Download II. General theory of locally compact groups

II. General theory of locally compact groups
5. Basic properties
In the following, we often use the abbreviation “locogroup” for “locally compact topogroup”.
5.1 Lemma A topogroup is a locogroup if, and only if, the neutral element has a compact
and closed neighborhood.
Proof. Necessity follows directly from Corollary 1.28. Sufficiency follows easily from Theorem 1.8.
Clearly, in a (locally) compact topogroup, every closed subtopogroup is (locally) compact. It follows, by Corollary 2.8, that a subtopogroup of a (locally) compact T1 -topogroup
is (locally) compact if, and only if, the subgroup is closed.
By the Tihonov Theorem, a product space formed by compact topological spaces is
compact. As a consequence, a product space formed by finitely many locally compact
spaces is locally compact. Therefore a product topogroup formed by compact topogroups
is compact and a product topogroup formed by finitely many locogroups is a locogroup.
Since a continuous image of a compact space is compact and the image of a locally
compact space under an open and continuous mapping is locally compact, Theorem 2.15
gives us the following resul.
5.2 Theorem A quotient group of a (locally) compact topogroup is (locally) compact.
Next we give a partial converse to the above theorem. We need the following topological result.
5.3 Lemma Let f : X → Y be a closed mapping between topological spaces such that,
for every y ∈ Y , the subset f −1 (y) of X is compact. Let K be a compact subset of Y .
Then the subset f −1 (K) of X is compact.
Proof. Let O be an open cover of the subspace f −1 (K) of X . For every y ∈ K , since
f −1 (y) ⊂ f −1 (K) and f −1 (y) is compact, the family O has a finite subfamily Oy such
S
S
that f −1 (y) ⊂ Oy . Since Oy is an open subset of the subspace f −1 (K) of X , there
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exists an open subset Uy of X such that Uy ∩ f −1 (K) =
S
Oy . Let Vy = Y \ f (X \ Uy ) ,
and note that, since f is a closed mapping and Uy is open in X , the set Vy is open in Y .
Since f −1 (y) ⊂ Uy , we have y ∈ Vy .
Since the family {Vy : y ∈ K} of open subsets of Y covers the compact set K , there
S
exists a finite set A ⊂ K such that K ⊂ y∈A Vy . Denote by O′ the finite subfamily
S
′
−1
(K) . Let x be a point of the set f −1 (K) .
y∈A Oy of O . We show that O covers f
S
Since K ⊂ y∈A Vy , there exists u ∈ A such that f (x) ∈ Vu . From the definition of the
set Vu it follows that f (x) ∈
/ f (X \ Uu ) , and from this it further follows that x ∈ Uu .
S
S
Hence x ∈ Uu ∩ f −1 (K) = Ou . Since Ou ⊂ O′ , we have x ∈ O′ . We have shown
that O′ is a finite subcover of O . By the foregoing, the set f −1 (K) is compact.
Corollary Let H be a compact normal subgroup of a topogroup G and let K be a
compact subset of the topogroup G/H . Then the subset ϕ−1
H (K) of G is compact.
Proof. The natural homomorphism ϕH : G → G/H is a closed mapping by Theorem
−1
2.17. For every gH ∈ G/H , we have ϕ−1
H (gH) = gH , and thus ϕH (gH) is a compact
subset of G. Hence we can apply the preceding lemma to the mapping ϕH , and we see
that the subset ϕ−1
H (K) of G is compact.
5.4 Theorem Let H be a compact normal subgroup of a topogroup G. If the quotient
group G/H is (locally) compact, then G is (locally) compact.
Proof. It follows directly from the previous corollary that if G/H is compact, then
G = ϕ−1
H (G/H) is compact. On the other hand, if a point gH of G/H has a compact neighborhood V , then the compact set ϕ−1
H (V ) is a neigborhood of g , by continuity
of ϕH . As a consequence, G is locally compact if G/H is locally compact.
Now we start to consider connectedness and disconnectedness of locogroups. The
following basic result is the starting point of our considerations.
5.5 Theorem Let K be an open and compact subset of a topogroup G such that e ∈ K .
Then K contains an open subgroup of G.
Proof. Since K is compact and open, it follows from Theorem 1.24 that there exists
U ∈ ηe such that U K ⊂ K . We have that U 2 K = U (U K) ⊂ U K ⊂ K . Using induction,
S∞
we see that U n K ⊂ K for every n > 0 . Let S = n=1 (U ∩ U −1 )n . By Theorem 2.13, the
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set S is an open subgroup of G. Moreover, we have
!
∞
∞
∞
[
[
[
−1 n
−1 n
S ⊂ SK =
(U ∩ U ) K =
U n K ⊂ K.
(U ∩ U ) K ⊂
n=1
n=1
n=1
Using Theorem 5.5, we can give an interesting characterization of total disconnectedness for locogroups. First we prove some auxiliary results.
A topological space is zero–dimensional if the topology has a base consisting of clopen
sets. Note that every zero–dimensional space is regular and every zero–dimensional T1 –
space is totally disconnected. We show now that the last result has a converse in the class
of locally compact regular spaces.
5.6 Lemma Every locally compact totally disconnected regular space is zero–dimensional.
Proof. First, let X be a compact totally disconnected regular space. We show that X is
zero–dimensional. Let x be a point of X . Set U = {U ⊂ X : U is clopen and x ∈ U } and
T
S = U . We note that S is closed. We show that the following condition is satisfied:
(*) When O ⊂ X is open and S ⊂ O , there exists U ∈ U such that U ⊂ O .
Let O open and S ⊂ O . For every y ∈ X \ O , we have y ∈
/ S and hence there exists a set
Uy ∈ U such that y ∈
/ Uy . Since the set Uy is clopen, the set X \ Uy on open. The closed
subset X \ O of X is compact, and it follows that there exists a finite set A ⊂ X \ O such
S
T
S
that X \ O ⊂ y∈A (X \ Uy ) . Note that y∈A Uy = X \ y∈A (X \ Uy ) ⊂ O . Moreover,
T
T
x ∈ y∈A Uy and the intersection y∈A Uy of finitely many clopen sets is clopen. Since
T
T
x ∈ y∈A Uy ∈ U and y∈A Uy ⊂ O , we have shown that condition (*) holds.
We show that S = {x} . Assume on the contrary that S 6= {x} . Then, since X is
totally disconnected, the subspace S of X is disconnected. It follows that there exist nonempty closed subsets M and N of the subspace S such that M ∪ N = S and M ∩ N = ∅ .
Since S is closed in X , the sets M and N are closed in X . The totally disconnected
space X is a T1 –space. As a regular T1 –space, X is a Hausdorff space. It follows, since
X is compact, that X is normal. As a consequence, there exist open subsets V and W
of X such that M ⊂ V , N ⊂ W and V ∩ W = ∅ . We have S = M ∪ N ⊂ V ∪ W , and
it follows by condition (*), that there exists U ∈ U such that U ⊂ V ∪ W . We show that
sets U ∩ V and U ∩ W are clopen. Since the sets U , V and W are open, also the sets
U ∩ V and U ∩ W are open. On the other hand, since V ∩ W = ∅ and U ⊂ V ∪ W ,
we have U ∩ V = U \ W and U ∩ W = U \ V . The set U is closed and the sets V and
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W are open, and hence it follows from the foregoing, that the sets U ∩ V and U ∩ W
are closed. We have shown that U ∩ V and U ∩ W are clopen. We have x ∈ S and
S ⊂ (U ∩ V ) ∪ (U ∩ W ) , and thus x belongs either to the set U ∩ V or to the set U ∩ W .
Assume that x ∈ U ∩ V . Then U ∩ V ∈ U and therefore S ⊂ U ∩ V . This, however,
is a contradiction, since 6= N ⊂ S and V ∩ N =. Similarly, we obtain a contradiction
in the case that x ∈ U ∩ W . We have shown that the assumption S 6= {x} leads to a
contradiction. Therefore S = {x} , and it follows, by condition (*), that the family U of
clopen sets is a neighborhood base of x . We have shown that X is zero–dimensional.
Now, let Z be a locally compact totally disconnected regular space. To show that Z
is zero–dimensional, let z be a point of Z and O a neighborhood of z . Since Z is locally
compact, the point z has a compact neighborhood K . By regularity of Z , the point z has
an open neighborhood V such that V ⊂ O ∩ K . The subspace V of Z is compact, totally
disconnected and regular. By the first part of the proof, there exists a clopen subset U of
the subspace V such that z ∈ U ⊂ V . Since V ⊂ O , we have U ⊂ O . We show that U
is clopen in Z . The set U is closed in V and the set V is closed Z . As a consequence,
U is closed in Z . On the other hand, U is open in V and contained in V . Hence U is
open in the open subspace V of Z , and therefore U is open in Z . We have shown that
space Z is zero–dimensional.
Using the above results, we can now prove the following theorem.
5.7 Theorem A locogroup G is totally disconnected if, and only if, G is T1 and the open
subgroups of G form a neighborhood base of e in G.
Proof. Necessity. Assume that G is totally disconnected. Then G is T1 . By Corollary
1.22 and Lemma 5.6, G is zero–dimensional. Let O be a neighborhood of e. Since G is a
locogroup, e has a compact neighborhood K . Let U be a clopen subset of G such that
e ∈ U ⊂ K ∩ O . The set U is compact and open, and it follows from Theorem 5.5 that
there exists an open subgroup S of G such that S ⊂ U . We have e ∈ S ⊂ U ⊂ O , and
we have thus shown that the open subgroups of G form a base of ηe .
Sufficiency. Assume that G is a T1 –locogroup and the open subgroups of G form a base
of ηe . By Theorem 2.9, open subgroups of G are clopen. Since G is T1 and e has a
clopen neighborhood base, we have that CG (e) = {e} . By Theorem 3.2, G is totally
disconnected.
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5.8 Corollary Every T1 quotient group of a totally disconneted locogroup is totally
disconnected.
Proof. Let G be a totally disconnected locogroup and let H be a normal subgroup of
G. We show that the open subgroups of G/H form a neighborhood base for the neutral
element H of G/H . From this it follows by the above theorem that G/H is totally
disconnected, provided that G/H is T1 .
Let O be a neighborhood of H in G/H . Then ϕ−1
H (O) is a neighborhood of e in G.
By the preceding theorem, there exists an open subgroup S of G such that S ⊂ ϕ−1
H (O) .
Since ϕH is an open mapping, the set ϕH (S) is open in G/H . Since ϕH is homomorphism,
the image ϕH (S) of the subgroup S of G is a subgroup of G/H . Moreover, since S ⊂
ϕ−1
H (O) , we have ϕH (S) ⊂ O . We have shown that ηH (G/H) has a base consisting of
open subgroups.
5.9 Theorem Let G be a locogroup, let C be the connected component of the neutral
element e of G and let U be a neighborhood of e. Then the set CU contains an open
subgroup of G.
Proof. By Theorem 3.2, C is a normal subgroup of G. By Theorems 5.2 and 3.9, the
quotient group G/C is a totally disconnected locogroup. Since ϕC is an open mapping
and U ∈ ηe (G) , we have ϕC (U ) ∈ ηC (G/C) . By Theorem 5.7, there exists an open
subgroup J of G/C such that J ⊂ ϕC (U ) . Since ϕC is a continuous homomorphism, the
−1
set ϕ−1 (J) is an open subgroup of G. Since J ⊂ ϕC (U ) , we have ϕ−1
C (J) ⊂ ϕC (ϕC (U )).
−1
Since ϕ−1
C (ϕC (U )) = U C = CU , it follows from the foregoing that ϕC (J) ⊂ CU .
Next we point out some consequences of the above theorem. The first result strengthens Theorem 5.5.
5.10 Corollary Every clopen neighborhood of the neutral element of a locogroup contains
an open subgroup.
Proof. Let U be a clopen neighborhood of the neutral element e of a locogroup G. Let
C = CG (e) . For every g ∈ U , we have CG (g) ∩ U 6= ∅ and it follows, by clopenness of U ,
that CG (g) ⊂ U . By Theorem 3.2, we have CG (g) = Cg for every g ∈ G. It follows from
the foregoing that CU ⊂ U . By Theorem 5.9, the set U contains an open subgroup of G.
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The next three results deal with the component of the neutral element of a locogroup.
5.11 Corollary The component of the neutral element of a locogroup is the intersection
of all open subgroups.
Proof. Let G be a locogroup and let C be the component of e. Set J = {J ⊂ G :
T
Jis an open subgroup of G} . Since e ∈ J and every J ∈ J is a clopen subset of G, we
T
T
have C ⊂ J . Since C is closed, it follows from Theorem 1.19 that C = U∈η CU . By
e
theorem 5.9, there exists, for every U ∈ ηe , a set J ∈ J such that that J ⊂ CU . Hence
T
T
J ⊂ U∈η CU = C .
e
5.12 Corollary The component of the neutral element of a locogroup is compact if, and
only if, the group has an open compact subgroup.
Proof. Sufficiency follows directly, since the component of the neutral element is closed
and it is contained in every open subgroup.
Necessity. Let G be a locogroup such that the component C of e is compact. Let U be
a compact neighborhood of e. By Theorem 5.9, there exists an open subgroup J of G
such that J ⊂ CU . The subgroup J is compact, because J is closed and the set CU is
compact by Theorem 1.27.
By Theorems 1.24 and 5.9, we have the following result.
5.13 Corollary Let G be a locogroup such that the component C of e is compact. If
an open subset O of G contains the set C , then O contains an open subgroup of G.
We show that the last result above does not remain valid without the assumption of
compactness of the component of e.
5.14 Example Let D be the set {0, 1} and define a binary operation + on D by setting
0 + 0 = 1 + 1 = 0 and 0 + 1 = 1 + 0 = 1 . Then D is a group with respect to the
operation +. We make D into a topogroup by equipping it with operation + and discrete
topology. Let G be the product topogroup DN . The topogroup G is compact by the
Tihonov Theorem. Denote the neutral element h0, 0, . . .i of G by 0. For every k > 0 , let
Jk = {hdn i∞
n=0 ∈ G : dn = 0 for every n ≤ k} . The sets Jk are open in G and the family
{Jk : k = 1, 2, ...} is a neighborhood base of 0 in G. Moreover, every Jk is a subgroup of
G. It follows from the foregoing that the compact group G is totally disconnected.
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Let P be the product topogroup G × R . Since G and R are locogroups, P is a
locogroup. Let C be the component of the neutral element (0, 0) of P . Since G is totally
disconnected and R is connected, we have C = {0} × R . For every k > 0 , denote by Ok
S∞
the open subset Jk × {r ∈ R : |r| < k} of P . Let O = k=1 Ok . Then O is an open
subset of P and C ⊂ O . We show that O does not contain any clopen neighborhood
of the neutral element. Let U be a clopen subset of P such that (0, 0) ∈ U . Since
U is a neighborhood of (0, 0) , there exists g ∈ G \ {0} such that (g, 0) ∈ U . The set
U ∩({g}×R) is a non-empty clopen subset of the subspace {g}×R of P . Note that {g}×R
is homeomorphic with R and hence connected. It follows that U ∩ ({g} × R) = {g} × R ,
in other words, that {g} × R ⊂ U . Since g 6= 0, there exists ℓ > 0 such that g 6∈ Jℓ .
By the foregoing we have (g, ℓ) ∈ U . However, (g, ℓ) does not belong to the set O . As a
consequence, U 6⊂ O . We have shown that O does not contain any clopen neighborhood
of the neutral element. It follows that no open subgroup of P is contained in O .
Next we consider connected locogroups. Note that Theorem 3.6 and Corollaries 3.7
and 5.11 give the following characterizations of connectedness for locogroups.
5.15 Theorem The following conditions are mutually equivalent for a locogroup G:
a) G is connected.
b) G has no proper open subgroup.
S∞
c) For every U ∈ ηe , we have n=1 U n = G.
Proof. a ⇒ c: This follows from Corollary 3.7.
b⇐⇒ c: This holds in any topogroup, by Theorem 3.6.
b⇒ a: This follows from Corollary 5.11.
Example 3.8 shows that the preceding result does not remain valid without the assumption that G is locally compact.
We say that a topogroup is compactly generated if the topogroup has a compact
generating set.
5.16 Corollary Every connected locogroup is compactly generated.
Proof. Let G be a connected locogroup. Let V be a compact neighborhood of the neutral
S∞
element of G. By the preceding theorem, we have n=1 V n = G. As a consequence,
S∞
−1 n
) = G.
n=1 (V ∪ V
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An uncountable discrete group, for example R with discrete topology, gives an example of a locogroup, which is not compactly generated (note that, in a discrete space,
compact subsets are finite and that every “finitely generated” group is countable). The
topogroup Q is an example of a compactly generated topogroup, which is not a locogroup:
the compact set {0} ∪ { n1 : n ∈ Z and n 6= 0} generates Q.
We close this chapter with some results on compact topogroups.
Every commutative topogroup and every discrete topogroup is (trivially) locally invariant. We now show that compact topogroups form a third important class of locally
invariant topogroups.
5.17 Theorem Every compact topogroup is locally invariant.
Proof. Let G be a compact topogroup and let U be a neighborhood of the neutral element
e of G. By Theorem 1.25, there exists a neighborhood V of e such that, for every g ∈ G,
S
we have gV g −1 ⊂ U . Let W = g∈G gV g −1 , and note that W ∈ ηe ja W ⊂ U . For
S
S
every h ∈ G, we have hW h−1 = g∈G hgV g −1 h−1 = g∈G (hg)V (hg)−1 ⊂ W ; It follows
that W is an invariant subset of G.
For compact groups, we can use the above result to strengthen earlier results on the
existence of open subgroups in locogroups. We also need the following result.
5.18 Lemma In a locally invariant topogroup, every open subgroup contains an open
normal subgroup.
Proof. Let J be an open subgroup of a locally invariant topogroup G. Then J ∈ ηe ,
and thus there exists an invariant set U ∈ ηe such that U ⊂ J . Since U is invariant, the
set U −1 , and hence also the set U ∪ U −1 , is invariant. For two invariant subsets A and
B of G, the set AB is invariant. It follows, that for every n > 0 , the set (U ∪ U −1 )n
S∞
is invariant. As a consequence, the set H = n=1 (U ∪ U −1 )n is invariant. By Theorem
2.13, H is an open subgroup of G. As an invariant set, the subgroup H is normal. Since
U ⊂ J and J is a subgroup, we have H ⊂ J .
The following is a consequence of Corollary 5.13, Theorem 5.17 and Lemma 5.18.
5.19 Theorem Let G be a compact topogroup and O an open subset of G which contains
the component of the neutral element. Then O contains an open normal subgroup.
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5.20 Corollary In a compact totally disconneted topogroup, the neutral element has a
neighborhood base consisting of open normal subgroups.
The last result above allows us to give an interesting characterization of totally disconnected compact topogroups. First we record a result which follows from Theorems 5.2
and 2.18 together with the observation that every compact discrete space is finite.
5.21 Lemma Let J be an open subgroup of a compact topogroup G. Then the quotient
group G/J is finite.
We now show that every compact totally disconnected topogroup can be represented
as a closed subgroup of the product of a collection of finite groups.
5.22 Theorem A topogroup G is compact and totally disconnected if, and only if, there
exists a family {Fi : i ∈ I} of finite discrete topogroups and a closed subtopogroup S of
the topogroup Πi∈I Fi such that G =S
e .
Proof. Sufficiency. Let Fi , i ∈ I , be finite discrete topogroups, and let S be a closed
subgroup of the topogroup Πi∈I Fi such that G =S
e . The topogroup Πi∈I Fi is compact,
by the Tihonov Theorem. It follows that the closed subtopogroup S , and hence also
the topogroup G, is compact. Using continuity of the projection maps, we see easily
any product of totally disconneted topological spaces is totally disconnected. Hence the
topogroup Πi∈I Fi is totally disconnected. It follows that the subtopogroup S of Πi∈I Fi ,
and hence also the topogroup G, is totally disconnected.
Necessity. We assume that the topogroup G is compact and totally disconnected. Then
G is T1 –space. By Corollary 5.20, there exists a collection {Hi : i ∈ I} of open normal
T
subgroups of G such that i∈I Hi = {e} . For every i ∈ I , denote by Fi the finite and
discrete quotient topogroup G/Hi and by ϕi the natural homomorphism G → G/Hi .
Define a mapping f : G → Πi∈I Fi by the formula f (g) = hϕi (g)ii∈I . Since f
the component mappings ϕi of f are continuous homomorphisms, the mapping f is a
continuous homomorphism. We show that f is one-to-one. Let g and h be two distinct
T
points of G. Then h−1 g 6= e, and it follows, since i∈I Hi = {e} , that there exists j ∈ I
such that h−1 g ∈
/ Hj . We have g ∈
/ hHj and, therefore, ϕj (g) 6= ϕj (h) . It follows that
f (g) 6= f (h) . We have shown that f is one-to-one. By the foregoing, f is a continuous
group isomorphism G → f (G) . The subspace f (G) of the Hausdorff space Πi∈I Fi is
Hausdorff, and it follows, by compactness of G, that the continuous one-to-one and onto
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mapping f : G → f (G) is a homeomorphism. Hence G =
e f (G) . Moreover, the compact
set f (G) is closed in Πi∈I Fi .
We close this chapter with an example, which shows that one can associate with every
(totally disconneted) compact metric space a (totally disconnected) compact topogroup.
5.23 Example Let (X, d) be a compact metric space. As in Example 4.16, we equip the
isometry group (Id , ◦) with the topology τ determined by the invariant length function
ρ , where ρ(f ) = supx∈X d(f (x), x) . In the following, we speak of “the topogroup Id ” to
refer to (Id , ◦, τ) .
We show that the topogroup Id is compact. Besides the topology τ , we need to
consider two other topologies which Id inherits from the space X X . For the definition
of these topologies and of other concepts used in the proof below, see the Appendix on
Ascoli’s Theorem. We denote by τp the relative topology of point-wise convergence on
Id and by τu the relative topology of uniform convergence (note that, by compactness
of X , the topology τu is the same as the relative compact–open topology on Id ). Since
every f ∈ Id is an isometry, Id is an equicontinuous set of mappings (X, d) → (X, d) . By
Lemma 1 of the Appendix, we have that τp = τu . It is easy to see that the set Id is closed
in the product space X X . By the Tihonov Theorem, the product space X X is compact.
It follows that Id is compact in the topology τp . By the foregoing, compactness of the
topogroup Id follows, if we show that τ = τu . Let f ∈ Id . In the topology τu , the sets
{g ∈ Id : supx∈X d(g(x), f (x)) < ǫ} , where ǫ > 0 , form a neighborhood base of f . On the
other hand, in the topology τd , the sets f Vρ,ǫ , where ǫ > 0 , form a neighborhood base of
f . Since f is an isometry, we have for every ǫ > 0 , that
f Vρ,ǫ = {f g : ρ(g) < ǫ} = {h : ρ(f −1 h) < ǫ}
= {h : sup d f −1 (h(x)) , x < ǫ} = {h : sup d (h(x), f (x)) < ǫ} .
x∈X
x∈X
We have shown that every f ∈ Id has a same neighborhood base in both topologies. Hence
τ = τu .
Assume that X is totally disconnected. Then the product space X X is totally disconnected. By the foregoing, the topology τ of Id coincides with the relative topology τp
of Id in X X . As a consequence, the topogroup Id is totally disconnected.
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