Download LECTURE NOTES IN TOPOLOGICAL GROUPS 1. Lecture 1

LECTURE NOTES IN TOPOLOGICAL GROUPS
MICHAEL MEGRELISHVILI
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.
(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.
Date: November 4, 2013.
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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.
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 a
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.
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(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.
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) ** Let X ∈ T1 . 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, 03/11
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 .
5. 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 5.1. 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.
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Remark 5.2. In particular, if f2 is an identity continuous map then it is a homeomorphism iff it is a quotient.
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 5.3. 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 5.4. 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 5.1 and Remark 5.2.
Lemma 5.5. (Properties of the quotient groups G/H)
(1)
(2)
(3)
(4)
(5)
G/H is Hausdorff iff H is closed in G.
G/H is discrete iff H is open in G.
Any closed subgroup H ≤ G of the finite index [G : H] < ∞ is clopen.
If G is LC then G/H is also LC.
(up to the Kakutani thm mentioned above)
If G is metrizable (=B1 ) 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.
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6. Closed subgroups of R and T
We will examine the following questions: What are the closed subgroups of R and
T ? We are going to prove that:
the closed subgroups of R are only:
R, aZ a ≥ 0
the closed subgroups of T are only:
T, Ωn n ∈ N
To be Continued ........
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6.1. Homework 3 (returning date 18.11.13).
Exercise 6.1. * Prove or disprove: the topological group T2 (two-dimensional torus)
is a topological factor group of the group C r {0}.
Exercise 6.2. 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 6.3. 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 6.4. 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 6.5. * 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).
Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
E-mail address: [email protected]
URL: http://www.math.biu.ac.il/∼ megereli