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Theorem 5.2.17 (Birkhoff-Kakutani theorem). A Hausdorff group \( G \) is metrizable if and only if \( \chi \left( G\right) \leq \omega \) . | Proof. The necessity is obvious as every metrizable space is first countable.\n\nSuppose now that \( \mathcal{V}\left( {e}_{G}\right) \) has a countable base. Then one can build a chain \( \left\{ {{U}_{n} : n \in }\right. \) \( \mathbb{N}\} \) as in (5.2) of symmetric neighborhoods of \( {e}_{G} \) in \( G \) with \( ... | Yes |
Let \( \left\{ {{G}_{i} : i \in I}\right\} \) be an infinite family of nontrivial metrizable Hausdorff groups. Then \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) satisfies \( \chi \left( G\right) = \left| I\right| \) | by Birkhoff-Kakutani theorem 5.2.17 and by Theorem 5.1.15. | No |
Every Hausdorff abelian group \( G \) embeds into a product of metrizable abelian groups. | Denote by \( \tau \) the topology of \( G \) and let \( \mathcal{B} = \left\{ {U \in {\mathcal{V}}_{G}\left( 0\right) : U}\right. \) open \( \} \) . For every \( U \in \mathcal{B} \), build a decreasing chain \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) of symmetric open neighborhoods of 0 with \( {U}_{0} \subset... | Yes |
Every Hausdorff abelian group \( \left( {G,\tau }\right) \) of countable pseudocharacter is submetrizable. In particular, every countable Hausdorff abelian group is submetrizable. | Proof. Let \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) be open neighborhoods of 0 in \( \tau \) with \( \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}{U}_{n} = \{ 0\} \) . It is not restrictive to assume that they form a decreasing chain as in (5.2) and Corollary 5.2.15. Call \( {\tau }_{m} \) the group topology ... | No |
Example 5.2.21. Following the counterpart of Example 5.2.7(iii) for \( {\mathbb{C}}^{n} \) in place of \( {\mathbb{R}}^{n} \) , here we extend the definition of the sup-norm from \( {\mathbb{C}}^{n} \) to the \( \mathbb{C} \) -algebra \( {C}^{ * }\left( X\right) \) of all bounded complex-valued functions of an arbitrar... | \[ \parallel f\parallel \mathrel{\text{:=}} \sup \{ \left| {f\left( x\right) }\right| : x \in X\} \;\text{ for every }f \in {C}^{ * }\left( X\right) . \] This norm gives rise to the invariant metric \( d = {d}_{\parallel - \parallel } \) described by \[ d\left( {f, g}\right) = \parallel f - g\parallel = \sup \{ \left| ... | Yes |
Let \( \left( {X, d}\right) \) be a compact metric space and consider on \( X \) the metric topology induced by \( d \) . Then the group Homeo \( \left( X\right) \) of all homeomorphisms of \( X \) admits a norm \( v \) defined by\n\n\[ \nv\left( f\right) = \sup \left\{ {d\left( {x, f\left( x\right) }\right) + d\left( ... | Since surjective isometries \( X \rightarrow X \) of a compact metric space \( X \) are homeomorphisms with respect to the metric topology on \( X \), it makes sense to consider the subgroup Iso \( \left( X\right) \) of all surjective isometries \( X \rightarrow X \) . The restriction of the Birkhoff topology on \( \op... | No |
(a) The sequences \( {\left\{ {n}^{2}\right\} }_{n \in \mathbb{N}} \) and \( {\left\{ {n}^{3}\right\} }_{n \in \mathbb{N}} \) are not \( T \) -sequences in \( \mathbb{Z} \) . | Indeed, suppose for a contradiction that some Hausdorff group topology \( \tau \) on \( \mathbb{Z} \) makes \( {\left\{ {n}^{2}\right\} }_{n \in \mathbb{N}} \) converge to 0 . Then \( {\left\{ {\left( n + 1\right) }^{2}\right\} }_{n \in \mathbb{N}} \) converges to 0 as well. Taking the difference of \( {\left\{ {\left(... | Yes |
Lemma 5.3.11. Let \( G \) be an abelian group and \( {\left\{ {a}_{n}\right\} }_{n \in \mathbb{N}} \) a \( T \) -sequence of \( G \) . Then the subgroup \( H \) of \( G \) generated by the countable set \( \left\{ {{a}_{n} : n \in \mathbb{N}}\right\} \) is \( {\tau }_{\left\{ {a}_{n}\right\} } \) -open. | Proof. Denote by \( {\tau }^{ * } \) the supremum of \( {\tau }_{\left\{ {a}_{n}\right\} } \) and the Alexandrov group topology on \( G \) with \( H \) as the smallest neighborhood of 0 . Then \( {\tau }^{ * } \geq {\tau }_{\left\{ {a}_{n}\right\} } \) and obviously \( {a}_{n} \rightarrow 0 \) in \( {\tau }^{ * } \) . ... | Yes |
Lemma 6.1.1. Let \( G \) be a topological group.\n\n(a) If \( {C}_{1},\ldots ,{C}_{n} \) are connected subsets of \( G \), then also \( {C}_{1}\cdots {C}_{n} \) is connected.\n\n(b) If \( C \) is a connected subset of \( G \), then \( {C}^{-1} \), as well as the subgroup \( \langle C\rangle \) generated by \( C \), is ... | Proof. (a) Consider the case \( n = 2 \), the general case easily follows by induction. The subset \( {C}_{1} \times {C}_{2} \) of \( G \times G \) is connected. Now the multiplication \( \mu : G \times G \rightarrow G,\left( {x, y}\right) \mapsto {xy} \) , is continuous and \( \mu \left( {{C}_{1} \times {C}_{2}}\right... | Yes |
Proposition 6.1.2. Let \( G \) be a topological group and \( N \) a closed normal subgroup of \( G \) . If \( N \) and \( G/N \) are connected (respectively, hereditarily disconnected), then also \( G \) is connected (respectively, hereditarily disconnected). | Proof. Let \( q : G \rightarrow G/N \) be the canonical projection.\n\nAssume that \( N \) and \( G/N \) are connected and let \( A \) be a nonempty clopen set of \( G \) . As every coset \( {aN} \) with \( a \in G \) is connected in view of Lemma B.6.4(a) and Lemma 2.1.6, one has either \( {aN} \subseteq A \) or \( {a... | Yes |
Corollary 6.1.3. A topological group \( G \) is connected if and only if \( \mathfrak{h}G \) is connected. | Proof. Since \( \mathfrak{h}G = G/\operatorname{core}\left( G\right) \) and \( \operatorname{core}\left( G\right) \) is connected (being indiscrete), Proposition 6.1.2 and Lemma B.6.4(a) apply. | No |
Corollary 6.1.4. If a connected group \( G \) is not indiscrete, then \( \left| {\mathfrak{h}G}\right| \geq \mathfrak{c} \) . | Proof. In view of Corollary 6.1.3, we can assume without loss of generality that \( G = \mathfrak{h}G \) is Hausdorff. Then \( G \) is also a Tichonov space by Theorem 5.2.14, so we can arrange for a nonconstant continuous function \( f : G \rightarrow \left\lbrack {0,1}\right\rbrack \) . Since \( G \) is connected, th... | Yes |
Every proper subgroup \( H \) of \( \mathbb{T} \) is zero-dimensional. | Indeed, \( H \) is either finite or dense, in view of Example 3.1.14(a). If \( H \) is finite, then it is clearly zero-dimensional. If \( H \) is dense, then for any fixed \( a \in \mathbb{T} \smallsetminus H \) also \( a + H \) is dense and disjoint with \( H \) . Hence, \( \left\{ {{\Gamma }_{b, c} \cap H : b, c \in ... | Yes |
Proposition 6.2.2. The connected component \( c\left( G\right) \) of a topological group \( G \) is a closed normal subgroup of \( G \) . The connected component of an element \( x \in G \) is simply the coset \( {xc}\left( G\right) = c\left( G\right) x. | Proof. To prove that \( c\left( G\right) \) is stable under multiplication, it suffices to note that \( c\left( G\right) c\left( G\right) \) is still connected by Lemma 6.1.1(a) and contains \( {e}_{G} \), so must be contained in the connected component \( c\left( G\right) \) . Similarly, an application of Lemma 6.1.1(... | Yes |
Lemma 6.2.3. For every topological group \( G \), the quotient \( G/c\left( G\right) \) is hereditarily disconnected. | Proof. Let \( q : G \rightarrow G/c\left( G\right) \) be the canonical projection and \( H = {q}^{-1}\left( {c\left( {G/c\left( G\right) }\right) }\right) \). Apply Proposition 6.1.2 to \( H \) and its connected quotient \( H/c\left( G\right) \cong c\left( {G/c\left( G\right) }\right) \) to conclude that \( H \) is con... | Yes |
Proposition 6.2.5. For a topological group \( G \), the arc component \( a\left( G\right) \) of \( G \) is a normal subgroup of \( G \) . | Proof. By Exercise 6.3.3(a) and the continuity of the multiplication \( \mu : G \times G \rightarrow G \) , \( \left( {x, y}\right) \mapsto {xy} \), we get \( a\left( G\right) a\left( G\right) \subseteq a\left( G\right) \) . Analogously, using the continuity of the inversion map \( t : G \rightarrow G, x \mapsto {x}^{-... | No |
For a topological group \( G \), the quasicomponent \( Q\left( G\right) \) is a closed normal subgroup of \( G \) . The quasicomponent \( {Q}_{x} \) of \( x \in G \) coincides with the coset \( {xQ}\left( G\right) = \) \( Q\left( G\right) x \) . | Let \( x, y \in Q\left( G\right) \) . To prove that \( {xy} \in Q\left( G\right) \), we need to verify that \( {xy} \in O \) for every clopen set \( O \) of \( G \) containing \( {e}_{G} \) . Let \( O \) be such a set, then \( x, y \in O \) . Obviously, \( O{y}^{-1} \) is a clopen set containing \( {e}_{G} \), hence \(... | Yes |
Lemma 7.1.4. Let \( G \) be a topological group.\n\n(a) A Cauchy net \( {\left\{ {g}_{\alpha }\right\} }_{\alpha \in A} \) of \( G \) is convergent if and only if it has a convergent subnet. | Proof. (a) Let \( {\left\{ {g}_{{\alpha }_{\gamma }}\right\} }_{\gamma \in \Gamma } \) be a subnet of \( {\left\{ {g}_{\alpha }\right\} }_{\alpha \in A} \) with \( {g}_{{\alpha }_{\gamma }} \rightarrow x \in G \) . We prove that \( {g}_{\alpha } \rightarrow x \) . Let \( U \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) ... | Yes |
Proposition 7.1.7. Let \( G \) be a topological group and \( q : G \rightarrow \mathfrak{h}G \) its Hausdorff reflection.\n\nThen:\n\n(a) a net \( {\left\{ {x}_{\alpha }\right\} }_{\alpha \in A} \) in \( G \) is convergent (respectively, Cauchy) if and only if \( {\left\{ q\left( {x}_{\alpha }\right) \right\} }_{\alpha... | Proof. (a) follows from Exercise 7.3.1(b) and Lemma B.3.2; (b) follows from (a). | No |
Proposition 7.1.8. Let \( G \) be a complete Hausdorff group and \( H \) a subgroup of \( G \) . Then \( H \) is complete if and only if \( H \) is closed. | Proof. Assume that \( H \) is a closed subgroup of the complete Hausdorff group \( G \) and let \( {\left\{ {h}_{\alpha }\right\} }_{\alpha \in A} \) be a Cauchy net of \( H \) . Since \( G \) is complete, \( {\left\{ {h}_{\alpha }\right\} }_{\alpha \in A} \) converges to some \( g \in G \) . Since \( H \) is closed, \... | Yes |
Proposition 7.1.9. Let \( \left\{ {{G}_{i} : i \in I}\right\} \) be a family of topological groups. Then \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) is complete if and only if \( {G}_{i} \) is complete for every \( i \in I \) . | Proof. Assume that \( {G}_{i} \) is complete for every \( i \in I \) and let \( {\left\{ {x}_{\alpha }\right\} }_{\alpha \in A} \) be a Cauchy net of \( G = \) \( \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) . Since for every \( i \in I \) the projection \( {p}_{i} : G \rightarrow {G}_{i} \) is continuous, the net \( {... | No |
Lemma 7.1.12. Let \( G \) be a topological group and \( \mathcal{B} \) a local base at \( {e}_{G} \) . Then \( G \) is complete if and only if every Cauchy \( \mathcal{B} \) -net of \( G \) converges in \( G \) . | Proof. Let \( {\left\{ {x}_{\alpha }\right\} }_{\alpha \in A} \) be a Cauchy net of \( G \) . For every \( U \in \mathcal{B} \), there exists \( {\alpha }_{U} \in A \) such that, for every \( \alpha ,\beta \geq {\alpha }_{U},{x}_{\alpha }^{-1}{x}_{\beta } \in U \) and \( {x}_{\beta }{x}_{\alpha }^{-1} \in U \) . We pro... | Yes |
Lemma 7.1.13. A metrizable group is complete if and only if every Cauchy sequence of G converges in G. | Proof. By Birkhoff-Kakutani theorem 5.2.17, there exists a countable base \( \mathcal{B} = \left\{ {{U}_{n} : n \in }\right. \) \( \mathbb{N}\} \) of \( {\mathcal{V}}_{G}\left( {e}_{G}\right) \) . For every \( n \in \mathbb{N} \) let \( {V}_{n} = {U}_{0} \cap \cdots \cap {U}_{n} \) . Then \( {\mathcal{B}}^{\prime } = \... | Yes |
Let \( \mathbb{Q} \) be endowed with the usual topology. For every \( k \in {\mathbb{N}}_{ + } \), let \( {U}_{k} = \) \( \left( {-\pi /k,\pi /k}\right) \cap \mathbb{Q} \) ; clearly, \( \left\{ {{U}_{k} : k \in {\mathbb{N}}_{ + }}\right\} \) is a base of \( {\mathcal{V}}_{\mathbb{Q}}\left( 0\right) \). Let \( {\left\{ ... | Now \( {\left\{ {x}_{n}\right\} }_{n \in \mathbb{N}} \in {U}_{1}^{ \sim } \), but there exists no \( m \in {\mathbb{N}}_{ + } \) such that \( {\left\{ {x}_{n}\right\} }_{n \in \mathbb{N}} + {U}_{m}^{ \sim } \subseteq {U}_{1}^{ \sim } \). | Yes |
Proposition 7.1.16. If a topological group \( G \) is metrizable, then its completion \( \widetilde{G} \) is metrizable as well. | Proof. Since \( \widetilde{G} \) is Hausdorff, by Birkhoff-Kakutani theorem 5.2.17 it suffices to prove that in case there exists a countable base \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) of \( {\mathcal{V}}_{G}\left( {e}_{G}\right) \), there exists a countable base of \( {\mathcal{V}}_{\widetilde{G}}\left( {... | No |
Corollary 7.1.17. The completion \( \widetilde{G} \) exists for every Hausdorff abelian group \( G \) . | Proof. According to Corollary 5.2.19, \( G \) is isomorphic to a subgroup of a product \( \mathop{\prod }\limits_{{i \in I}}{M}_{i} \), where each abelian group \( {M}_{i} \) is metrizable. By Theorem 7.1.14, every \( {M}_{i} \) has a completion \( {\widetilde{M}}_{i} \) . Then \( P = \mathop{\prod }\limits_{{i \in I}}... | Yes |
Corollary 7.1.19. Let \( \left\{ {{G}_{i} : i \in I}\right\} \) be a family of Hausdorff groups and \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) . Then \( \widetilde{G} \cong \mathop{\prod }\limits_{{i \in I}}{\widetilde{G}}_{i} \) | Proof. By Proposition 7.1.9, the product \( \mathop{\prod }\limits_{{i \in I}}{\widetilde{G}}_{i} \) is complete. Since \( G \) is a dense subgroup of \( \mathop{\prod }\limits_{{i \in I}}{\widetilde{G}}_{i} \), Theorem 7.1.18 applies. | Yes |
Proposition 7.1.22. A Hausdorff group \( G \) is complete if and only if for every embedding \( j : G \hookrightarrow H \) into a Hausdorff group \( H \) the subgroup \( j\left( G\right) \) of \( H \) is closed. | Proof. Assume that there exists an embedding \( j : G \hookrightarrow H \) into a Hausdorff group \( H \) such that \( j\left( G\right) \) is not a closed subgroup of \( H \) . Then there exists a net \( {\left\{ {y}_{\alpha }\right\} }_{\alpha \in A} \) in \( j\left( G\right) \) converging to some element \( h \in H \... | Yes |
Lemma 7.1.26. Let \( G \) be a Hausdorff group. Every left (respectively, right) Cauchy net of \( G \) with a convergent subnet is convergent. | Proof. Let \( {\left\{ {g}_{\alpha }\right\} }_{\alpha \in A} \) be a left Cauchy net of \( G \) and let \( {\left\{ {g}_{\beta }\right\} }_{\beta \in B} \) be a subnet convergent to \( x \in G \), where \( B \) is a cofinal subset of \( A \) . Let \( U \) be a neighborhood of \( {e}_{G} \) in \( G \) and \( V \) a sym... | Yes |
Proposition 7.1.29. For an infinite set \( X \), let \( S\left( X\right) \) be equipped with \( {\mathrm{T}}_{X} \): (a) \( S\left( X\right) \) is Raïkov complete; | Proof. (a) Let \( {\left\{ {f}_{\alpha }\right\} }_{\alpha \in A} \) be a Cauchy net of \( S\left( X\right) \) . For every finite subset \( E \) of \( X \), there exists \( {\alpha }_{0} \in A \) such that for every \( \alpha ,\beta \geq {\alpha }_{0},{f}_{\beta }^{-1}{f}_{\alpha } \in {S}_{E} \) and \( {f}_{\alpha }{f... | Yes |
Lemma 7.2.2. Let \( G \) be a Hausdorff group.\n\n(a) For a filter \( \mathcal{F} \) on \( G \), the following conditions are equivalent:\n\n\( \left( {\mathrm{a}}_{1}\right) \) the filter \( \mathcal{F} \) is Cauchy;\n\n\( \left( {\mathrm{a}}_{2}\right) \) the filter \( {\mathcal{F}}^{-1} \mathrel{\text{:=}} \left\{ {... | Proof. The verification of (a)-(c) is a straightforward application of the definitions.\n\n(d) Pick \( U \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) and \( V \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) with \( {VV} \subseteq U \) . According to \( \left( {\mathrm{a}}_{4}\right) \), there exist \( H \in \mathcal{H} ... | No |
Proposition 7.2.6. Let \( G \) be a Hausdorff group and \( \mathcal{F} \) a Cauchy filter on \( G \) . The following conditions are equivalent:\n\n(a) \( \mathcal{F} \) is minimal;\n\n(b) if \( {\mathcal{F}}_{1} \) is a Cauchy filter on \( G \) such that \( {\mathcal{F}}_{1} \subseteq \mathcal{F} \), then \( {\mathcal{... | Proof. (a) \( \Rightarrow \) (b) Let \( F \in \mathcal{F} \) . By the minimality of \( \mathcal{F} \), there exist \( {F}^{\prime } \in \mathcal{F} \) and \( U \in \mathcal{V}\left( {e}_{G}\right) \) such that \( U{F}^{\prime }U \subseteq F \) . Moreover, there exists \( {F}_{1} \in {\mathcal{F}}_{1} \) such that \( {F... | Yes |
Lemma 7.2.7. Let \( G \) be a topological subgroup of a Hausdorff group \( H \) . Let \( \mathcal{F} \) be a Cauchy filter on \( H \) such that the restriction \( \mathcal{F}{ \upharpoonright }_{G} \mathrel{\text{:=}} \{ G \cap U : U \in \mathcal{F}\} \) is a filter base on \( G \) . Then \( \mathcal{F}{ \upharpoonrigh... | Proof. To verify that \( \mathcal{F}{ \upharpoonright }_{G} \) is Cauchy, let \( V \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) ; so \( V = U \cap G \) for some \( U \in {\mathcal{V}}_{H}\left( {e}_{H}\right) \) . Let \( B \in \mathcal{F} \) such that \( B{B}^{-1} \cup {B}^{-1}B \subseteq U \) . Then \( C = B \cap G \... | Yes |
Lemma 7.2.8. Let \( G \) be a topological group. If \( \mathcal{F},\mathcal{H} \) are minimal Cauchy filters on \( G \), then also \( \mathcal{F} \cdot \mathcal{H} \) and \( {\mathcal{F}}^{-1} \) are minimal Cauchy filters on \( G \) . | Proof. We already noticed that \( \mathcal{F} \cdot \mathcal{H} \) and \( {\mathcal{F}}^{-1} \) are Cauchy filters in Lemma 7.2.2. The proof that they are minimal is straightforward. | No |
Lemma 7.2.9. If \( G \) is a topological subgroup of a Hausdorff group \( H \) and \( h \in {\bar{G}}^{H} \), then \( \mathcal{F} = {\mathcal{V}}_{H}\left( h\right) { \upharpoonright }_{G} \) is a minimal Cauchy filter on \( G \) . Consequently, a Hausdorff group \( G \) is complete if and only if every minimal Cauchy ... | Proof. The first assertion follows from Lemma 7.2.7 applied to \( \mathcal{F} = {\mathcal{V}}_{H}\left( h\right) \) . To prove the second assertion, in view of Proposition 7.2.3 we only need to check that if every minimal Cauchy filter on \( G \) converges then \( G \) is complete. To this end, argue by contradiction a... | Yes |
Proposition 7.2.11. Let \( G \) be a Hausdorff linearly topologized group and let \( \left\{ {{N}_{i} : i \in I}\right\} \) be a base of \( {\mathcal{V}}_{G}\left( {e}_{G}\right) \) consisting of open normal subgroups of \( G \) . Then the completion \( \widetilde{G} \) of \( G \) is isomorphic to the inverse limit \( ... | Proof. Since \( \mathop{\bigcap }\limits_{{i \in I}}{N}_{i} \) is trivial, there is a natural embedding of \( G \) in the product \( P = \mathop{\prod }\limits_{{i \in I}}G/{N}_{i} \) of the discrete quotients \( G/{N}_{i} \) . Clearly, \( P \) is complete, by Proposition 7.1.9. Hence, the closure \( {\bar{G}}^{P} \) i... | Yes |
Lemma 7.2.13. Closed subgroups and continuous homomorphic images (provided they are Hausdorff and linearly topologized) of a linearly compact abelian group \( G \) are linearly compact. | Proof. Obviously, closed subgroups of linearly compact groups are linearly compact.\n\nAssume that \( H \) is a Hausdorff linearly topologized abelian group and that \( f : G \rightarrow \) \( H \) is a continuous surjective homomorphism. If \( \mathcal{F} \) is a filter base of closed cosets of subgroups of \( H \), t... | Yes |
Theorem 7.2.14. A linearly compact abelian group G is complete. | Proof. To see that \( G \) is complete, it suffices to check that every minimal Cauchy filter \( \mathcal{F} \) on \( G \) converges. From the definition of minimal Cauchy filter and the fact that \( G \) is linearly topologized, we deduce that \( \mathcal{F} \) has a base consisting of cosets of open subgroups. Now th... | Yes |
Every power \( {\mathbb{T}}^{I} \) of \( \mathbb{T} \), as well as every closed subgroup of \( {\mathbb{T}}^{I} \), is compact. | It becomes clear in the sequel that this is the most general instance of a compact abelian group: every compact abelian group is topologically isomorphic to a closed subgroup of a power of \( \mathbb{T} \) (see Corollary 11.5.2). | No |
For every abelian group \( G \), the group \( {G}^{ * } = \operatorname{Hom}\left( {G,\mathbb{T}}\right) \) of all characters of \( G \) is closed in the product \( {\mathbb{T}}^{G} \). | In fact, considering the projections \( {\pi }_{x} : {\mathbb{T}}^{G} \rightarrow \mathbb{T} \) for every \( x \in G \), \n\n\[ \n{G}^{ * } = \mathop{\bigcap }\limits_{{h, g \in G}}\left\{ {f \in {\mathbb{T}}^{G} : f\left( {h + g}\right) = f\left( h\right) + f\left( g\right) }\right\} \n\] \n\n\[ \n= \mathop{\bigcap }\... | Yes |
Lemma 8.1.3. Let \( G \) be an abelian group and \( N = {\left\{ {\chi }_{\alpha }\right\} }_{\alpha \in A} \) a net in \( {G}^{ * } \) . Then there exist \( \chi \in {G}^{ * } \) and a subnet \( S = {\left\{ {\chi }_{{\alpha }_{\beta }}\right\} }_{\beta \in B} \) of \( N \) such that \( {\chi }_{{\alpha }_{\beta }}\le... | Proof. By Example 8.1.1, the group \( {\mathbb{T}}^{G} \) endowed with the product topology is compact. Since \( {G}^{ * } \) is a topological subgroup of \( {\mathbb{T}}^{G} \), there exist \( \chi \in {\mathbb{T}}^{G} \) and a subnet \( S = {\left\{ {\chi }_{{\alpha }_{\beta }}\right\} }_{\beta \in B} \) of \( N \) t... | Yes |
For \( n \in {\mathbb{N}}_{ + } \) the set \( U\left( n\right) \) of all \( n \times n \) unitary matrices over \( \mathbb{C} \) (a matrix is unitary if its inverse coincides with its conjugate transposed) is a subgroup of \( {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \). | As a subset of \( {\mathbb{C}}^{{n}^{2}}, U\left( n\right) \) is closed and bounded. So, \( U\left( n\right) \) is compact, by Example B.5.6. It is easy to see that \( U\left( 1\right) \cong \mathbb{S} \). | No |
The Hilbert space \( \left( {{\ell }_{2},\parallel - \parallel }\right) \) of square summable real sequences is not locally compact. | Indeed, the closed unit disk is not compact: since \( {\ell }_{2} \) is metrizable, it is enough to observe that the sequence \( {\left\{ {e}_{n}\right\} }_{n \in \mathbb{N}} \) of the vectors of the canonical base has no Cauchy subsequences (so no convergent subsequences), as \( \begin{Vmatrix}{{e}_{n} - {e}_{m}}\end{... | Yes |
Lemma 8.2.1. Let \( G \) be a topological group and \( C, K \) closed sets of \( G \). (a) If \( K \) is compact, then both \( {CK} \) and \( {KC} \) are closed. | Proof. (a) Let \( {\left\{ {x}_{\alpha }\right\} }_{\alpha \in A} \) be a net in \( {CK} \) such that \( {x}_{\alpha } \rightarrow {x}_{0} \in G \) . One has to show that \( {x}_{0} \in {CK} \) . For every \( \alpha \in A \), there exist \( {y}_{\alpha } \in C \) and \( {z}_{\alpha } \in K \) such that \( {x}_{\alpha }... | Yes |
Lemma 8.2.2. Let \( G \) be a topological group and \( K \) a compact normal subgroup of \( G \) . Then the canonical projection \( q : G \rightarrow G/K \) is closed. | Proof. Let \( C \) be a closed set of \( G \) . As \( {q}^{-1}\left( {q\left( C\right) }\right) = {CK} \) is closed by Lemma 8.2.1(a), we may conclude that \( q\left( C\right) \) is closed. | Yes |
Lemma 8.2.3. Let \( H \) be a closed normal subgroup of a topological group \( G \). (a) If \( G \) is compact, then \( G/H \) is compact. | Proof. (a) is obvious, since the canonical projection \( q : G \rightarrow G/H \) is continuous. | No |
Lemma 8.2.5. Let \( G \) be a locally compact group, \( H \) a closed normal subgroup of \( G \), and \( q : G \rightarrow G/H \) the canonical projection. Then:\n\n(a) \( G/H \) is locally compact, too;\n\n(b) if \( C \subseteq G/H \) is compact, there exists \( K \subseteq G \) compact with \( q\left( K\right) = C \)... | Proof. Let \( U \) be an open neighborhood of \( {e}_{G} \) in \( G \) with compact closure.\n\n(a) Consider the open neighborhood \( q\left( U\right) \) of \( {e}_{G/H} \) in \( G/H \) . By the continuity of \( q \) , \( q\left( \overline{U}\right) \subseteq \overline{q\left( U\right) } \) and \( q\left( \overline{U}\... | Yes |
Proposition 8.2.6. A locally compact group \( G \) is Weil complete. | Proof. Let \( U \) be a neighborhood of \( {e}_{G} \) in \( G \) with compact closure and let \( {\left\{ {g}_{\alpha }\right\} }_{\alpha \in A} \) be a left Cauchy net of \( G \) . Then there exists \( {\alpha }_{0} \in A \) such that \( {g}_{\alpha }^{-1}{g}_{\beta } \in U \) for every \( \alpha ,\beta \geq {\alpha }... | Yes |
Proposition 8.2.7. The character and the pseudocharacter of a locally compact group G coincide. | Proof. Clearly, \( \psi \left( G\right) \leq \chi \left( G\right) \) . Let \( U \) be an open neighborhood of \( {e}_{G} \) such that \( \bar{U} \) is compact. To prove that \( \chi \left( G\right) \leq \) \( \psi \left( G\right) \), pick a family \( \mathcal{B} = \left\{ {{V}_{i} : i \in I}\right\} \) of neighborhoods... | Yes |
Any infinite compact group \( G \) satisfies \( \psi \left( G\right) = \chi \left( G\right) = w\left( G\right) \) . | Proof. We are going to prove that \( d\left( G\right) \leq \chi \left( G\right) \) . Let \( \mathcal{B} \) be a local base at \( {e}_{G} \) of cardinality \( \chi \left( G\right) \) consisting of symmetric sets. For every \( U \in \mathcal{B} \), choose a finite subset \( {F}_{U} \) of \( G \) such that \( G = {F}_{U}U... | Yes |
Lemma 8.3.2. If \( G \) is a compactly generated group, then \( G \) is \( \sigma \) -compact. | Proof. There exists a compact subset \( K \) of \( G \) such that \( G = \mathop{\bigcup }\limits_{{n \in {\mathbb{N}}_{ + }}}{\left( K \cup {K}^{-1}\right) }^{n} \) . Since \( K \) is compact, \( {\left( K \cup {K}^{-1}\right) }^{n} \) is compact for every \( n \in {\mathbb{N}}_{ + } \) . | Yes |
Corollary 8.3.3. A locally compact group is a normal space. | Proof. By hypothesis and Lemma 8.3.2, \( G \) contains a \( \sigma \) -compact open subgroup \( N \) . Then \( N \) is Lindelöff by Lemma B.5.18, so a normal space by Theorem B.5.10(b). Now \( G = \mathop{\bigsqcup }\limits_{{g \in G}}{gN} \) is a normal space as well. | Yes |
Lemma 8.3.4. Let \( G \) be a locally compact group.\n\n(a) If \( K \) is a compact subset of \( G \) and \( U \) is an open set of \( G \) such that \( K \subseteq U \), then there \( \\textit{exists an open neighborhood}\\;V\\;{\\textit{ofe}}_{G}\\;\\textit{in}\\;G\\;\\textit{such that}\\;\\left( {KV}\\right) \\cup \... | Proof. (a) By Lemma 8.2.1(c), there exists an open neighborhood \( V \) of \( {e}_{G} \) in \( G \) such that \( \\left( {KV}\\right) \\cup \\left( {VK}\\right) \\subseteq U \) . Since \( G \) is locally compact, one can choose \( V \) with compact closure. Thus, \( K\\bar{V} \) is compact by Lemma 8.2.1(b). Since \( {... | Yes |
Theorem 8.4.1 (Open mapping theorem). Let \( G, H \) be locally compact groups and \( f : G \rightarrow H \) a continuous homomorphism. If \( G \) is \( \sigma \) -compact and \( f \) is surjective, then \( f \) is open. | Proof. Let \( U \) be a neighborhood of \( {e}_{G} \) in \( G \) . There exists a symmetric open neighborhood \( V \) of \( {e}_{G} \) in \( G \) such that \( \bar{V}\bar{V} \subseteq U \) and \( \bar{V} \) is compact. Since \( G = \mathop{\bigcup }\limits_{{x \in G}}{xV} \) and \( G \) is Lindelöff by Lemma B.5.18, th... | Yes |
The question when an infinite abelian group \( G \) may carry a minimal group topology has been studied thoroughly. | It is known that none of the groups \( {\mathbb{Q}}^{n},\mathbb{Z}\left( {p}^{\infty }\right) ,\mathbb{Z}{\left( {p}^{\infty }\right) }^{n},\mathbb{Z}\left( {p}_{1}^{\infty }\right) \oplus \cdots \oplus \mathbb{Z}\left( {p}_{n}^{\infty }\right) \), where \( n \in {\mathbb{N}}_{ + } \) and \( p,{p}_{1},\ldots ,{p}_{n} \... | Yes |
Theorem 8.5.2. Let \( G \) be a locally compact group. Then:\n\n(a) if \( G \) is hereditarily disconnected, every neighborhood of \( {e}_{G} \) contains a compact open subgroup of \( G \) ;\n\n(b) \( c\left( G\right) \) coincides with the intersection of all open subgroups of \( G \) .\n\nIf \( G \) is compact, then t... | Proof. (a) By Vedenissov theorem B.6.10, there is a neighborhood base \( \mathcal{O} \) at \( {e}_{G} \) consisting of compact symmetric clopen sets. Let \( U \in \mathcal{O} \) . Then, by Lemma 3.1.1(a),\n\n\[ U = \bar{U} = \bigcap \{ {UV} : V \in \mathcal{O}, V \subseteq U\} \]\n\nwhere every set \( {UV} \) is compac... | Yes |
Let \( p \) be a prime and \( G = {\mathbb{Q}}_{p} \rtimes \mathbb{Z} \) such that \( \mathbb{Z} \cong \left\{ {{p}^{n} : n \in \mathbb{Z}}\right\} \) acts on \( {\mathbb{Q}}_{p} \) by multiplication by \( p \), and the subgroup \( O = {\mathbb{Q}}_{p} \rtimes \{ 1\} \) of \( G \) is taken to be open carrying its natur... | Nevertheless, the only compact open normal subgroups of \( G \) are those containing \( O = {\mathbb{Q}}_{p} \rtimes \{ 1\} \) . Indeed, if \( V \) is a compact open subgroup of \( G \), there exists \( n \in \mathbb{N} \) such that \( {U}_{n} \subseteq V \) . Since the normal closure of \( {U}_{n} \) (i. e., the small... | Yes |
Corollary 8.5.4. Let \( G \) be a locally compact group. Then \( o\left( G\right) = Q\left( G\right) = c\left( G\right) \) . So, \( G \) is hereditarily disconnected if and only if it is totally disconnected. | Proof. It is always true that \( c\left( G\right) \subseteq Q\left( G\right) \subseteq o\left( G\right) \) . By Theorem 8.5.2(b), \( c\left( G\right) \) is the intersection of open subgroups, so \( c\left( G\right) \supseteq o\left( G\right) \). | Yes |
Corollary 8.5.9. A quotient of a hereditarily disconnected locally compact group \( G \) is hereditarily disconnected. | Proof. Let \( N \) be a closed normal subgroup of \( G \) . It follows from Theorem 8.5.2(a) that \( G \) has a local base at \( {e}_{G} \) formed by compact open subgroups. This yields that the quotient \( G/N \) has the same property. In particular, \( G/N \) is hereditarily disconnected, too. | Yes |
Corollary 8.5.10. Let \( G, H \) be locally compact groups and \( f : G \rightarrow H \) a continuous surjective homomorphism. If \( G \) is \( \sigma \) -compact, then:\n\n(a) \( f\left( {c\left( G\right) }\right) = c\left( H\right) \), provided \( c\left( G\right) \) is compact;\n\n(b) if \( G \) is hereditary discon... | Proof. (a) Since \( c\left( G\right) \) is a compact normal subgroup of \( G, f\left( {c\left( G\right) }\right) \) is a compact (so, closed) normal subgroup of \( H \), as \( f \) is surjective. The group \( G/c\left( G\right) \) is hereditarily disconnected by Corollary 8.5.9 and \( f \) induces a continuous surjecti... | Yes |
For every topological group \( G \), there exist a compact group \( {bG} \) and \( a \) continuous homomorphism \( {b}_{G} : G \rightarrow {bG} \) with \( \overline{{b}_{G}\left( G\right) } = {bG} \) such that for every continuous homomorphism \( f : G \rightarrow K \), where \( K \) is a compact group, there exists a ... | Proof. Let \( \left\{ {{N}_{j} : j \in J}\right\} \) be the family of all kernels of continuous homomorphisms \( G \rightarrow C \) where \( C \) is a compact group. For every \( j \in J \), let \( {q}_{j} : G \rightarrow G/{N}_{j} \) be the canonical projection and let \( {\mathcal{F}}_{j} = \left\{ {{\tau }_{\left( j... | Yes |
Lemma 9.1.1. Let \( n \in {\mathbb{N}}_{ + }, H \) a topological abelian group, and \( \varepsilon > 0 \) . Then every map \( f : {B}_{\varepsilon }\left( 0\right) \rightarrow H \) such that \( f\left( {x + y}\right) = f\left( x\right) + f\left( y\right) \) whenever \( x, y \in {B}_{\varepsilon /2}\left( 0\right) \) ca... | Proof. Put \( U = {B}_{\varepsilon /2}\left( 0\right) \) . For \( x \in {\mathbb{R}}^{n} \), there exists \( m \in {\mathbb{N}}_{ + } \) such that \( \frac{1}{m}x \in U \), and we put\n\n\[ \n{f}^{\prime }\left( x\right) = {mf}\left( {\frac{1}{m}x}\right) \n\]\n\nTo see that this definition is correct, assume that \( \... | No |
Lemma 9.2.4. If \( H \) is a closed subgroup of \( {\mathbb{R}}^{n}, L \cong \mathbb{R} \) is a one-dimensional subspace of \( {\mathbb{R}}^{n} \), and \( H \cap L \neq \{ 0\} \), then denoting by \( p : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}/L \) the canonical projection, \( p\left( H\right) \) is a closed subg... | Proof. If \( n = 1 \), then \( {\mathbb{R}}^{n}/L \) is trivial, so we are done.\n\nAssume that \( n > 1 \) and consider the nonzero closed subgroup \( {H}_{1} = H \cap L \) of \( L \cong \mathbb{R} \) . If \( {H}_{1} = L \) (i. e., \( L \subseteq H \) ), then the assertion follows from Theorem 3.2.8(b). Now assume tha... | Yes |
Corollary 9.2.6. For every \( n \in {\mathbb{N}}_{ + } \), the only compact subgroup of \( {\mathbb{R}}^{n} \) is \( \{ 0\} \) . | Proof. Let \( K \) be a compact subgroup of \( {\mathbb{R}}^{n} \) . By Theorem 9.2.2, \( K = V \times D \), where, for some \( s, m \in \mathbb{N}, V \cong {\mathbb{R}}^{s} \) is a linear subspace of \( {\mathbb{R}}^{n} \) and \( D \cong {\mathbb{Z}}^{m} \) is a discrete subgroup of \( {\mathbb{R}}^{n} \) . The compac... | Yes |
Lemma 9.3.1. Let \( H \) be a discrete subgroup of \( {\mathbb{R}}^{n} \). If the elements \( {v}_{1},\ldots ,{v}_{m} \) of \( H \) are independent, then they are also \( \mathbb{R} \)-linearly independent. | Proof. Let \( D = \left\langle {{v}_{1},\ldots ,{v}_{m}}\right\rangle \cong {\mathbb{Z}}^{m} \), and let \( V \cong {\mathbb{R}}^{k} \) be the linear subspace of \( {\mathbb{R}}^{n} \) generated by \( H \). We need to prove that \( k \geq m \). We can assume without loss of generality that \( V = {\mathbb{R}}^{n} \) (i... | Yes |
Proposition 9.3.2. For a discrete subgroup \( H \) of \( {\mathbb{R}}^{n}, H \) is free and \( {r}_{0}\left( H\right) \leq n \) . | Proof. Since by Lemma 9.3.1 there are at most \( n \) independent vectors in \( H \), we have \( m : = {r}_{0}\left( H\right) \leq n \) . Then there exist \( m \) independent vectors \( {v}_{1},\ldots ,{v}_{m} \) of \( H \) . By Lemma 9.3.1, the vectors \( {v}_{1},\ldots ,{v}_{m} \) are also \( \mathbb{R} \) -linearly ... | Yes |
Lemma 9.3.3. Let \( H \) be a discrete subgroup of \( {\mathbb{R}}^{n} \) and \( L \cong \mathbb{R} \) a one-dimensional linear subspace of \( {\mathbb{R}}^{n} \) with \( H \cap L \neq \{ 0\} \) . Then, denoting by \( p : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}/L \) the canonical projection, \( p\left( H\right) \... | Proof. If \( n = 1 \), then \( L = \mathbb{R} \), so this case is trivial. Assume \( n > 1 \) in the sequel. Since \( \{ 0\} \neq {H}_{1} = H \cap L \) is a discrete subgroup of \( L \cong \mathbb{R} \), we conclude that \( {H}_{1} = \langle a\rangle \) is cyclic, by Proposition 3.1.11. Making use of an appropriate lin... | Yes |
Corollary 9.4.1. A quotient of \( {\mathbb{R}}^{n} \) is isomorphic to \( {\mathbb{R}}^{k} \times {\mathbb{T}}^{m} \), where \( k + m \leq n \) . In particular, a compact quotient of \( {\mathbb{R}}^{n} \) is isomorphic to \( {\mathbb{T}}^{m} \) for some \( m \leq n \) . | Proof. Let \( H \) be a closed subgroup of \( {\mathbb{R}}^{n} \) . By Theorem 9.2.2, \( H = V \times D \), with \( V \cong {\mathbb{R}}^{s} \) , \( D \cong {\mathbb{Z}}^{m} \) discrete and \( s + m \leq n \) . Let \( {V}_{1} \) be the linear subspace of \( {\mathbb{R}}^{n} \) spanned by \( D \) . Pick a complementing ... | Yes |
Corollary 9.4.3. Every closed subgroup \( C \) of \( {\mathbb{R}}^{n} \times {\mathbb{T}}^{k} \) is elementary locally compact. Consequently, every closed subgroup of \( {\mathbb{T}}^{S} \) is elementary compact. | Proof. Let \( q : {\mathbb{R}}^{n + k} \rightarrow {\mathbb{R}}^{n} \times {\mathbb{T}}^{k} \) be the canonical projection. Then \( H = {q}^{-1}\left( C\right) \) is a closed subgroup of \( {\mathbb{R}}^{n + k} \) . By Theorem 9.2.2, \( H \) is a direct product \( H = V \times D \) with \( V \cong {\mathbb{R}}^{s} \) a... | Yes |
Lemma 9.4.4. Let \( H,{H}_{1} \) be subgroups of \( {\mathbb{R}}^{n} \). Then:\n\n(a) \( {H}^{ \dagger } \) is closed subgroup of \( {\mathbb{R}}^{n} \) and the correspondence \( H \mapsto {H}^{ \dagger } \) is monotone decreasing;\n\n(b) \( {\left( \bar{H}\right) }^{ \dagger } = {H}^{ \dagger } \); | Proof. The map \( {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) defined by \( \left( {x, y}\right) \mapsto \left( {x \mid y}\right) \) is continuous.\n\n(a) Let \( {q}_{0} : \mathbb{R} \rightarrow \mathbb{T} = \mathbb{R}/\mathbb{Z} \) be the canonical projection. For every \( a \in {\mathbb{R}}^{n}... | Yes |
Proposition 9.4.5. For every subgroup \( H \) of \( {\mathbb{R}}^{n},\bar{H} = {\left( {H}^{ \dagger }\right) }^{ \dagger } \) . In particular, \( H \) is dense in \( {\mathbb{R}}^{n} \) if and only if \( {H}^{ \dagger } = \{ 0\} \) . | Proof. By Lemma 9.4.4(b), \( {\left( \bar{H}\right) }^{ \dagger } = {H}^{ \dagger } \), so we can assume without loss of generality that \( H = \bar{H} \) is closed. According to Theorem 9.2.2, there exist a base \( \left( {{v}_{1},\ldots ,{v}_{n}}\right) \) of \( {\mathbb{R}}^{n} \) and \( k \leq n \) such that \( H =... | Yes |
Proposition 9.4.6. Let \( {v}_{1},\ldots ,{v}_{n} \in \mathbb{R} \) . Then, for \( v = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \in {\mathbb{R}}^{n} \), the subgroup \( \langle v\rangle + {\mathbb{Z}}^{n} \) of \( {\mathbb{R}}^{n} \) is dense if and only if \( {v}_{0} = 1,{v}_{1},\ldots ,{v}_{n} \in \mathbb{R} \) are li... | Proof. Assume that \( {v}_{0} = 1,{v}_{1},\ldots ,{v}_{n} \in \mathbb{R} \) are linearly independent and let \( H = \langle v\rangle + {\mathbb{Z}}^{n} \) . Then \( {H}^{ \dagger } \subseteq {\mathbb{Z}}^{n} = {\left( {\mathbb{Z}}^{n}\right) }^{ \dagger } \) . Therefore, some \( z \in {\mathbb{Z}}^{n} \) belongs to \( ... | Yes |
Corollary 9.4.7. Let \( {q}_{0} : \mathbb{R} \rightarrow \mathbb{T} \) be the canonical projection. For \( n \in {\mathbb{N}}_{ + } \) and \( {v}_{1},\ldots ,{v}_{n} \in \) \( \mathbb{R} \) such that \( 1,{v}_{1},\ldots ,{v}_{n} \in \mathbb{R} \) are \( \mathbb{Q} \) -linearly independent in \( \mathbb{R},\left\langle ... | Proof. By Proposition 9.4.6, with \( v = \left( {{v}_{1},\ldots ,{v}_{n}}\right) \in {\mathbb{R}}^{n} \), the subgroup \( H = \langle v\rangle + {\mathbb{Z}}^{n} \) of \( {\mathbb{R}}^{n} \) is dense. Consider the canonical projection \( \pi : {\mathbb{R}}^{n} \rightarrow {\mathbb{T}}^{n} \cong {\mathbb{R}}^{n}/{\mathb... | Yes |
Theorem 9.4.8. The group \( {\mathbb{T}}^{\mathfrak{c}} \) is monothetic. | Proof. Let \( B \) be a Hamel base of \( \mathbb{R} \) on \( \mathbb{Q} \) that contains 1 and let \( {B}_{0} = B \smallsetminus \{ 1\} \) ; in particular, \( \left| {B}_{0}\right| = \left| B\right| = \mathfrak{c} \) . To see that the element \( x = {\left( {x}_{b}\right) }_{b \in {B}_{0}} \in {\mathbb{T}}^{{B}_{0}} \)... | Yes |
Let \( G \) be the free product of a cyclic group \( A = \langle a\rangle \) of order 2 and an infinite cyclic group \( C = \langle b\rangle \) . Every element of \( G \smallsetminus C \) can be uniquely written as a product\n\n\[ w = {b}^{{n}_{0}} \cdot a \cdot {b}^{{n}_{1}} \cdot a \cdot {b}^{{n}_{2}}\cdots a \cdot {... | Set\n\n\[ Y = \left\{ {w \in G:k \in {\mathbb{N}}_{ + }\text{ and }{n}_{k} = 0\text{ in (10.1)}}\right\} \;\text{and}\;X = G \smallsetminus Y.\]\n\nNote that \( Y = {Xa} \), so that \( X = {Ya} \), too. Thus, both \( X \) and \( Y \) are right big, since \( G = \) \( X \cup {Xa} = Y \cup {Ya} \) . Let us see now that n... | Yes |
Lemma 10.1.4. (a) Assume that \( {B}_{j} \) is a left big subset of the group \( {G}_{j} \), for \( j \in \{ 1,\ldots, n\} \) . Then \( {B}_{1} \times \cdots \times {B}_{n} \) is a left big subset of \( {G}_{1} \times \cdots \times {G}_{n} \) . | Proof. (a) and \( \left( {\mathrm{b}}_{2}\right) \) follow directly from the definition. | No |
Proposition 10.1.5. Let \( G \) be a group and \( B \) a left big subset of \( G \). Then:\n\n(a) for every subgroup \( H \) of \( G,{B}^{-1}B \cap H \) is a big subset of \( H \);\n\n(b) for every \( a \in G \), there exists \( n \in {\mathbb{N}}_{ + } \) such that \( {a}^{n} \in {B}^{-1}B \). | Proof. (a) Let \( F \) be a finite subset of \( G \) such that \( {FB} = G \). For \( f \in F \), if \( {fB} \cap H \neq \varnothing \), choose \( {a}_{f} \in {fB} \cap H \) and let \( E = \left\{ {{a}_{f} : f \in F,{fB} \cap H \neq \varnothing }\right\} \). For every \( h \in H \), there exists \( f \in F \) such that... | Yes |
Lemma 10.1.8. Let \( G \) be an infinite group and \( S \) a subset of \( G \). (a) If \( S \) is finite, then \( S \) is left and right small. | Proof. (a) obviously follows from (e). | No |
Lemma 10.2.3. If \( f : G \rightarrow H \) is a continuous surjective homomorphism of topological groups, then \( H \) is totally bounded whenever \( G \) is totally bounded. If \( G \) carries the initial topology off and \( H \) is totally bounded, then also \( G \) is totally bounded. In particular, \( G \) is total... | Proof. To prove the first assertion, it suffices to recall that the homomorphic image of a left big subset under a surjective homomorphism is left big, by Lemma 10.1.4(b,). The second assertion follows from the fact that the open sets of \( G \) are preimages of the open sets of \( H \) . So, Lemma 10.1.4( \( {\mathrm{... | Yes |
Proposition 10.2.4. If \( \left\{ {{G}_{i} : i \in I}\right\} \) is a family of topological groups, then \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) is totally bounded if and only if each \( {G}_{i} \) is totally bounded. | Proof. If \( G \) is totally bounded, then each \( {G}_{i} \) is totally bounded by Lemma 10.2.3.\n\nAssume that each \( {G}_{i} \) is totally bounded and let \( U \) be a nonempty open set of \( G \) . Then there exist a finite subset \( J \) of \( I \) and a nonempty open set \( V \) of \( {G}_{J} = \mathop{\prod }\l... | Yes |
Proposition 10.2.5. All subgroups of totally bounded groups are totally bounded. In particular, all subgroups of compact groups are precompact. | Proof. Let \( H \) be a subgroup of \( G \) . If \( U \in {\mathcal{V}}_{H}\left( {e}_{G}\right) \), there exists \( W \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) such that \( U = W \cap H \) . Pick \( V \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) such that \( {V}^{-1}V \subseteq W \) . Since \( V \) is left big in... | Yes |
Theorem 10.2.6. (a) A Hausdorff group \( G \) having a dense precompact subgroup \( H \) is necessarily precompact. | Proof. (a) For every \( U \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \), choose an open \( V \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) with \( {VV} \subseteq U \) . By the precompactness of \( H \), there exists a finite subset \( F \) of \( H \) such that \( H = F\left( {V \cap H}\right) \) . Then \( G = {HV} = F\... | Yes |
Lemma 10.2.7. For a topological group \( G \), the following are equivalent:\n\n(a) \( G \) is not totally bounded;\n\n(b) G has a left small nonempty open set;\n\n(c) G has a right small nonempty open set. | Proof. (b) \( \Leftrightarrow \) (c) since a subset \( S \) of \( G \) is left small if and only if \( {S}^{-1} \) is right small, while \( \left( \mathrm{b}\right) \Rightarrow \left( \mathrm{a}\right) \) is a consequence of Lemma 10.1.8(c).\n\n\( \left( a\right) \Rightarrow \left( b\right) \) If \( U \in {\mathcal{V}}... | Yes |
Lemma 10.2.8. If \( G \) is a totally bounded group, then for every \( U \in \mathcal{V}\left( {e}_{G}\right) \), there exists \( V \in \mathcal{V}\left( {e}_{G}\right) \) such that \( {g}^{-1}{Vg} \subseteq U \) for all \( g \in G \) . | Proof. Let \( W \in \mathcal{V}\left( {e}_{G}\right) \) be symmetric and such that \( {WWW} \subseteq U \) . By hypothesis, \( G = {FW} \) for some finite subset \( F \) of \( G \) . For every \( a \in F \), pick \( {V}_{a} \in \mathcal{V}\left( {e}_{G}\right) \) such that \( {a}^{-1}{V}_{a}a \subseteq W \) , and let \... | Yes |
Corollary 10.2.11. Let \( G \) be a locally compact group such that \( {M}_{G} \) is dense in \( G \) . Then \( G \) is compact, connected and \( w\left( G\right) \leq \mathfrak{c} \) . | Proof. The compactness of \( G \) and \( w\left( G\right) \leq \mathfrak{c} \) follow, respectively, from Theorem 10.2.9 and Exercise 5.4.4 (the weaker hypothesis \( {M}_{G} \neq \varnothing \) is sufficient). Now assume that \( G \) is not connected. Then \( G \) has a nontrivial proper open subgroup \( U \), by Theor... | Yes |
Proposition 10.2.13. For every topological group \( \left( {G,\tau }\right) \), there exists the finest totally bounded group topology \( {\tau }^{ + } \) on \( G \) coarser than \( \tau \) . | Proof. Let \( \left\{ {{\tau }_{i} : i \in I}\right\} \) be the family of all totally bounded group topologies on \( G \) coarser than \( \tau \), and let \( {\tau }^{ + } = \sup \left\{ {{\tau }_{i} : i \in I}\right\} \) . Then \( \left( {G,{\tau }^{ + }}\right) \) is topologically isomorphic to the diagonal subgroup ... | Yes |
Proposition 10.2.14. For every topological group \( \left( {G,\tau }\right) \), the quotient group\n\n\[ \n{G}^{ + } \mathrel{\text{:=}} G/{\overline{\left\{ {e}_{G}\right\} }}^{\tau + } = \mathfrak{h}\left( {G,{\tau }^{ + }}\right) \n\]\n\nequipped with the quotient topology of \( {\tau }^{ + } \) is precompact, and e... | Proof. The precompactness of the quotient \( {G}^{ + } \) with the quotient topology of \( {\tau }^{ + } \) follows from Lemma 10.2.3. Let \( {\tau }_{1} \) be the initial topology of \( G \) with respect to \( f : G \rightarrow P \) . According to Proposition 10.2.5, we may assume that \( f \) is surjective. Then \( {... | Yes |
Theorem 10.2.15. For every topological group \( G \), there exists a topological isomorphism \( i : {bG} \rightarrow \widetilde{{G}^{ + }} \) such that \( i \circ {b}_{G} = {j}_{G} \) . | Proof. In view of Theorem 8.6.1, it suffices to prove that \( {j}_{G} : G \rightarrow \widetilde{{G}^{ + }} \) has the universal property of \( {b}_{G} : G \rightarrow {bG} \) . Let \( f : G \rightarrow K \) be a continuous homomorphism, where \( K \) is a compact group. By Proposition 10.2.14, \( f \) factorizes throu... | Yes |
Proposition 10.2.16. If \( G \) is an abelian group, \( \delta > 0 \), and \( {\chi }_{1},\ldots ,{\chi }_{s} \in {G}^{ * } \) with \( s \in {\mathbb{N}}_{ + } \) , then \( U\left( {{\chi }_{1},\ldots ,{\chi }_{s};\delta }\right) \) is a big subset of \( G \) . Moreover, for every \( a \in G \), there exists \( n \in {... | Proof. Let \( h : G \rightarrow {\mathbb{S}}^{s}, x \mapsto \left( {{\chi }_{1}\left( x\right) ,\ldots ,{\chi }_{s}\left( x\right) }\right) \), and let\n\n\[ B = \left\{ {\left( {{z}_{1},\ldots ,{z}_{s}}\right) \in {\mathbb{S}}^{s} : \left| {\operatorname{Arg}\left( {z}_{i}\right) }\right| < \frac{\delta }{2}\text{ for... | Yes |
Theorem 10.3.2 (Gel’fand-Raïkov theorem). For every locally compact group \( G \) and \( a \in G \smallsetminus \left\{ {e}_{G}\right\} \), there exists an irreducible unitary representation \( V : G \rightarrow U\left( \mathcal{H}\right) \) of \( G \) by unitary operators of some Hilbert space \( \mathcal{H} \) such t... | The proof of this theorem can be found in [174, 22.12]. | No |
Theorem 10.3.3 (Peter–Weyl–van Kampen theorem). Let \( G \) be a compact group. For every \( a \in G \smallsetminus \left\{ {e}_{G}\right\} \), there exist \( n \in {\mathbb{N}}_{ + } \) and a continuous homomorphism \( f : G \rightarrow U\left( n\right) \) such that \( f\left( a\right) \neq {e}_{G} \) . | The proof of the Peter–Weyl–van Kampen theorem can be found in [174, 22.13]. | No |
Corollary 10.3.4. If \( G \) is a compact group, then \( G \) is isomorphic to a (closed) subgroup of some power \( {\mathbb{U}}^{I} \) of the group \( \mathbb{U} \) . | Proof. In view of Theorem 10.3.3, there exist a nonempty set \( I \) and a family \( \left\{ {{f}_{i} : G \rightarrow }\right. \) \( \mathbb{U} : i \in I\} \) of continuous homomorphisms that separate the points of \( G \) . Then the diagonal map determined by \( \left\{ {{f}_{i} : G \rightarrow \mathbb{U} : i \in I}\r... | Yes |
Proposition 10.3.5. The Bohr compactification of a topological group \( G \) can be equivalently defined by the property \( \left( \mathrm{{Bc}}\right) \) . | Proof. First, we verify that if a continuous homomorphism \( h : G \rightarrow K \), where \( K \) is a compact group, has the property (Bc) (i. e., every continuous homomorphism \( f : G \rightarrow U\left( n\right) \) for some \( n \in \mathbb{N} \) factorizes through \( h \) by means of a unique continuous homomorph... | Yes |
Theorem 10.5.1 ([239]). Minimal abelian groups are precompact. | A simplified proof of this theorem can be found in [99, Theorem 2.7.7]. Combining it with Exercise 8.7.6, one can deduce that minimal abelian groups are precisely the dense essential subgroups of compact abelian groups. A significant generalization of Theorem 10.5.1 was recently obtained by Banakh [15]. | No |
Proposition 11.1.1. Let \( G \) be a finite abelian group and \( \varphi ,\chi \in {G}^{ * }, x, y \in G \) . Then:\n\n(a) \( \left( {\varphi \mid \chi }\right) = \left\{ \begin{array}{ll} 1 & \text{ if }\varphi = \chi \\ 0 & \text{ if }\varphi \neq \chi \end{array}\right. \)\n\n(b) \( \frac{1}{\left| {G}^{ * }\right| ... | Proof. (a) If \( \varphi = \chi \), then \( \chi \left( x\right) \overline{\chi \left( x\right) } = \chi \left( x\right) \chi {\left( x\right) }^{-1} = 1 \) . If \( \varphi \neq \chi \), there exists \( z \in G \) such that \( \varphi \left( z\right) \neq \chi \left( z\right) \) . Therefore, the equalities\n\n\[ \matho... | No |
Proposition 11.1.3. Let \( G \) be a finite abelian group and \( f \in {\mathbb{C}}^{G} \) with Fourier coefficients \( \left\{ {{c}_{\chi } : \chi \in {G}^{ * }}\right\} \). Then:\n\n(a) \( {G}^{ * } \) is an orthonormal base of \( {\mathbb{C}}^{G} \);\n\n(b) \( f\left( x\right) = \mathop{\sum }\limits_{{\chi \in {G}^... | Proof. (a) According to Proposition 11.1.1(a), \( {G}^{ * } \) is an orthonormal set of cardinality \( \left| G\right| \) in \( {\mathbb{C}}^{G} \), hence an orthonormal base.\n\n(b) Since \( {G}^{ * } \) is an orthonormal base of \( {\mathbb{C}}^{G} \) by item (a), we obtain for \( f \in {\mathbb{C}}^{G} \) that \( f ... | Yes |
Corollary 11.1.4. Let \( G \) be a finite abelian group, \( f : G \rightarrow {\mathbb{R}}_{ \geq 0} \) a function, and \( E = \{ x \in \) \( G : f\left( x\right) > 0\} \) . Then, for \( g \mathrel{\text{:=}} f * f \) and \( x \in G \) :\n\n(a) \( g\left( x\right) > 0 \) if and only if \( x \in {E}_{\left( 2\right) } \... | Proof. (a) For \( x \in G \), by definition \( g\left( x\right) > 0 \) if and only if there exists \( y \in E \) with \( x + y \in E \) , that is, \( x \in E - E = {E}_{\left( 2\right) } \) .\n\n(b) follows from Proposition 11.1.3(d). | Yes |
Lemma 11.2.3. Let \( A \) be an abelian group and \( {\left\{ {A}_{n}\right\} }_{n \in {\mathbb{N}}_{ + }} \) a sequence of finite subsets of \( A \) such that, for every \( a \in A \) , \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{\left| \left( {A}_{n} - a\right) \cap {A}_{n}\right| }{\left| {A}_{n}\right| ... | Proof. Let \( {a}_{1},\ldots ,{a}_{k} \in A \) be such that \( A = \mathop{\bigcup }\limits_{{i = 1}}^{k}\left( {{a}_{i} + V}\right) \) . Let \( \varepsilon > 0 \) . By hypothesis, there exists \( N > 0 \) such that, for every \( n \geq N \) and every \( i \in \{ 1,\ldots, k\} \) , \[ \left| {\left( {{A}_{n} - {a}_{i}}... | Yes |
Corollary 11.2.6. For a subset \( E \) of an infinite abelian group \( A \), the following conditions are equivalent:\n\n(a) \( E \) contains \( {V}_{\left( 8\right) } \) for some big subset \( V \) of \( A \) ;\n\n(b) for every \( n \in {\mathbb{N}}_{ + } \) , \( E \) contains \( {V}_{\left( 2n\right) } \) for some bi... | Proof. (a) \( \Rightarrow \) (c) follows from Følner lemma 11.2.5, while (c) \( \Rightarrow \) (b) follows from Corollary 10.2.17 and Proposition 10.2.16, and (b) \( \Rightarrow \) (a) is obvious. | Yes |
For a subgroup \( H \) of an abelian group \( G \), the Bohr topology \( {\mathfrak{B}}_{G/H} \) of \( G/H \) coincides with the quotient topology of the Bohr topology \( {\mathfrak{B}}_{G} \). | Proof. Let \( q : G \rightarrow G/H \) be the canonical projection. The quotient topology \( {\overline{\mathfrak{B}}}_{G} \) of the Bohr topology \( {\mathfrak{B}}_{G} \) is a precompact group topology on \( G/H \) (as \( H \) is closed in \( {G}^{\# } \) by Proposition 3.1.9). Hence, \( {\overline{\mathfrak{B}}}_{G} ... | Yes |
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