G is said to be minimal if there is
no strictly coarser Hausdorff group topology on G.
Let us explain why R is not minimal (one of the homework exercises).
We use the following reformulation of the minimality.
Fact. A Hausdorff topological group G is minimal if and only if for every continuous injective onto homomorphism f : G → Y is a homeomorphism (or, in other
words, if any morphism in TGr which is an isomorphism in the category Gr is an
isomorphism in the category TGr).
Consider the following injective continuous not onto homomorphism
√
f : R → T × T, f (t) = (e2πit , e2πi
2t
)
24
By a corollary of Kronecker’s theorem f (R) is dense in T × T. f is not a topological
embedding. Otherwise, f (R) is a locally compact subgroup in T × T. By a theorem
(we have proved earlier) we obtain that f (R) is closed in T × T. Since f (R) is dense
we obtain f (R) = T × T, a contradiction.
Below all topological groups are assumed to be Hausdorff.
Remark 6.19. (Zoo of minimal groups)
(1) Every compact group is minimal.
(2) (Stephenson) Every LCA minimal group is necessarily compact. Prove it as
an exercise.
(3) (Prodanov, Stoyanov) Every abelian minimal topological group is precompact.
(Observe that one may easily derive Stephenson’s theorem using this result).
(4) (Doitchinov) (Z, τp ) is minimal.
(5) (Stoyanov) The unitary group Isolin (H) is minimal for every Hilbert space H.
(6) (Dierolf and Schwanengel) The topological semidirect product R n R+ is minimal.
(7) (Gaughan) The symmetric group SX in the pointwise topology (for every set
X).
(8) (Remus and Stoyanov) SLn (R) for every n ≥ 2.
(9) (Me) Every LCA group G is a group retract of a LC minimal group (T×G∗ )nG
which in fact is the generalized Heisenberg group modelled on the biadditive
mapping G × G∗ → T.
(10) (Gamarnik) The homeomorphisms group (in the compact open topology)
Homeo([0, 1]N ) and Homeo({0, 1}N ) are minimal.
Note that [0, 1]N is the Hilbert cube and {0, 1}N is the Cantor cube (homeomorphic to the usual Cantor set).
Theorem 6.20. Every compact abelian group is a subgroup of some (maybe infinitedimensional) torus TS .
Proof. As an exercise.
What about nonabelian case ? Peter-Weyl theorem asserts that for every compact
group G the homomorphisms into finitely dimensional orthogonal groups separate the
points. Conclude that every compact group ,→ product of orthogonal groups.
25
7. Lecture 8
Recall
Proposition 7.1. For
(1) ∀U ∈ γ ∃V ∈ γ
(2) ∀U ∈ γ ∃V ∈ γ
(3) ∀U ∈ γ ∀a ∈ G
(T2 )
every topological group G and every local base γ at e we have:
: V 2 ⊂ U;
: V −1 ⊂ U ;
∃V ∈ γ : aV a−1 ⊂ U .
G is Hausdorff if and only if
∩{U : U ∈ γ} = {e}.
Theorem 7.2. (Group topologization)
Let G be a group and γ be a nonempty family of subsets containing e and satisfying
the conditions (1), (2), (3) of Proposition 7.1. Then there exists a unique topology τ
on G such that: (G, τ ) is a topological group and γ is a local topological base at e.
(T2 ) (G, τ ) is Hausdorff if and only if ∩{U : U ∈ γ} = {e}.
Sketch of the proof: Define the desired topology τ on G as follows:
O ∈ τ iff ∀x ∈ O ∃U ∈ γ xU ⊂ O.
Remark 7.3. Suppose that in Theorem 7.2, in addition, the following condition is
satisfied:
(*)
∀U ∈ γ ∀x ∈ U ∃V ∈ γ xV ⊂ U
Then γ ⊂ τ (that is, every nbd U ∈ γ is open). Observe that if U is a subgroup of
G then this condition is trivially holds (take V := U ) and hence, U is open.
Example 7.4. Let G := Z and γ := {pn Z}n∈N with a given prime p. Then the
conditions of γ Theorem 7.2 are satisfied. The corresponding group topology τp is
Hausdorff because ∩n pn Z = {0} and every pn Z is an open subgroup (hence, clopen)
in Z. It is easy to see that τp = top(dp ), where dp is the standard p-adic topology on
Z.
Exercise 7.5. Show that the topologies τp 6= τq on Z are different for different p, q.
Remark 7.6. Let G be a group, (Gi , τi ) is a family of topological groups and fi : G →
Gi be a family of homomorphisms. Then the so-called weak topology τw on G is a
group topology. It is the smallest topology on G such that all fi are continuous. One
way to see that it is a group topology is to use Theorem 7.2. Indeed, define γ as the
collection {fi−1 (Ne ) : i ∈ I}f in (where S f in means the family of all finite intersections
of members from the family S).
An important and well known particular case of the weak topology is the topological
product defined for the collection of the projections.
It is easy to interpret the p-adic topology τp as the weak topology Z with respect to
the collection of homomorphisms into the discrete (cyclic finite) groups fn : Z → Zn ,
n ∈ N.
Example 7.7. Let G is a group and γ is a collection of subsets in G such that:
(1) Every H ∈ γ is a normal subgroup of G.
(2) H1 ∩ H2 ∈ γ for every H1 , H2 ∈ γ.
26
Then γ is satisfies the conditions of Theorem 7.2.
Example 7.8. On every group G one may define the corresponding profinite topology
as the weak topology with respect to all possible homomorphisms G → F , where F
is a finite group. Another description is as follows. Consider γ as a collection of all
normal subgroups with the finite index.
γ := {H / G : [G : H] < ∞}
Then γ satisfies the conditions of Theorem 7.2.
Warning: the profinite topology is not always Hausdorff.
Definition 7.9. A group G is said to be residually finite if the profinite topology
on G is Hausdorff. It is equivalent to saying that homomorphisms into finite groups
separate the points of G.
Other equivalent form of this definitions is: the intersection of all normal subgroups
with finite index is {e}. Moreover, one may say also: the intersection of all (not
necessarily normal) subgroups with finite index is {e}. The reason is that if H is
a finite index subgroup of G then there exists a finite index normal subgroup of G
such that N ⊂ H. Indeed, if G/H is finite then consider the standard left action
G × G/H × G/H → G/H and the corresponding homomorphism h : G → SG/H . Now
observe that the kernel of this action N := ker(h) is the desired subgroup.
Some examples of residually finite groups: finite groups, free groups, free abelian
groups, finitely generated linear groups, finitely generated nilpotent groups. There
exist finitely generated not residually finite grouos. One may take for example a
subgroup G =< a, b >≤ SZ of the symmetric group SZ , where G is generated by two
elements a, b, where a(n) = n + 1 and b(0) = 1, b(1) = 0, b(n) = n for n ∈
/ {0, 1}.
Example 7.10. (Furstenberg’s proof of the infiniteness of prime numbers using the
profinite topology on Z)
Let (Z, τpro−f in ) be Z in its profinite topology. Clearly, the subset {1, −1} is not
open (in fact, every open subset is infinite (because every nbd of 0 contains an infinite
subgroup)).
We have to prove that the set P of all primes is infinite. Assuming the contrary
let P := {p1 , · · · , pn }. Then
Z \ {1, −1} = ∪ni=1 pi Z
Every pi Z is an open subgroup of (Z, τpro−f in ) (being the kernel of the homomorphism
Z → Zpi into the discrete group Zpi ). Then, as we already know, every open subgroup
is closed. So, the finite union ∪ni=1 pi Z is closed, too. Then we obtain that its
complement {1, −1} is open, a contradiction.
Example 7.11. (Symmetric groups) For every set X denote by SX the symmetric
group of all bijections X → X. Consider the pointwise topology τp on SX ⊂ X X .
Then SX is a Hausdorff topological group. One may verify this directly. Another way
is to use Theorem 7.2. Indeed, define
γ := {HA : A ⊂ X, A is finite} HA := {f ∈ SX : f (a) = a ∀a ∈ A}
27
It is trivial to see the conditions (1), (2) and (T2 ) of Theorem 7.2. As to (3), observe
that for every f ∈ SX and every finite A ⊂ X we have
f Hf −1 (A) f −1 ⊂ HA .
∗
SX
:= {f ∈ SX : f (x) = x for almost all x ∈ X} is a dense normal subgroup of
SX .
Example 7.12. (I-adic topology on rings) Let R be a commutative unital ring and
I ⊂ R be an ideal in R. Consider
γ := {I n : n ∈ N}
Then γ satisfies the conditions (1), (2), (3) of Theorem 7.2. The corresponding
topology τI is a topological ring topology on R.
Clearly, O ∈ τI if and only if ∀x ∈ O ∃n ∈ N x + I n ⊂ O.
Definition 7.13. A topological group G is said to be non-archimedean if there exists
a nbd base γ at e such that every U ∈ γ is a (necessarily, open, see Remark 7.3)
subgroup of G. Notation G ∈ NA.
Note that the class NA is important in many applications. NA is closed under
formation of: subgroups, factor-groups, products, completions. NA contains the class
of all profinite groups and also all symmetric groups SX . It is a remarkable fact
that there is an NA analog of Cayley’s theorem: every NA group is embedded (as a
topological subgroup) into some SX .
Definition 7.14. A topological
group G is said to be profinite if G is a closed subQ
group of the product i Fi of some finite groups.
It is equivalent to say that G is a zero-dimensional compact Hausdorff topological
group, or, G is an inverse limit of finite groups. For example, the completion of (Z, dp )
is a profinite group.
28
8. Lecture 9
Theorem 8.1. 1 Let G be a topological group. Suppose that {Un }n∈N is a sequence
3
of symmetric nbds of e in G such that Un+1
⊂ Un for every n ∈ N. Then there is a
right invariant pseudometric ρ on G such that
1
1
1 (e) := {x ∈ G : ρ(e, x) <
1 [e] := {x ∈ G : ρ(e, x) ≤
(∗) B n+1
}
⊂
U
⊂
B
}
n
n
2
2
2n+1
2n
Proof. For every r > 0 define Wr ∈ N (e) as follows. Let Wr := G for every r ≥ 1.
For every 0 < r < 1 define
1
1
Wr := ∪{Ui1 · · · Uik : i1 + · · · + i < r}
2
2k
Observe that x ∈ Wr iff there are x0 , x1 , · · · , xk ∈ G such that x0 = 1, xk = x and
x−1
j−1 xj ∈ Uij . It is trivial to see that
Wr−1 = Wr and
Wr Ws ⊂ Wr+s for every r, s ∈ (0, 1].
Define
ρ(x, y) := inf{r ∈ (0, 1] : xy −1 ∈ Wr }
Then
1) ρ(x, x) = 0 (because, e ∈ Wr for every r).
2) ρ(x, y) = ρ(y, x) (because, Wr−1 = Wr ).
3) ρ(x, z) ≤ ρ(x, y) + ρ(y, z).
Because if ρ(x, y) < r, ρ(y, z) < s then xy −1 ∈ Wr , yz −1 ∈ Ws . Therefore, xz −1 ∈
Wr Ws ⊂ Wr+s . So, ρ(x, z) < r + s.
4) ρ(xg, yg) = ρ(x, y) (because (xg)(yg −1 ) = xy −1 ).
Lemma. If
1
2i1
+ ··· +
1
2ik
<
1
2n+1
then Ui1 · · · Uik ⊂ Un .
We use induction wrt k.
1
For k = 1. Let 21i1 < 2n+1
. Then i1 > n + 1. So, Ui1 ⊂ Un+1 ⊂ Un (because Un is a
decreasing sequence).
Suppose that the inclusion holds for every k ∈ {1, · · · , k − 1}. Now we prove the
induction step for k.
Let m be the largest natural number such that
1
1
1
+ ··· + i < m.
i
1
k
2
2
2
Then we have il > m (trivial).
Let l < k be the smallest natural number that satisfies
1
1
1
< i1 + · · · + i .
m+1
2
2
2l
1
Then m ≥ n + 1 (because the induction assumption 21i1 + · · · + 21ik < 2n+1
and the
choice of m) and
1
1
1
+ · · · + i < m+1
i
1
l−1
2
2
2
1
1
1
+ · · · + i < m+1
i
2 l+1
2k
2
1present
proof is from [2]
29
Using the inductive hypothesis we obtain
Ui1 · · · Uil−1 ⊂ Um
and
Uil+1 · · · Uik ⊂ Um
Therefore,
(Ui1 · · · Uil−1 )Uil (Uil+1 · · · Uik ) ⊂ Um Um Um ⊂ Um−1 ⊂ Un
Here we use the following inequalities il > m ≥ n + 1 mentioned above.
This proves the lemma.
Now we check the inclusions
1
}
2
2n
1 (e) then by definition of ρ and Wr thee exists k ∈ N such that
(I) If x ∈ B n+1
(∗) B
1
2n+1
(e) := {x ∈ G : ρ(e, x) <
1
2n+1
} ⊂ Un ⊂ B 1n [e] := {x ∈ G : ρ(e, x) ≤
2
1
1
1
<
.
+
·
·
·
+
i
i
n+1
21
2k
2
By the lemma above this implies that x ∈ Un .
(II) Observe that Un ⊂ Wr for every r > 21n . This implies that ρ(x, e) ≤
x ∈ Ui1 · · · Uik and
1
.
2n
Similarly can be obtained left invariant pseudometric with the property (*). However, in general we cannot require that ρ simultaneously left and right invariant.
Theorem 8.2. (Pontryagin) Let G be a topological group such that G ∈ T0 . Then
G ∈ T3 1 .
2
Proof. Since G ∈ T0 we already know that G ∈ T1 . We have to prove that continuous
real valued functions on G separate points and closed subsets. That is, for every
g0 ∈ G and every closed subset g0 ∈
/ F ⊂ G there exists a continuous function
f : G → R such that f (g0 ) ∈
/ cl(f (F )). Since G is a (semi)topological group it is
equivalent to show this property only for g0 = e. Let F be a closed subset of G such
that e ∈
/ F . Consider O := G \ F . By standard properties of topological groups
there exists a sequence {Un }n∈N of symmetric nbds of e in G with U1 = O such that
3
Un+1
⊂ Un for every n ∈ N. By Theorem 8.1 there is a right invariant pseudometric
ρ on G such that
1 (e) ⊂ Un ⊂ B 1 [e].
B n+1
n
2
2
In particular, for n = 1 we have
B 1 (e) ⊂ O ⊂ B 1 [e].
4
2
Now define the desired function f : G → R as f (x) := ρ(e, x). Then f separates e
and F because f (e) = 0 and f (x) ≥ 14 for every x ∈ F = G \ O. Finally note that
f : G → R is continuous. Indeed, f is continuous at e by (*) of Theorem 8.1. It is
enough for the continuity of f (at arbitrary g0 ∈ G) because ρ is right invariant. Theorem 8.3. (Birkhoff-Kakutani Theorem) Let G be a Hausdorff topological group.
The following are equivalent:
(1) G is metrizable.
(2) G ∈ B1 (first countable, that is has a countable basis at every point).
30
Proof. (1) ⇒ (2) is always true for topological spaces.
(2) ⇒ (1) There exists a countable basis γ := {Un }n∈N of open nbds at e. Again,
by the standard properties of topological groups one may assume that every Un is
3
symmetric and Un+1
⊂ Un for every n ∈ N. By Theorem 8.1 there exists a right
invariant pseudometric ρ on G which satisfies (*). Since (G, τ ) is Hausdorff then
1
∩n Un = {e}. By (*) we know that B n+1
⊂ Un . This implies that ∩n B 1n (e) = {e}.
2
2
Hence, ρ(e, x) > 0 for every x 6= e. Therefore, ρ is a metric. The property (*) also
implies that the topology of ρ is the given topology of G. Indeed, top(ρ) induces the
same nbd system at e as the topology τ of G. At the same time, N (g0 ) = N (e)g0 for
both of these topologies top(ρ) and τ (for the first, recall that ρ is right invariant).
So, ρ is a metric on G which generates the original topology. This means that G is
metrizable.
So, every Hausdorff topological group G is a Tychonoff space. Note that during
many years it was an open question (of Pontryagin) if G is even normal. This question
negatively was answered by Markov. His approach was much more important than
above mentioned question. Markov introduced a concept of a free topological group
F (X) over a given Tychonoff space X. He showed that X is closely embedded into
F (X). This answers the question because the normality is a hereditary property with
respect to closed subsets. It is enough to take G := F (X) for a Tychonoff space X
which is not normal (for example, X= Sorgenfrey plane). Much later was proved that
the Tychonoff topological group ZR is not normal.
8.1. Uniform spaces.
Definition 8.4. (A. Weil) Let X be a set. A nonempty subset µ ⊂ P (X × X) (so,
µ is a collection of relations on X) is said to be an uniform structure on X if the
following conditions are satisfied:
(1)
(2)
(3)
(4)
(5)
(6)
∆ ⊂ ∀ε ∈ µ.
ε−1 ∈ µ ∀ε ∈ µ.
∀ ε ∈ µ ∃δ ∈ µ δ ◦ δ ⊂ ε.
∀ ε1 , ε2 ∈ µ ε1 ∩ ε2 ∈ µ.
δ ∈ µ and δ ⊂ ε ⇒ ε ∈ µ.
(”Hausdorff property”) ∩{ε : ε ∈ µ} = ∆.
Then (X, µ) is said to be a uniform space.
The concept of the uniform spaces is a far reaching simultaneous generalization of
two important concepts: metric space and topological group.
Let us say that a subset γ ⊂ µ is a uniform base of µ if for every ε ∈ µ there exists
δ ∈ γ such that δ ⊂ ε.
For every ε ∈ µ and every x0 ∈ X define an analogue of a ball in metric spaces as
follows:
Bε (x0 ) = ε(x0 ) := {x ∈ X : (x0 , x) ∈ ε}
The induced topology top(µ) is defined as follows:
O ∈ top(µ) ⇔ (x0 ∈ O ⇒ ∃ε ∈ µ Bε (x0 ) ⊂ O)
31
Exercise 8.5. Show that (X, top(µ)) is a topological space for every uniform space
(X, µ).
Example 8.6.
(1) For every metric space (X, d) the pair (X, µ(d)) is a uniform space, where
µ(d) := {R ⊂ X × X : ∃ε > 0 Rε ⊂ R}, Rε := {(x, y) ∈ X 2 : d(x, y) < ε}
(2) For every Hausdorff topological group G the pair (G, µr (G)) (the right uniformity) is a uniform space, where
µr (G) := {R ⊂ G2 : ∃U ∈ N (e) RU ⊂ R}, RU := {(x, y) ∈ G2 : xy −1 ∈ U }
(3) Similarly can be defined the left uniformity µl on G.
µl (G) := {R ⊂ G2 : ∃U ∈ N (e) LU ⊂ R}, LU := {(x, y) ∈ G2 : x−1 y ∈ U }
32
9. Lectures 10,11,12
Let (X, µ) be a uniform space. We say that a subset γ ⊂ µ is a base of the uniform
structure (or, a uniform base) of µ if for every ε ∈ µ there exists δ ∈ γ such that
δ ⊂ ε.
We say that γ is uniform prebase if γ ∩f in is a base.
For example, {RU }U ∈N (e) , RU := {(x, y) ∈ G2 : xy −1 ∈ U } is a uniform base of
the right uniformity µr on the topological group G.
The union γ := {RU }U ∈N (e) ∪ {LU }U ∈N (e) is a prebase of the, so-called, two-sided
uniformity on G.
Example 9.1. Additional examples of uniform spaces:
(1) On every set X the set
µ∆ := {ε ⊂ X × X : ∆ ⊂ ε}
defines the maximal possible uniform structure on X which has the single
point uniform base γ := {∆}. Clearly, top(µ∆ ) is discrete.
(2) For every compact Hausdorff space (K, τ ) the set µK of all nbds of ∆ in K ×K
is a uniformity. Then top(µK ) = τ . It is the unique possible uniformity on
K which is compatible with the topology. Note that for the natural metric
uniformity µd on R not every nbd of ∆ is an element of µd .
(3) For every system of pseudometrics F := {ρi }i∈I (which separates the points)
on a set X we have the corresponding (Hausdorff) weak uniformity µF , where
the uniform prebase is the collection
[i, ε] := {(x, y) ∈ X 2 : ρi (x, y) < ε}
where i ∈ I and ε > 0.
(4) More generally. Let X be a set and fi : X → (Y, µi ) is a set of functions into
(pseudo)uniform spaces. Then the collection
[i, ε] := {(x, y) ∈ X 2 : (fi (x), fi (y)) ∈ ε}
with i ∈ I and ε ∈ µi is a prebase of a (pseudo)uniformity on X which is
called a weak uniformity.
(5) Subspace and product of uniform spaces can be naturally defined (as a kind of
weak uniformity). Note that the corresponding topologies are exactly subspace
and product topologies.
(6) Let (Y, µ) be uniform space, X a topological space. On F ⊂ Y X consider the
uniformity µco (of compact convergence). Its uniform subbase is a system of
subsets
[K, ε] := {(f1 , f2 ) ∈ F × F : (f1 (x), f2 (x)) ∈ ε ∀x ∈ K}
where ε ∈ µ and K is a compact subset of X. Then if F ⊂ C(X, Y ) we
have top(µco ) = τco . For details see for example [1].
Similarly can be defined the uniformity of the pointwise convergence. Here
for any F ⊂ Y X we have top(µp ) = τp .
The collection [X, ε] defines the uniformity of uniform convergence on X.
33
The class of all uniform spaces define a category U nif of uniform spaces. Morphisms in U nif are the uniformly continuous functions. A function f : (X1 , µ1 ) →
(X2 , µ2 ) is said to be uniformly continuous if
∀ε ∈ µ2 ∃δ ∈ µ1 : (f (x), f (y)) ∈ ε ∀(x, y) ∈ δ.
On a compact space K every continuous map f : K → (Y, µ) into a uniform space
is uniformly continuous with respect to the natural (uniquely defined) compatible
uniformity on X.
We have a forgetful functor U nif → T op. What is the range of this functor ? The
following Proposition 9.3 gives the expected answer.
Example 9.2. Let X be a Tychonoff space. for every continuous bounded function
f ∈ Cb (X) define the induced pseudometric ρf (x, y) := |f (x) − f (y)|. The system
{ρf : f ∈ Cb (X)} is a subbase of some uniformity µβ on X (see Examples 9.1 items
(3) or (4)). Observe that since X is Tychonoff the corresponding topology top(µbeta )
is the original topology of X.
Proposition 9.3. A topological space (X, τ ) admits a compatible uniform structure
iff X is Tychonoff (i.e., X ∈ T3 1 ).
2
first proof:
for every continuous function f : X → R define the induced pseudometric ρf (x, y) :=
|f (x) − f (y)|. Observe that the corresponding weak uniformity on X induces the original topology of X.
second proof: X ,→ K ∈ Comp2 iff X ∈ T3 1 .
2
(Comp2 ⊂ T4 ⊂ T3 1 . On the other hand, if X ∈ T3 1 then there exists a collection
2
2
S ⊂ C(X) which separates points and closed subsets of X. We can suppose that
S ⊂ C(X, [0, 1]. Then the diagonal function fS : X → [0, 1]S =: K ∈ Comp2 is a
topological embedding as it follows by a classical lemma 9.4 of Tychonoff.
Lemma 9.4. Let S := {fi : X →QYi } be a set of some continuous functions. Consider
the diagonal function fS : X → i∈I Yi . Then
Q
(1) If S separates points of X then fS : X → i∈I Yi is a continuous injection.
(2) If
Q S, in addition, separates points and closed subsets of X then fS : X →
i∈I Yi is a topological embedding.
Proof. (1) is trivial.
(2) We have to show that the restricted map fS : X → fS (X) is open. It is enough
to show that for every x0 ∈ X and every U ∈ N (x0 ) the
Q image fS (U ) contains an
intersection of fS (X) with a neighborhood O of fS (x0 ) in i∈I Yi . By our assumption
there exists fi0 : X → Yi0 such that fi0 (x0 ) ∈
/ cl(fi0 (X \ U )). Now the desired
c
O ∈ N (fS (x0 )) can be defined as O := (πi−1
(cl(f
i0 (X \ U )))) .
0
Q
Remark 9.5. Note that for every topological product Y := i∈I Yi the set of all
natural maps (generalized projections)
Y
Y
πJ :
Yi :→
Yj
i∈I
j∈J
34
with finite subsets J ⊂ I separates points and closed subsets of Y (as it follows
from the definition of product topology).
At the same time, observe that the set of all usual projections πi : Y → Yi i ∈ I does
not separate the points and closed subsets in Y . For example, for R2 the projections
do not separate the points and closed subsets. Indeed, take for example any circle
and its center in R2 .
We have also another natural functor M etr → U nif . Its range is the class of all
metrizable uniform spaces. The following remark can be treated as a far reaching
generalization of Pontryagin’s theorem.
Remark 9.6. (Alexandrov-Urysohn) A uniform space (X, µ) is metrizable iff µ has a
countable uniform base.
Hint: One direction is trivial because for every metric space (X, d) the countable
collection
1
R 1 := {(x, y) ∈ X 2 : d(x, y) < }
n
n
is a base of the corresponding uniformity generated by d. Second direction is similar
to the proof of Pontryagin’s theorem.
µl (G) and µr (G) are the same for: abelian groups, compact groups.
In general, they are different.
Example 9.7. Let G := GL2 (R). Then µl (G) 6= µr (G).
Proof. Sketch:
Consider twosequences
of matrices:
1
1
n 1
xn :=
and yn := n n2
0 1
0 1
1
1
−n
n
Then x−1
and we have
n =
0 1
1
1 1
1 n + n12
−1 −1
yn xn = yn (xn ) =
→E=
0 1
0
1
1
1
1
1
+
1
−1
n
→A=
xn yn = (x−1
n ) yn =
0 1
0
1
For uniform spaces one may define a very successful analogue of Cauchy sequences,
Cauchy filters, completeness ...
For example, a sequence xn in a topological group is a Cauchy sequence for µr if
for every U ∈ N (e) there exists n0 ∈ N such that xn x−1
m ∈ U for all n, m ≥ n0 . More
generally, a filter α on G is a Cauchy filter for µr if for every U ∈ N (e) there exists
A ∈ α such that AA−1 ⊂ U .
Completion of a Hausdorff topological group G with respect to the two-sided uniformity admits a natural structure of a topological group.
Every compactification is a particular case of a completion. Namely, a completion
of a uniform space (X, µ) is a compactification iff µ is totally bounded. Meaning that
for every ε ∈ µ and ε-covering {ε(x)}x∈X there exists a finite subcovering.
For every Tychonoff space X the uniformity µβ defined in Example 9.2 is totally
bounded and its completion leads to the maximal (the so-called Chech-Stone) compactification X → β(X).
35
Remark 9.8.
(1) For every uniform space (X, µ) denote by U nifb (X) the set of all bounded
uniformly continuous functions X → R. Then U nifb (X) is a closed subalgebra
of Cb (X) such that U nifb (X, µ) separates points and closed subsets.
(2) For a topological group G and its right uniformity µr denote by RUC(G) the
set U nifb (G, µr ). Note that f ∈ RUC(G) iff
∀ ε > 0 ∃U ∈ N (e) |f (ux) − f (x)| < ε ∀u ∈ U, x ∈ G
9.1. Theorems of Teleman. When a topological group G can be represented on a
Banach space by linear isometries. That is, when G ,→ Iso(V ) ? In fact Teleman’s
theorem below 9.10 shows that always.
Every finite group G can be represented on Rn with n = |G|. Indeed,take
G ,→ Sn ,→ Isolin (Rn ) = On (R).
Exercise 9.9. Every discrete group G admits an effective isometric representation on
lG and also on the Hilbert space l2 (G).
For example, on l∞ := l(N) and on l2 := l2 (N) if G is countable. Note that for
|G| = n the Banach space lG is a copy of (Rn , || · ||max ).
What happens for nondiscrete topological groups ? Our aim is to prove the following theorem of Teleman (rediscovered by many authors).
Theorem 9.10. (Teleman’s theorems) Let G be a Hausdorff topological group. Then
(1) G is embedded into Isolin (V ) for some Banach space V .
(2) G is embedded into Iso(M, d) for some metric space (M, d).
(3) G is embedded into Homeo(K) for some compact space K.
Clearly, (1) ⇒ (2).
Teleman’s theorem suggests more challenging question “how good” can be the
target Banach space V ? This questions leads to a natural hierarchy of topological
groups closely related to a hierarchy of main-stream Banach space classes like: Hilbert,
reflexive, Asplund, Rosenthal, etc.
Remark 9.11. For example, the topological group Homeo+ ([0, 1]) (of all orientation preserving homeomorphisms) is not reflexively, or, even, Asplund, representable.
However it is Rosenthal representable.
It is an open question if every separable metrizable topological group (enough to
examine G := Homeo([0, 1]N )) is Rosenthal representable.
First we provide here some related definitions and results. With every Banach
space V one may naturally associate several structures which are related to: Analysis,
topological dynamics, topological (semi)groups.
Let V be a Banach space. By V ∗ we denote its dual space. That is, the vector
space of all continuous linear functionals f : V → R. For every f ∈ V ∗ one may
define its operator norm
||f || := sup{|f (v)| : ||v|| ≤ 1}.
36
Then (V ∗ , || · ||) becomes a Banach space and the canonical bilinear function
V × V ∗ → R, (v, f ) 7→< v, f >:= f (v)
satisfies the following
| < v, f > | ≤ ||v|| · ||f ||
for every (v, f ) ∈ V × V . In particular, that function is continuous.
By L(V, V ) (or, L(V )) we denote the set of all continuous linear maps V → V .
∗
Definition 9.12. The weak topology τw on V is the weak topology on the set V with
respect to the set V ∗ of all continuous linear functionals f : V → R.
A net vi converges to v in this topology iff the net f (vi ) in R converges to f (v) for
all f ∈ V ∗ .
The collection
{[v0 ; f1 , · · · , fn ; ε] : ε > 0, fi ∈ V ∗ ∀ i ∈ {1, · · · , n}}
is a local topological base at v0 , where, as usual
[v0 ; f1 , · · · , fn ; ε] := {v ∈ V : |fi (v0 ) − fi (v)| < ε ∀ i ∈ {1, · · · , n}}.
Weak topology is weaker than the norm topology on V and these two topologies
coincide iff V is finite dimensional.
Definition 9.13. The weak-star topology on V ∗ is the weak topology on the set V ∗
with respect to the set V ⊂ V ∗∗ of continuous linear functionals
v : V ∗ → R, f 7→ f (v),
v ∈ V.
A net fi converges to f in this topology iff the net fi (v) in R converges to f (v) for
all v ∈ V .
The collection
{[f0 ; v1 , · · · , vn ; ε] : ε > 0, vi ∈ V, ∀ i ∈ {1, · · · , n}}
is a local topological base at f0 , where,
[f0 ; v1 , · · · , vn ; ε] := {f ∈ V ∗ : |f0 (vi ) − f (vi )| < ε ∀ i ∈ {1, · · · , n}}.
Weak-star topology is weaker than the weak topology on V ∗ and these two topologies coincide iff V is finite dimensional.
One of the important properties of the weak-star topology is Alaouglu theorem
which asserts that every bounded weak-star closed subset of the dual V ∗ is compact
for every Banach space V . In particular, it is true for the closed unit ball
B ∗ := {f ∈ V ∗ : ||f || ≤ 1}.
Definition 9.14. Let V be a Banach space.
(1) The strong operator topology τsot on L(V, V ) is the pointwise topology inherited
from (V, || · ||)V . That is, a net si converges to s iff si (v) converges to s(v) in
the norm topology for every v ∈ V .
The collection
{[s0 ; v1 , · · · , vn ; ε] : ε > 0, vi ∈ V, ∀ i ∈ {1, · · · , n}}
37
is a local topological base at s0 ∈ L(V, V ), where, as usual
[s0 ; v1 , · · · , vn ; ε] := {s ∈ L(V, V ) : ||svi − s0 vi || < ε} ∀ i ∈ {1, · · · , n}}
(2) Replacing the norm topology of V by its weak topology we obtain the weak
operator topology τwot on L(V, V ).
A net si τw -converges to s in L(V, V ) iff f (si (v)) converges to f (s(v)) in R
for every v ∈ V .
The collection
{[s0 ; v1 , · · · , vn ; f1 , · · · , fn ; ε] : ε > 0, vi ∈ V, fi ∈ V ∗ ∀ i ∈ {1, · · · , n}}
is a local topological base at s0 ∈ L(V, V ), where, as usual
[s0 ; v1 , · · · , vn ; f1 , · · · , fn ; ε] := {s ∈ L(V, V ) : ||fi (svi )−fi (s0 vi )|| < ε} ∀ i ∈ {1, · · · , n}}
We use the notation: Θ(V )s , Iso(V )s (respectively, Θ(V )w , Iso(V )w ) or simply
Θ(V ) and Iso(V ), where the topology is understood.
Proposition 9.15.
(1) For every compact space Y the semigroup C(Y, Y ) endowed with the compact
open topology τco is a topological monoid.
(2) Note also that the subset Homeo (Y ) in C(Y, Y ) of all homeomorphisms Y →
Y is a topological group.
(3) For every submonoid (S, τco ) ⊂ C(Y, Y ) the natural monoid action (S, τco ) ×
Y → Y is continuous.
(4) Furthermore, it satisfies the following remarkable minimality property. If τ
is an arbitrary topology on S such that (S, τ ) × Y → Y is continuous then
τco ⊂ τ .
Proof. (1) and (2) are parts of a homework.
(3) The continuity of the action (S, τco ) × Y → Y is easy taking into account that
the compact open topology on S ⊂ C(Y, Y ) is the same as the topology of compact
convergence. So, s0 and s in S are ε-close means that (s0 y, sy) is ε-close for every
y ∈ Y (where ε is an element of the uniformity µY on Y ). Since s0 ∈ C(Y, Y ), we
can choose δ ∈ µY such that (s0 y0 , s0 y) ∈ ε for every (y0 , y) ∈ δ. Therefore, we
can “control” (s0 y0 , sy) in the entourage ε ◦ ε (“triangle equality” axiom of uniform
spaces).
(4) Let (S, τ ) × Y → Y be continuous. Then by the compactness of Y it is easy to
see the following
∀s0 ∈ S ∀ε ∈ µY ∃U ∈ Nτ (s0 ) : (s0 y, sy) ∈ ε ∀y ∈ Y.
This proves that the topology of compactness convergence τco is weaker than τ .
Proposition 9.16.
(1) For every metric space (M, d) the semigroup Θ(M, d) of all d-contractive maps
f : X → X (that is, d(f (x), f (y)) ≤ d(x, y)) is a Hausdorff topological monoid
with respect to the topology of pointwise convergence. The group Iso(M ) of all
onto isometries is a topological group.
(2) Furthermore, the evaluation map S × M → M is a jointly continuous action
for every subsemigroup S ⊂ Θ(M, d).
38
(3) For every normed space (V, || · ||) the semigroup Θ(V ) of all contractive linear
operators V → V endowed with the strong operator topology (being a topological submonoid of Θ(V, d) where d(x, y) := ||x − y||) is a topological monoid.
The subspace Iso(V ) of all linear onto isometries is a topological group.
Proof. (1) is a part of a homework.
(2) The continuity of the action S × M → M at the point (s0 , v0 ) is straightforward
using the following inequality
d(s0 v0 , sv) ≤ d(s0 v0 , sv0 ) + d(sv0 , sv) ≤ d(s0 v0 , sv0 ) + d(v0 , v)
(3) easily follows from (1).
An action S × X → X on a metric space (X, d) is contractive if every s-translation
s̃ : X → X lies in Θ(X, d). It defines a natural homomorphism h : S → Θ(X, d).
Remark 9.17. If an action of S on (X, d) is contractive then it is easy to show that
the following conditions are equivalent:
(i) The action is continuous.
(ii) The action is separately continuous.
(iv) The natural homomorphism h : S → Θ(X, d) of monoids is continuous.
Proposition 9.18. Θ(V ) is a semitopological monoid with respect to the weak operator topology.
Proof. An exercise.
Remark 9.19. The semitopological semigroup Θ(V )w is compact iff V is a reflexive
Banach space.
Hint: Observe that Θ is closed in (B, w)B for every V . Then use the following
characterization of reflexivity: V is reflexive iff every bounded weakly closed subset
is weakly compact iff B := BV is weakly compact.
For every semigroup (S, ·) one may define its opposite semigroup S op with the
“opposite multiplication” (x, y) 7→ y · x. Clearly, the usual inverse map on any group
G defines an isomorphism between G and Gop . For any left action S × X → X we
have the induced right action X × S → X and vice versa.
Theorem 9.20. For every normed space V and a strongly continuous homomorphism
h : S ⊂ Θ(V )op the induced action S × B ∗ → B ∗ on the compact space B ∗ is jointly
continuous.
Proof. For the continuity of the action S × B ∗ → B ∗ at the point (s0 , f0 ) use the
following inequaity
|(s0 f0 )(vi ) − (sf )(vi )| ≤ |(s0 f0 )(vi ) − (s0 f )(vi )| + |(s0 f )(vi ) − (sf )(vi )| ≤
|f0 (vi s0 ) − f (vi s0 )| + ||f || · ||vi s0 − vi s||
39
Every left action π : S × X → X induces the right action C(X) × S → C(X)
(where (f s)(x) = f (sx)) and a co-homomorphism hπ : S → Θ(C(X)) and. While the
translations are continuous, the orbit maps
f˜ : S → C(X), s 7→ f s
are not necessarily norm (even weakly) continuous and require additional assumptions
for their continuity. It turns out that this happens iff f mimics the property of right
uniform continuity like RUC(G) (see Remark 9.8).
For every normed space V the usual adjoint map
adj : L(V ) → L(V ∗ ), σ 7→ σ ∗ , < vs, u >=< v, s∗ u >
is an injective co-homomorphism of monoids. For simplicity we write s instead of s∗ .
Theorem 9.21. (Teleman’s fisrt theorem) Every Hausdorff topological group G can
be embedded into Isolin (V ) for some Banach space V .
Proof. Consider the Banach space V := RUC(G) of all right uniformly continuous
bounded functions which is a closed linear subspace of (Cb (G), ||·||sup ). As we already
mentioned in Remark 9.8 the set RU C(G) separates points and closed subsets of G.
The usual left action G × G → G induces the following right action
V × G → V, (f, g) 7→ f g
on V := RU C(G). Then this action is linear contractive (by linear isometries) and
continuous. The linearity is trivial. For the contractivity observe that clearly, ||f g|| ≤
||f || for all g ∈ G. Since G is a group in fact we get ||f g|| ≤ ||f ||. It is easy to
show the continuity of that linear action using Remark 9.8.2. Indeed the condition
f ∈ RU C(G) means that
∀ ε > 0 ∃U ∈ N (e) |f0 (ux) − f0 (x)| < ε ∀u ∈ U, x ∈ G
it implies that
||f0 u − f0 || ≤ ε ∀u ∈ U
Now in order to see the continuity of that action at point (f0 , g0 ) use the following
inequality
||f0 g0 − f g|| = ||f0 g0 − f g0 u|| ≤ ||f0 g0 − f0 g0 u|| + ||f0 g0 u − f g0 u|| =
||f0 g0 − f0 g0 u|| + ||(f0 − f )g0 u|| = ||f0 g0 − f0 g0 u|| + ||f0 − f ||.
Consider the natural homomorphism
h : G → Isolin (V )op , h(g)(v) := vg.
This map is well defined because the action is contractive. Also h is a homomorphism. Clearly, (Isolin (V )op , τsot ) is a Hausdorff topological group. So, it is enough to
show that h is a topological embedding.
Continuity of h. By the trivial “transport argument” it is enough to show the
continuity of h at e ∈ G. Consider the following typical basic nbd of h(e) = id in the
topological group (Isolin (V )op , τsot ):
[id; f1 , f2 , · · · , fn ; ε] := {s ∈ Isolin (V )op : ||fi − fi s|| < ε ∀i ∈ {1, · · · , n}}
40
Note that here fi ∈ V play the role of vi in the definition of strong operator topology.
Since each fi is in RU C(G) we can find U ∈ N (e) such that ||fi u − fi || < ε for
every u ∈ U and every i ∈ {1, · · · , n}. Then clearly, g ∈ U implies that h(g) ∈
[id; f1 , f2 , · · · , fn ; ε]. This proves the continuity of h (at e).
This map h is injective because if g1 6= g2 then f (g1 ) 6= f (g2 ) for some f ∈ RUC(G).
Therefore, h(g)(f1 ) = f1 g 6= f2 g = h(g)(f2 ).
Now we show that, in fact, h is a topological embedding. It is enough to show that
the homomorphism h−1 : G0 → G is continuous (at id), where G0 := h(G). For a
given U ∈ N (e) there exists f ∈ RUC(G) such that f (e) ∈
/ cl(f (G \ U )) in R. Then
there exists ε > 0 (take ε := d(e, G \ U )) such that
|f (g) − f (e)| < ε ⇒ g ∈ U.
Now we get
||h(g)(f ) − f || = ||f g − f || < ε ⇒ g ∈ U.
Or, equivalently,
h(g) ∈ [id; f ; ε] ⇒ g ∈ U.
This means that the function h−1 : G0 → G is continuous at id.
9.2. Appendix.
Lemma 9.22. For every normed space V the injective map
∗
∗
γ : Θ(V )op
s ,→ C(B , B )
induced by the adjoint map adj : L(V ) → L(V ∗ ), is a topological (even uniform)
monoid embedding. In particular (by Proposition 9.15.3),
Θ(V )op × B ∗ → B ∗
∗
is a jointly continuous monoidal action of Θ(V )op
s on the compact space B .
Proof. The strong uniformity on Θ(V ) is generated by the family of pseudometrics
{pv : v ∈ V },where pv (s, t) = ||sv − tv||. On the other hand the family of pseudometrics {qv : v ∈ V },where qv (s, t) = sup{|(f s)(v) − (f t)(v)| : f ∈ B ∗ } generates
the natural uniformity inherited from C(B ∗ , B ∗ ). Now observe that pv (s, t) = qv (s, t)
by the Hahn-Banach theorem. This proves that γ is a uniform (and hence, also,
topological) embedding.
For every compact space K we have a topological embedding (Gelfand representation)
δ : K ,→ C(K)∗ , x 7→ δx
where δx : C(K) → R is the standard “evaluation at x functional” defined as δ(f ) :=
f (x). It can be identified with the “point measure” on K.
Theorem 9.23. Let S × K → K be a continuous action. Then
(1) the induced action S ×B ∗ → B ∗ is continuous on B ∗ ⊂ V ∗ , where V := C(K).
(2) Moreover, if S carries the compact open topology inherited from C(K, K) then
the homomorphisms S → C(B ∗ , B ∗ ) and h : S → Θ(V )op
s are topological
embeddings.
41
Proof. (1) The induced right linear action C(K) × S → C(K) is continuous (because
the orbit maps are norm continuous). This action is contractive. It follows that
h : S → Θ(V )op is a well defined strongly continuous homomorphism. Moreover, B ∗
is an S-subset under the dual action. By Theorem 9.20 we obtain that the action
S × B ∗ → B ∗ is continuous.
(2) Let S ⊂ C(K, K). Denote by τ0 the induced topology on S. The action
S × K → K can be treated as a restriction of the bigger action S × B ∗ → B ∗ , where
K naturally is embedded into B ∗ via Gelfand’s map. Then the topology τ on S
inherited from C(B ∗ , B ∗ ) majors the original topology τ0 . Hence, τ0 ⊂ τ .
On the other hand, the continuity of S × B ∗ → B ∗ easily implies by Proposition
9.15 that τ ⊂ τ0 on S. Summing up we conclude that τ = τ0 on S.
Theorem 9.24. (Teleman’s second theorem) Every Hausdorff topological group G is
embedded into Homeo(K) for some compact space K.
Proof. Combine Teleman’s first theorem 9.21 and Lemma 9.22. It follows that the
desired compact space K we can choose as (B ∗ , τw∗ ), where B ∗ is the closed unit
ball of the dual space V ∗ of V := RU C(G), where B ∗ is endowed with the weakstar topology. Recall that by Alaouglu theorem, (B ∗ , τw∗ ) is always compact for any
Banach space V .
References
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J. Kelley, General Topology.
G. Lukacs, Compact-like topological groups, 2009.
J. Munkres, Topology.
S. Morris, Topology without tears, 2011.
S. Morris, Pontryagin duality and the structure of locally compact abelian groups, 1977.
V. Pestov, Topological groups: where to from here, 2000.
A. Wilansky, Topology for Analysis, 1998.
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
E-mail address: [email protected]
URL: http://www.math.biu.ac.il/∼ megereli