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Lemma 2.1.6. Let \( G \) be a topological group. For every \( a \in G \), the left translation \( {}_{a}t : G \rightarrow \) \( G, x \mapsto {ax} \), the right translation \( {t}_{a} : G \rightarrow G, x \mapsto {xa} \), and the inner automorphism \( {\phi }_{a} : G \rightarrow G, x \mapsto {ax}{a}^{-1} \), are homeomo... | Proof. Since \( \mu : G \times G \rightarrow G,\left( {x, y}\right) \mapsto {xy} \) is continuous by Remark 2.1.2, both \( {}_{a}t = \mu \left( {a, - }\right) \) and its inverse \( {a}^{-1}t \), are continuous, for every \( a \in G \) . Hence, \( {a}^{t} \) is a homeomorphism for every \( a \in G \) . A similar proof w... | Yes |
Lemma 2.1.7. Let \( G \) be a countable topological group. If \( G \) is of second category, then \( G \) is discrete. Consequently, if \( G \) is a Baire space then \( G \) is discrete. | Proof. The second assertion follows from the first, since every Baire space is of second category, by Lemma B.5.19. Assume that \( G \) is of second category. As the union \( G = \mathop{\bigcup }\limits_{{g \in G}}\{ g\} \) is countable, there exists \( g \in G \) with \( \operatorname{Int}\{ g\} \neq \varnothing \), ... | Yes |
Example 2.1.13. Consider the group \( \mathbb{R} \) with the Euclidean topology. Then\n\n\[ \mathcal{V}\left( 0\right) = \left\{ {U \subseteq \mathbb{R} : \exists \varepsilon \in {\mathbb{R}}_{ > 0},\left( {-\varepsilon ,\varepsilon }\right) \subseteq U}\right\} . \] | The base \( \mathcal{B} = \left\{ {\left( {-\varepsilon ,\varepsilon }\right) : \varepsilon \in {\mathbb{R}}_{ > 0}}\right\} \) of \( \mathcal{V}\left( 0\right) \) consisting of symmetric open neighborhoods of 0 has size \( \mathfrak{c} \) . One may choose also the countable base \( {\mathcal{B}}_{1} = \left\{ {\left( ... | Yes |
Lemma 2.1.16. For a topological group \( G \) , \( \operatorname{core}\left( G\right) \) is a normal subgroup of \( G \) . | Proof. Let \( N = \operatorname{core}\left( G\right) \) . Clearly, \( {e}_{G} \in N \) . If \( x, y \in N \), then \( {xy} \in N \) by (gt1), while \( {x}^{-1} \in N \) for \( x \in N \) can be deduced from (gt2). Finally, for \( g \in G \), the inclusion \( {gN}{g}^{-1} \subseteq N \) follows from (gt3). | Yes |
Lemma 2.1.20. Let \( G, H \) be topological groups and \( f : G \rightarrow H \) a homomorphism. Then the following conditions are equivalent:\n\n(a) \( f \) is continuous;\n\n(b) \( f \) is continuous at \( {e}_{G} \) ;\n\n(c) for every \( U \in {\mathcal{V}}_{H}\left( {e}_{H}\right) \) there exists \( V \in {\mathcal... | Proof. (a) \( \Leftrightarrow \) (b) immediately follows from the homogeneity of topological groups (see Exercise 2.4.1), while (c) and (d) are clearly equivalent forms of (b). | No |
If \( G, H \) are Alexandrov groups, a homomorphism \( f : G \rightarrow H \) is continuous precisely when \( f\left( {\operatorname{core}\left( G\right) }\right) \subseteq \operatorname{core}\left( H\right) \). | Therefore, the category of Alexandrov groups is isomorphic to the category of pairs \( \left( {G, N}\right) \) of a group \( G \) and a normal subgroup \( N \) of \( G \), where the morphisms \( \left( {G, N}\right) \rightarrow \left( {H, L}\right) \) are group homomorphisms \( f : G \rightarrow H \) with \( f\left( N\... | No |
Let \( p \) be a prime. The basic open neighborhoods of 0 in the topology of \( {\mathbb{J}}_{p} \) described in Example 2.1.5 are the subgroups \( {p}^{n}{\mathbb{J}}_{p} \) of \( \left( {{\mathbb{J}}_{p}, + }\right) \) for \( n \in \mathbb{N} \), that is, the topology of \( {\mathbb{J}}_{p} \) is its \( p \) -adic to... | Actually, each \( {p}^{n}{\mathbb{J}}_{p} \) is an ideal of the ring \( {\mathbb{J}}_{p} \) (see Claim A.4.16). | No |
Lemma 2.2.4. Let \( G, H \) be groups and let \( f : G \rightarrow H \) be a homomorphism. Then \( f \) is continuous when both groups \( G, H \) are equipped with their profinite (respectively, pro-p-finite, pro-countable, p-adic, natural) topology. | Proof. If \( N \) is a subgroup of \( H \) of finite index, then \( {f}^{-1}\left( N\right) \) is a subgroup of finite index of \( G \) . The other cases are similar. | No |
Lemma 2.2.11. Let \( G \) be an abelian group and \( H \) a subset of \( {G}^{ * } \) . The assignment \( \mathcal{P}\left( {G}^{ * }\right) \rightarrow \mathfrak{L}\left( G\right), H \mapsto {\mathcal{T}}_{H} \), is monotone increasing and \( {\mathcal{T}}_{\langle H\rangle } = {\mathcal{T}}_{H} \) . | Proof. If \( H \subseteq {H}^{\prime } \subseteq {G}^{ * } \), then \( {\mathcal{T}}_{H} \leq {\mathcal{T}}_{{H}^{\prime }} \) ; this proves the first assertion and that \( {\mathcal{T}}_{H} \leq {\mathcal{T}}_{\langle H\rangle } \) . The converse inclusion \( {\mathcal{T}}_{\langle H\rangle } \leq {\mathcal{T}}_{H} \)... | Yes |
Lemma 2.2.12. Let \( G, H \) be abelian groups and let \( f : G \rightarrow H \) be a homomorphism. Then \( f : {G}^{\# } \rightarrow {H}^{\# } \) is continuous. | Proof. Let \( {\chi }_{1},\ldots ,{\chi }_{n} \in {H}^{ * } \) and \( \delta > 0 \) . Then\n\n\[
{f}^{-1}\left( {{U}_{H}\left( {{\chi }_{1},\ldots ,{\chi }_{n};\delta }\right) }\right) = {U}_{G}\left( {{\chi }_{1} \circ f,\ldots ,{\chi }_{n} \circ f;\delta }\right) .
\]\n\nTo conclude apply Lemma 2.1.20. | No |
Proposition 2.2.14. For every group \( G \) and every prime \( p \) , \[ {\varpi }_{G}^{p} \leq {\varpi }_{G} \leq {v}_{G} \geq {v}_{G}^{p}\;\text{ and }\;{\varpi }_{G}^{p} = \inf \{ {\varpi }_{G},{v}_{G}^{p}\} . \] | Proof. The first and the last inequality are obvious. To prove the inequality \( {\varpi }_{G} \leq {v}_{G} \) , it suffices to note that if \( N \) is a finite-index normal subgroup of \( G \), with \( \left\lbrack {G : N}\right\rbrack = m \) , then \( N \) contains the subgroup \( {M}_{m} = \left\langle {{g}^{m} : g ... | Yes |
Proposition 2.2.15 ([86]). For every abelian group \( G,{\varpi }_{G} \leq \inf \left\{ {{\mathfrak{B}}_{G},{v}_{G}}\right\} \) . | Proof. In view of the inequality \( {\varpi }_{G} \leq {v}_{G} \) established in Proposition 2.2.14, it is enough to prove that \( {\varpi }_{G} \leq {\mathfrak{B}}_{G} \) .\n\nLet \( H \) be a subgroup of \( G \) of finite index; we show that \( H \) is open in \( {G}^{\# } \) . Being a finite abelian group, \( G/H \)... | Yes |
Lemma 2.2.16. Let \( G \) be an abelian group and \( H \) a subset of \( {G}^{ * } \) . Then \( {\mathcal{T}}_{H} \leq {\varpi }_{G} \) if and only if every \( \chi \in H \) is torsion (i. e., \( H \subseteq t\left( {G}^{ * }\right) \) ). | Proof. Let \( \chi \in H \) be torsion and \( \delta > 0 \) . Then \( {U}_{G}\left( {\chi ;\delta }\right) \) contains \( \ker \chi \) . Since \( \chi \left( G\right) \cong \) \( G/\ker \chi \) is finite, \( \ker \chi \) is an open neighborhood of \( {e}_{G} \) in \( {\varpi }_{G} \), and so \( {U}_{G}\left( {\chi ;\de... | Yes |
Theorem 2.2.17. For an abelian group \( G,{\mathfrak{B}}_{G} = {\varpi }_{G} \) if and only if \( G \) is bounded. | Proof. By Proposition 2.2.15, \( {\varpi }_{G} \leq {\mathfrak{B}}_{G} \) . If \( G \) is bounded, then every character of \( G \) is torsion, so Lemma 2.2.16 gives \( {\mathfrak{B}}_{G} \leq {\varpi }_{G} \) . Now assume that \( {\mathfrak{B}}_{G} = {\varpi }_{G} \) . According to Lemma 2.2.16, the group \( {G}^{ * } ... | Yes |
Example 2.2.18. Let \( G \) be an abelian group and \( p \) a prime. The \( p \) -Bohr topology is \( {\mathfrak{B}}_{G}^{p} \mathrel{\text{:=}} \)\n\n\( {\mathcal{T}}_{\operatorname{Hom}\left( {G,\mathbb{Z}\left( {p}^{\infty }\right) }\right) } \) where we consider \( \mathbb{Z}\left( {p}^{\infty }\right) \leq \mathbb... | Remark 2.2.19. It can be proved that:\n\n(a) \( \varpi = \inf \{ \mathfrak{B}, v\} \) (see Exercise 3.5.17);\n\n(b) analogously, \( {\varpi }^{p} = \inf \left\{ {{v}^{p},{\mathfrak{B}}^{p}}\right\} = \inf \left\{ {{v}^{p},\varpi }\right\} = \inf \left\{ {{\mathfrak{B}}^{p},\varpi }\right\} \). | No |
For a nonempty set \( X \), let \( S\left( X\right) \) denote the group of all permutations of \( X \) . The stabilizer \( {S}_{x} = \{ f \in S\left( X\right) : f\left( x\right) = x\} \) of \( x \in X \) in \( S\left( X\right) \) is a subgroup of \( S\left( X\right) \) . Consider on \( S\left( X\right) \) the filter ba... | \[ {\mathcal{B}}_{X} \mathrel{\text{:=}} \left\{ {{S}_{F} : F \subseteq X\text{ finite }}\right\} ,\;\text{ where }{S}_{F} \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{x \in F}}{S}_{x} = \{ f \in S\left( X\right) : \forall x \in F, f\left( x\right) = x\} . \] Since the subgroups \( {S}_{x} \) are pairwise conjugated,... | Yes |
Lemma 3.1.1. Let \( G \) be a topological group. Then:\n\n(a) for a subset \( A \) of \( G \) ,\n\n\[ \bar{A} = \mathop{\bigcap }\limits_{{U \in \mathcal{V}\left( {e}_{G}\right) }}{UAU} = \mathop{\bigcap }\limits_{{U \in \mathcal{V}\left( {e}_{G}\right) }}{UA} = \mathop{\bigcap }\limits_{{U \in \mathcal{V}\left( {e}_{G... | Proof. Let \( \mathcal{V} = \mathcal{V}\left( {e}_{G}\right) \).\n\n(a) For \( x \in G \), one has \( x \notin \bar{A} \) if and only if there exists a neighborhood \( W \) of \( x \) such that \( W \cap A = \varnothing \) . Since \( G \times G \rightarrow G,\left( {a, b}\right) \mapsto {axb} \), is continuous, we can ... | Yes |
Corollary 3.1.3. If \( A, B \) are nonempty subsets of a topological group \( G \) , then \( \overline{A}\overline{B} \subseteq \overline{AB} \) . If one of the sets \( A, B \) is a singleton, then \( \bar{A}\bar{B} = \overline{AB} \) . | Proof. The inclusion follows from Lemma 3.1.1(a), as \( \bar{A}\bar{B} \subseteq {UABU} \) for every \( U \in \mathcal{V}\left( {e}_{G}\right) \) . In case \( B = \{ b\} \) is a singleton, \( {AB} = {Ab} = {t}_{b}\left( A\right) \) . Since \( {t}_{b} \) is a homeomorphism by Lemma 2.1.6, \( \overline{AB} = \overline{Ab... | Yes |
Proposition 3.1.7. Let \( G \) be a topological group and \( H \) a subgroup of \( G \). Then:\n\n(a) \( H \) is open in \( G \) if and only if \( H \) has nonempty interior;\n\n(b) if \( H \) is open, then \( H \) is also closed;\n\n(c) if \( \left\lbrack {G : H}\right\rbrack < \infty \), then \( H \) is closed if and... | Proof. (a) Let \( \varnothing \neq V \subseteq H \) be an open set of \( G \), and let \( {h}_{0} \in V \). Then \( {e}_{G} \in {h}_{0}^{-1}V \subseteq H = \) \( {h}_{0}^{-1}H \). Now \( U \mathrel{\text{:=}} {h}_{0}^{-1}V \subseteq H \) is open, contains \( {e}_{G} \), and \( h \in {hU} \subseteq H \) for every \( h \... | Yes |
Proposition 3.1.9. Let \( G \) be an abelian group and \( H \) a subgroup of \( G \) . Then \( H \) is (dually) closed in \( \left( {G,{\mathfrak{B}}_{G}}\right) \) and \( {\left. {\mathfrak{B}}_{G}\right| }_{H} = {\mathfrak{B}}_{H} \) . | Proof. To see that \( H \) is dually closed, pick \( x \in G \smallsetminus H \) . Now it suffices to apply Corollary A.2.6 to the quotient \( G/H \) and deduce that for the nonzero element \( x + H \) of \( G/H \) there exists a character \( \xi : G/H \rightarrow \mathbb{T} \) with \( \xi \left( {x + H}\right) \neq 0 ... | Yes |
Proposition 3.1.11. For a proper subgroup \( H \) of \( \mathbb{R} \), the following are equivalent:\n\n(a) \( H \) is cyclic;\n\n(b) \( H \) is discrete;\n\n(c) \( H \) is closed;\n\n(d) \( H \) is not dense in \( \mathbb{R} \). | Proof. (a) \( \Rightarrow \) (b) It is easy to see that a cyclic subgroup of \( \mathbb{R} \) is discrete.\n\n(b) \( \Rightarrow \) (c) Let \( \varepsilon > 0 \) and let the neighborhood \( U = \left( {-\varepsilon ,\varepsilon }\right) \) of 0 witness the discreteness of \( H \), that is, \( U \cap H = \{ 0\} \) . The... | Yes |
According to Proposition 3.1.11, a proper subgroup of \( \mathbb{R} \) is dense if and only if it is not cyclic. | This gives easy examples of closed subgroups \( {H}_{1},{H}_{2} \) of \( \mathbb{R} \) such that \( {H}_{1} + {H}_{2} \) is not closed in \( \mathbb{R} \) . Since such \( {H}_{1},{H}_{2} \) are necessarily cyclic, we can take \( {H}_{1} = \mathbb{Z} \) and \( {H}_{2} \) any cyclic subgroup generated by an irrational nu... | Yes |
Let \( {q}_{0} : \mathbb{R} \rightarrow \mathbb{T} \) be the canonical projection.\n\n(a) A proper subgroup \( H \) of \( \mathbb{T} \) is either closed (precisely when \( H \) is finite and cyclic), or dense (when \( H \) is infinite). | Indeed, \( L = {q}_{0}^{-1}\left( H\right) \) is a proper subgroup of \( \mathbb{R} \) containing \( \mathbb{Z} \) . So, if \( H \) is closed then \( L \) is closed, hence cyclic by Proposition 3.1.11 and generated by a rational since \( \mathbb{Z} \subseteq L \) ; therefore, \( H \) is a finite cyclic subgroup of \( \... | Yes |
Proposition 3.1.15. Let \( G \) be a topological group. Then \( G \) is a regular topological space and the following conditions are equivalent:\n\n(a) \( G \) is \( {T}_{0} \) ;\n\n(b) \( G \) is \( {T}_{1} \) (i.e., \( \overline{\left\{ {e}_{G}\right\} } = \left\{ {e}_{G}\right\} \) );\n\n(c) \( G \) is Hausdorff;\n\... | Proof. Since \( G \) is a homogeneous topological space, to prove regularity of \( G \) it suffices to check that for every \( U \in \mathcal{V}\left( {e}_{G}\right) \) there exists \( V \in \mathcal{V}\left( {e}_{G}\right) \) such that \( \bar{V} \subseteq U \) . According to Lemma 3.1.1, it suffices to pick a \( V \i... | Yes |
Proposition 3.1.17. Let \( G \) be a topological group and \( H \) a subgroup of \( G \) . If \( H \) is discrete and \( G \) is Hausdorff, then \( H \) is closed. | Proof. Since \( H \) is discrete there exists \( U \in \mathcal{V}\left( {e}_{G}\right) \) with \( U \cap H = \left\{ {e}_{G}\right\} \) . Choose \( V \in \mathcal{V}\left( {e}_{G}\right) \) with \( {V}^{-1}V \subseteq U \) . Then \( \left| {{xV} \cap H}\right| \leq 1 \) for every \( x \in G \), as \( {h}_{1} = x{v}_{1... | Yes |
Proposition 3.1.20. For an infinite abelian group \( G \) and a subgroup \( H \) of \( {G}^{ * } \): (a) \( {\mathcal{T}}_{H} \) is Hausdorff if and only if the characters of \( H \) separate the points of \( G \) ; (b) \( {\mathcal{T}}_{H} \) is nondiscrete. | Proof. (a) Assume that \( H \) separates the points of \( G \) and pick a nonzero \( a \in G \) . Then there exists \( \chi \in H \) such that \( \chi \left( a\right) \neq 1 \) . Let \( \delta = \frac{1}{2}\left| {\operatorname{Arg}\left( {\chi \left( a\right) }\right) }\right| \) . Then \( U\left( {\chi ;\delta }\righ... | Yes |
Proposition 3.1.21. For a group \( G \) the following conditions are equivalent:\n\n(a) the profinite topology \( {\varpi }_{G} \) of \( G \) is Hausdorff;\n\n(b) \( G \) is residually finite.\n\nIf \( G \) is abelian, then they are equivalent also to:\n\n(c) the natural topology \( {v}_{G} \) of \( G \) is Hausdorff;\... | Proof. (a) \( \Leftrightarrow \) (b) and (c) \( \Leftrightarrow \) (d) are obvious, in view of Corollary 3.1.16. Since \( {\varpi }_{G} \leq {v}_{G} \) , the first pair of conditions imply those of the second one. (d) \( \Rightarrow \) (a) follows from the fact that \( {G}^{1} \) coincides with the intersection of all ... | Yes |
Theorem 3.1.22. If \( G \) is a Hausdorff group containing a dense abelian subgroup \( H \), then \( G \) is abelian. | Proof. Take \( x, y \in G \) . Then by Remark 3.1.4, there exist nets \( {\left\{ {h}_{\alpha }\right\} }_{\alpha \in A},{\left\{ {g}_{\alpha }\right\} }_{\alpha \in A} \) in \( H \) such that \( x = \mathop{\lim }\limits_{{\alpha \in A}}{h}_{\alpha }\;\mathrm{{and}}\;y = \mathop{\lim }\limits_{{\alpha \in A}}{g}_{\alp... | No |
Theorem 3.1.24. If \( G \) is a Hausdorff group containing a dense nilpotent subgroup \( H \) of class \( s \), then \( G \) is nilpotent of class \( s \) . | Proof. For \( {g}_{0},{g}_{1},\ldots ,{g}_{s} \in G \), by the density of \( H \) in \( G \) and Remark 3.1.4, we can write, for every \( n \in \{ 0,1,\ldots, s\} ,{g}_{n} = \mathop{\lim }\limits_{{\alpha \in A}}{h}_{n,\alpha } \) where \( {\left\{ {h}_{n,\alpha }\right\} }_{\alpha \in A} \) is a net in \( H \) . Then,... | Yes |
For a Hausdorff group \( G \), the centralizer \( {c}_{G}\left( g\right) \) of each \( g \in G \) and so the center \( Z\left( G\right) \) are closed subgroups of \( G \). | Indeed, for every \( g \in G \), by Exercise B.7.12 we conclude that \( {c}_{G}\left( g\right) \) is closed in \( G \). Then also \( Z\left( G\right) = \mathop{\bigcap }\limits_{{g \in G}}{c}_{G}\left( g\right) \) is closed in \( G \). | No |
Lemma 3.2.1. Let \( G \) be a topological group, \( H \) a normal subgroup of \( G \), and let \( G/H \) be equipped with the quotient topology. Then:\n\n(a) the canonical projection \( q : G \rightarrow G/H \) is open;\n\n(b) iff: \( G/H \rightarrow {G}_{1} \) is a homomorphism to a topological group \( {G}_{1} \), th... | Proof. (a) Let \( U \neq \varnothing \) be an open set of \( G \) . Then \( {q}^{-1}\left( {q\left( U\right) }\right) = {HU} = \mathop{\bigcup }\limits_{{h \in H}}{hU} \) is open, since each \( {hU} \) is open. Therefore, \( q\left( U\right) \) is open in \( G/H \) .\n\n(b) If \( f \) is continuous, then the compositio... | Yes |
Theorem 3.2.3 (Frobenius theorem). Let \( G, H \) be topological groups, \( f : G \rightarrow H \) a continuous surjective homomorphism, and \( q : G \rightarrow G/\ker f \) the canonical projection, and let \( {f}_{1} : G/\ker f \rightarrow H \) be the unique homomorphism with \( f = {f}_{1} \circ q \): Then \( {f}_{1... | Proof. It follows immediately from the definitions of quotient topology and open map, and from Lemma 3.2.1. | No |
Corollary 3.2.4. Let \( G, H \) be topological groups and \( f : G \rightarrow H \) a topological isomorphism. Then for every normal subgroup \( N \) of \( G \) the quotient \( H/f\left( N\right) \) is topologically isomorphic to \( G/N \) . | Proof. Obviously, \( f\left( N\right) \) is a normal subgroup of \( H \) and the canonical projection \( q : H \rightarrow \) \( H/f\left( N\right) \) is continuous and open by Lemma 3.2.1. Therefore, \( h = q \circ f : G \rightarrow H/f\left( N\right) \) is an open continuous surjective homomorphism with \( \ker h = N... | Yes |
Proposition 3.2.5. Let \( G,{H}_{1},{H}_{2} \) be topological abelian groups and \( {f}_{i} : G \rightarrow {H}_{i}, i = 1,2 \) , open continuous surjective homomorphisms. Then there exists a continuous homomorphism \( \eta : {H}_{1} \rightarrow {H}_{2} \) such that \( {f}_{2} = \eta \circ {f}_{1} \) if and only if \( ... | Proof. The necessity is obvious. So, assume that \( \ker {f}_{1} \subseteq \ker {f}_{2} \) . By Frobenius theorem 3.2.3 applied to \( {f}_{i} \), for \( i = 1,2 \), there exists a topological isomorphism \( {j}_{i} : G/\ker {f}_{i} \rightarrow \) \( {H}_{i} \) such that \( {f}_{i} = {j}_{i} \circ {q}_{i} \), where \( {... | Yes |
Lemma 3.2.6. Let \( X, Y \) be topological spaces and let \( f : X \rightarrow Y \) be an open continuous map. Then for every subspace \( P \) of \( Y \) with \( P \cap f\left( X\right) \neq \varnothing \), the restriction \( {f}^{\prime } = f{ \upharpoonright }_{{H}_{1}} : {H}_{1} \rightarrow P \) to the subspace \( {... | Proof. To see that \( {f}^{\prime } \) is open, choose a point \( x \in {H}_{1} \) and a neighborhood \( U \) of \( x \) in \( {H}_{1} \) . Then there exists a neighborhood \( W \) of \( x \) in \( X \) such that \( U = {H}_{1} \cap W \) . To see that \( {f}^{\prime }\left( U\right) \) is a neighborhood of \( {f}^{\pri... | Yes |
For \( G = \mathbb{T} \) the continuous homomorphism \( {\mu }_{2} : G \rightarrow G, x \mapsto {2x} \), is surjective and open. Let now \( H = \mathbb{Z}\left( {3}^{\infty }\right) \leq \mathbb{T} \) . The restriction \( {\mu }_{2}^{\prime } : H \rightarrow {2H} = H \) of \( {\mu }_{2} \) is a continuous isomorphism. | To see that \( {\mu }_{2}^{\prime } \) is not open, it suffices to notice that the sequence \( {\left\{ {x}_{n}\right\} }_{n \in {\mathbb{N}}_{ + }} \) in \( H \) , defined by \( {x}_{n} = \mathop{\sum }\limits_{{k = 1}}^{n}1/{3}^{k} \) for every \( n \in {\mathbb{N}}_{ + } \), is not convergent in \( H \) (as it conve... | Yes |
Theorem 3.2.8. Let \( G \) be a topological group, \( N \) a normal subgroup of \( G \), and \( q : G \rightarrow \) \( G/N \) the canonical projection.\n\n(a) If \( H \) is a subgroup of \( G \), then the homomorphism \( {q}_{1} : {HN}/N \rightarrow q\left( H\right) \), defined by \( {q}_{1}\left( {xN}\right) = q\left... | Proof. (a) As \( {HN} = {q}^{-1}\left( {q\left( H\right) }\right) \), we can apply Lemma 3.2.6 and conclude that the restriction \( {q}^{\prime } : {HN} \rightarrow q\left( H\right) \) of \( q \) is open. Now Frobenius theorem 3.2.3 applies to \( {q}^{\prime } \) and implies that the unique homomorphism \( {q}_{1} : {H... | Yes |
For a topological group \( G \) and a normal subgroup \( H \) of \( G \): (a) \( G/H \) is discrete if and only if \( H \) is open; | Proof. Let \( q : G \rightarrow G/H \) be the canonical projection. (a) If \( G/H \) is discrete, then \( H = {q}^{-1}\left( {e}_{G/H}\right) \) is open since the singleton \( \left\{ {e}_{G/H}\right\} \) is open. If \( H \) is open, then \( \left\{ {e}_{G/H}\right\} = q\left( H\right) \) is open since the map \( q \) ... | Yes |
Proposition 3.2.12. Let \( G, H \) be abelian groups and \( f : G \rightarrow H \) a surjective homomorphism. Then \( f \) is (continuous and) open when both \( G, H \) are equipped with their profinite (respectively, pro-p-finite, p-adic, natural, pro-countable) topology. | Proof. For the profinite and for the pro-countable topology use that fact that for a subgroup \( N \) of \( G \) the homomorphism \( {f}_{1} : G/N \rightarrow H/f\left( N\right) \) induced by \( f \) is surjective. The remaining cases are trivial. | No |
Proposition 3.3.1. Let \( \left\{ {{G}_{i} : i \in I}\right\} \) be a family of topological groups. The direct product \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \), equipped with the product topology, is a topological group. | Proof. The filter \( \mathcal{V}\left( {e}_{G}\right) \) in the product topology of \( G \) has as a base the family of neighborhoods\n\n\[ \mathop{\bigcap }\limits_{{k = 1}}^{n}{p}_{{j}_{k}}^{-1}\left( {U}_{{j}_{k}}\right) = {U}_{{j}_{1}} \times \cdots \times {U}_{{j}_{n}} \times \mathop{\prod }\limits_{{i \in I \smal... | Yes |
Proposition 3.3.6. Let \( \left\lbrack {{G}_{i},\left( {\nu }_{ij}\right), I}\right\rbrack \) be an inverse system of topological groups.\n\n(a) In the product \( H = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \), consider the subgroup\n\n\[ G = \left\{ {x = {\left( {x}_{i}\right) }_{i \in I} \in H : {v}_{ij}\left( {x}_... | Proof. (a) This is straightforward, since \( {p}_{i} : G \rightarrow {G}_{i} \) is continuous for every \( i \in I \) . | No |
Let \( \left\{ {{G}_{n} : n \in \mathbb{N}}\right\} \) be a family of topological groups and let \( \left\{ {{\phi }_{n} : n \in \mathbb{N}}\right\} \) be a family of continuous homomorphisms \( {\phi }_{n} : {G}_{n + 1} \rightarrow {G}_{n} \) . Putting, for every pair \( m, n \in \) \( \mathbb{N} \) with \( m > n,{\va... | In these terms, if \( p \) is a prime and \( {\phi }_{n} : \mathbb{Z}\left( {p}^{n + 1}\right) \rightarrow \mathbb{Z}\left( {p}^{n}\right) \) is the canonical projection for \( n \in \mathbb{N} \), the inverse limit of the inverse system \( \left\lbrack {\mathbb{Z}\left( {p}^{n}\right) ,\left( {\phi }_{n}\right) ,\math... | Yes |
Lemma 3.4.1. Let \( \left( {G,\tau }\right) \) be a topological group, \( N \mathrel{\text{:=}} \operatorname{core}\left( G\right) = \overline{\left\{ {e}_{G}\right\} } \), and \( q : G \rightarrow G/N \) the canonical projection. Then:\n\n(a) \( N \) is an indiscrete closed normal subgroup of \( G \) and \( G/N \) is ... | Proof. (a) Since \( N \subseteq U \) for every \( U \in {\mathcal{V}}_{\tau }\left( {e}_{G}\right), N \) is indiscrete. The last assertion follows from Lemma 3.2.10(b).\n\n(b) Let \( V \in {\mathcal{V}}_{\tau }\left( {e}_{G}\right) \) be open. Then \( N \subseteq V \) . Fix arbitrarily \( x \in V \) . Then there exists... | Yes |
Proposition 3.4.3. Let \( G, H \) be topological groups and \( f : G \rightarrow H \) a continuous homomorphism. Then \( \mathfrak{h}f : \mathfrak{h}G \rightarrow \mathfrak{h}H \), defined by \( \mathfrak{h}f\left( {x\operatorname{core}\left( G\right) }\right) = f\left( x\right) \operatorname{core}\left( H\right) \) fo... | Proof. Since \( f\left( {e}_{G}\right) = {e}_{H} \), Proposition 3.4.2(a) implies that \( f\left( {\operatorname{core}\left( G\right) }\right) \subseteq \operatorname{core}\left( H\right) \) . This proves the correctness of the definition of \( \mathfrak{h}f \) and the commutativity of the following diagram:\n\n![e3c29... | Yes |
Theorem 4.2.1. Let \( X \) be an infinite set. Then \( {\mathfrak{M}}_{S\left( X\right) } = {\mathrm{T}}_{X} \) . | This theorem follows immediately from an old result due to Gaughan: | No |
Theorem 4.2.2 (Gaughan theorem). Let \( X \) be an infinite set. Every Hausdorff group topology on \( S\left( X\right) \) is finer than \( {\mathrm{T}}_{X} \) . | The proof of this theorem that we give below follows more or less the line of the proof exposed in \( \left\lbrack {{99},§{7.1}}\right\rbrack \) with several simplifications. The final stage of the proof is preceded by a number of claims (and their corollaries) and two facts about purely algebraic properties of the gro... | No |
Lemma 4.2.4. For an infinite set \( X \) and \( x \in X \), the subgroup \( {S}_{x} \) of \( S\left( X\right) \) is maximal. | Proof. Assume that \( H \) is a subgroup of \( S\left( X\right) \) properly containing \( {S}_{x} \) . To show that \( H = \) \( S\left( X\right) \), take any \( f \in S\left( X\right) \) . If \( f\left( x\right) = x \), then \( f \in {S}_{x} \subseteq H \), and we are done. Assume that \( y \mathrel{\text{:=}} f\left(... | Yes |
Corollary 4.2.6. If \( X \) is an infinite set and \( T \) is a Hausdorff group topology on \( S\left( X\right) \) such that \( {S}_{x} \) is \( T \) -closed in \( S\left( X\right) \) for some \( x \in X \), then \( {\mathrm{T}}_{X} \leq T \) . | Proof. Since all \( {S}_{x} \) are conjugated, the hypothesis implies that \( {S}_{x} \) is \( T \) -closed for every \( x \in X \) . By Claim 4.2.5, \( {S}_{x} \) is \( T \) -open for every \( x \in X \) . As the subgroups \( {S}_{x} \) of \( S\left( X\right) \) form a prebase of the filter of neighborhoods of \( i{d}... | Yes |
Corollary 4.2.10. Let \( X \) be an infinite set and \( T \) a Hausdorff group topology on \( S\left( X\right) \) . Then for every pair \( x, y \in X \) with \( x \neq y \), the stabilizer \( {S}_{x, y} \) is not \( T \) -dense in \( S\left( X\right) \) . | Proof. By Claims 4.2.8 and 4.2.9, there exist distinct \( {x}^{\prime },{y}^{\prime } \in X \) such that \( {S}_{{x}^{\prime },{y}^{\prime }} \) is not \( T \) -dense. The assertion follows from the fact that all stabilizers of the form \( {S}_{x, y} \) with distinct \( x, y \) are conjugated. | Yes |
Corollary 4.2.14. If \( G \) is a nontrivial Hausdorff group, \( X \) is an infinite set, and \( f : \left( {S\left( X\right) ,{\mathrm{T}}_{X}}\right) \rightarrow G \) is a continuous surjective homomorphism, then \( f \) is a topological isomorphism. | Proof. Since \( \left( {S\left( X\right) ,{\top }_{X}}\right) \) is topologically simple (see Exercise 8.7.7) and \( f \) is surjective and nontrivial, we deduce that \( f \) is a continuous isomorphism. Theorem 4.2.2 implies that \( f \) is open. | No |
Proposition 4.3.1. Every group with infinite center admits a nondiscrete Hausdorff group topology. | Proof. The center \( H = Z\left( G\right) \) of the group \( G \) has a nondiscrete Hausdorff group topology \( \tau \), by Proposition 3.1.20. Obviously, \( {\mathcal{V}}_{\left( H,\tau \right) }\left( {e}_{H}\right) \) is a filter base satisfying conditions (gt1), (gt2), and (gt3), so it forms a local base at \( {e}_... | Yes |
Theorem 4.3.4. Let \( m \) and \( n \) be odd integers \( \geq {665} \) and \( A = \mathcal{A}\left( {m, n}\right) \) . The group \( G = \) \( A/{C}_{m} \) has discrete Zariski topology. | Proof. Let us see that (b), (c), and (d) jointly imply that the Zariski topology of the infinite quotient \( G = A/{C}_{m} \) is discrete (so \( G \) is a countable Markov group). Let \( d \) be a generator of \( C \) . Then \( {x}^{m} \in C \smallsetminus {C}_{m} \) for every \( x \in A \smallsetminus C \) . Indeed, i... | Yes |
Under CH, there exists a group \( G \) of size \( {\omega }_{1} \) satisfying the following conditions (a) (with \( m = {10000} \) ) and (b) (with \( n = 2 \) ):\n\n(a) there exists \( m \in \mathbb{N} \) such that \( {A}^{m} = G \) for every subset \( A \) of \( G \) with \( \left| A\right| = \left| G\right| \);\n\n(b... | To see that \( G \) is a Markov group (i. e., \( {\mathfrak{M}}_{G} \) is discrete), assume that \( T \) is a Hausdorff group topology on \( G \) . There exists a \( T \) -neighborhood \( V \) of \( {e}_{G} \) with \( V \neq G \) . Choose a \( W \in {\mathcal{V}}_{T}\left( {e}_{G}\right) \) with \( \underset{m}{\underb... | Yes |
Lemma 4.4.2 ([103]). Let \( H \) be a subgroup of a group \( G \) such that \( G = H{c}_{G}\left( H\right) \) . Then for every group topology \( \tau \) on \( H \), the above described topology \( {\tau }^{ * } \) is a group topology on \( G \) such that \( \left( {H,\tau }\right) \) is an open topological subgroup of ... | Proof. The first two axioms, (gt1) and (gt2), on the neighborhood base are easy to check. For (gt3), pick a basic \( {\tau }^{ * } \) -neighborhood \( U \) of \( {e}_{G} \) in \( G \) . Since \( H \) is \( {\tau }^{ * } \) -open, we can assume without loss of generality that \( U \subseteq H \), so \( U \) is a \( \tau... | Yes |
Theorem 4.4.3 ([103]). Let \( H \) be a normal subgroup of the group \( G \) and let \( \tau \) be a group topology on \( H \). Then the following conditions are equivalent:\n\n(a) the extension \( {\tau }^{ * } \) is a group topology on \( G \);\n\n(b) \( \tau \) can be extended to a group topology of \( G \);\n\n(c) ... | Proof. (a) \( \Rightarrow \) (b) is obvious, while (b) \( \Rightarrow \) (c) follows from the fact that the conjugations are continuous in any topological group (see Lemma 2.1.6).\n\n(c) \( \Rightarrow \) (a) Assume that all automorphisms of \( H \) induced by the conjugation by elements of \( G \) are \( \tau \) -cont... | Yes |
Corollary 4.4.5. For a Hausdorff group \( \left( {H,\tau }\right) \), the following conditions are equivalent:\n\n(a) every automorphism of \( H \) is \( \tau \) -continuous;\n\n(b) for every group \( G \) containing \( H \) as a normal subgroup, \( \tau \) can be extended to a group topology on G;\n\n(c) \( \tau \) ca... | Proof. (a) \( \Rightarrow \) (b) follows from Corollary 4.4.4 and (b) \( \Rightarrow \) (c) is trivial.\n\n(c) \( \Rightarrow \) (a) Extend \( \tau \) to a group topology \( {\tau }^{\prime } \) on \( G \) and note that the automorphisms of \( H \) act as restrictions of inner automorphisms of \( G \) on \( H \) . As t... | Yes |
Let \( p \) be a prime number. If the group of \( p \) -adic integers \( N = {J}_{p} \) is a normal subgroup of some group \( G \), then the \( p \) -adic topology of \( N \) can be extended to a group topology on \( G \). | Indeed, it suffices to note that if \( \xi : N \rightarrow N \) is an automorphism of \( N \), then obviously \( \xi \left( {{p}^{n}N}}\right) = {p}^{n}N \) . Since the subgroups \( {p}^{n}N \) define the topology of \( N \) , this proves that every automorphism of \( N \) is continuous. Now Theorem 4.4.3 applies. | Yes |
Lemma 4.4.7. The only topological automorphisms \( \chi :\mathbb{T} \rightarrow \mathbb{T} \) are \( \pm i{d}_{\mathbb{T}} \) . | Proof. For \( n \in {\mathbb{N}}_{ + } \), let \( {c}_{n} = {q}_{0}\left( {1/{2}^{n}}\right) \) be the generators of \( \mathbb{Z}\left( {2}^{\infty }\right) \leq \mathbb{T} \) . Then \( {c}_{1} = {q}_{0}\left( {1/2}\right) \) is the only element of \( \mathbb{T} \) of order 2, hence \( \chi \left( {c}_{1}\right) = {c}... | Yes |
Lemma 5.1.3. If \( H \) is a dense subgroup of a Hausdorff group \( G \) and \( \mathcal{B} \) is an open local base at \( {e}_{G} \) in \( H \), then \( \left\{ {{\bar{U}}^{G} : U \in \mathcal{B}}\right\} \) is a local base at \( {e}_{G} \) in \( G \) . | Proof. Since \( G \) is a regular space by Proposition 3.1.15, the closed neighborhoods of \( {e}_{G} \) in \( G \) form a local base at \( {e}_{G} \) in \( G \) . Hence, for \( V \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \), one can find \( {V}_{0} \in {\mathcal{V}}_{G}\left( {e}_{G}\right) \) such that \( \overline{... | Yes |
Proposition 5.1.4. Let \( H \) be a dense subset of a topological group \( G \) and let \( \mathcal{B} \) be a local base at \( {e}_{G} \) in \( G \) consisting of symmetric neighborhoods. Then \( \{ {hU} : U \in \mathcal{B}, h \in H\} \) is a base of the topology of G. | Proof. Let \( x \in G \) and let \( O \) be an open set of \( G \) containing \( x \) . Then there exists a symmetric \( U \in \mathcal{B} \) with \( {xUU} \subseteq O \) . Pick \( h \in H \cap {xU} \) . Then \( {x}^{-1}h \in U \), so \( {h}^{-1}x \in {U}^{-1} = U \) . Therefore, \( x \in {hU} = x{x}^{-1}{hU} \subseteq... | Yes |
Lemma 5.1.5. Let \( G \) be a topological group. Then:\n\n(a) \( d\left( G\right) \leq w\left( G\right) \leq {2}^{d\left( G\right) } \); | Proof. (a) To see that \( d\left( G\right) \leq w\left( G\right) \), choose a base \( \mathcal{B} \) of the topology on \( G \) with \( \left| \mathcal{B}\right| = \) \( w\left( G\right) \), and for every \( \varnothing \neq U \in \mathcal{B} \) pick a point \( {d}_{U} \in U \) . Then the set \( D = \left\{ {{d}_{U} : ... | Yes |
Lemma 5.1.7. Let \( G \) be a topological group. Then \( w\left( G\right) = \chi \left( G\right) \cdot d\left( G\right) \). | Proof. The inequality \( w\left( G\right) \geq \chi \left( G\right) \) is obvious. The inequality \( w\left( G\right) \geq d\left( G\right) \) was proved in Lemma 5.1.5(a). This gives \( w\left( G\right) \geq \chi \left( G\right) \cdot d\left( G\right) \). To prove the inequality \( w\left( G\right) \leq \chi \left( G\... | Yes |
Lemma 5.1.8. Let \( H \) be a subgroup of a topological group \( G \). Then:\n\n(a) \( w\left( H\right) \leq w\left( G\right) \) and \( \chi \left( H\right) \leq \chi \left( G\right) \) ; moreover, \( \psi \left( H\right) \leq \psi \left( G\right) \) if \( G \) is \( {T}_{2} \) ;\n\n(b) if \( H \) is dense in \( G \), ... | Proof. (a) This follows from Lemma 5.1.2, as \( \left| {\{ U \cap G : U \in \mathcal{B}\} }\right| \leq \left| \mathcal{B}\right| \) for every base (respectively, local base at \( {e}_{G} \) ) \( \mathcal{B} \) of the topology on \( G \) . A similar argument applies for \( \psi \).\n\n(b) We prove first \( \chi \left( ... | Yes |
Lemma 5.1.9. Let \( G, H \) be topological groups. If \( f : G \rightarrow H \) is a continuous surjective homomorphism, then \( d\left( H\right) \leq d\left( G\right) \). If \( f \) is open, then also \( w\left( H\right) \leq w\left( G\right) \) and \( \chi \left( H\right) \leq \chi \left( G\right) \). | Proof. If \( D \) is a dense subset of \( G \), then \( f\left( D\right) \) is a dense subset of \( H \) with \( \left| {f\left( D\right) }\right| \leq \left| D\right| \). This proves the first assertion. The second assertion follows from the fact that if \( \mathcal{B} \) is a base (respectively, a local base at \( {e... | Yes |
Since in Lemma \( {5.1.9}\mathrm{H} \) is a continuous image of \( G \) by means of the continuous surjective homomorphism \( f : G \rightarrow H \), the canonical projection \( G \rightarrow \) \( G/\ker f \) is open, and there exists a continuous isomorphism \( G/\ker f \rightarrow H \) (which is open if and only if ... | (a) Let \( G \) be an infinite abelian group and \( {\tau }_{2} = {\delta }_{G} \), so that \( w\left( {G,{\tau }_{2}}\right) = \left| G\right| \) . Moreover, let \( {\tau }_{1} = {\mathfrak{B}}_{G} \) . Then \( w\left( {G,{\tau }_{1}}\right) = w\left( {G}^{\# }\right) = {2}^{\left| G\right| } > \left| G\right| = w\lef... | Yes |
Theorem 5.1.11. If \( \left\{ {{G}_{i} : i \in I}\right\} \) is a family of topological groups and \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \), then \( \sup \{ d\left( {G}_{i}\right) :i \in I\} \leq d\left( G\right) \leq \left| I\right| \cdot \sup \{ d\left( {G}_{i}\right) :i \in I\} . \) | Proof. Applying Lemma 5.1.9 to the projection \( {p}_{i} : G \rightarrow {G}_{i} \), we deduce \( d\left( {G}_{i}\right) \leq d\left( G\right) \) . Hence, \( \sup \left\{ {d\left( {G}_{i}\right) : i \in I}\right\} \leq d\left( G\right) \) . \n\nFor each \( i \in I \), let \( {D}_{i} \) be a dense subset of \( {G}_{i} \... | Yes |
The topological groups \( G \) with \( d\left( G\right) \leq \omega \) are precisely the separable groups. | If \( \left\{ {{G}_{i} : i \in I}\right\} \) is a family of separable groups, then \( d\left( {\mathop{\prod }\limits_{{i \in I}}{G}_{i}}\right) \leq \max \{ \omega ,\left| I\right| \} \), by Theorem 5.1.11. | No |
Lemma 5.1.14. Let \( n \in {\mathbb{N}}_{ + } \), let \( {G}_{1},\ldots ,{G}_{n} \) be topological groups and \( G = {G}_{1} \times \cdots \times {G}_{n} \) . Then\n\n\[ \chi \left( G\right) = \chi \left( {G}_{1}\right) \cdots \chi \left( {G}_{n}\right) \;\text{ and }\;w\left( G\right) = w\left( {G}_{1}\right) \cdots w... | Proof. We prove the assertions for \( n = 2 \) . Then one can proceed by induction. Let \( {\mathcal{B}}_{1},{\mathcal{B}}_{2} \) be local bases at \( {e}_{{G}_{1}},{e}_{{G}_{2}} \), respectively in \( {G}_{1},{G}_{2} \) . Then\n\n\[ \mathcal{B} = \left\{ {U \times V : U \in {\mathcal{B}}_{1}, V \in {\mathcal{B}}_{2}}\... | Yes |
Theorem 5.1.15. Let \( \left\{ {{G}_{i} : i \in I}\right\} \) be an infinite family of nonindiscrete topological groups and \( G = \mathop{\prod }\limits_{{i \in I}}{G}_{i} \) . Then\n\n\[ \chi \left( G\right) = \left| I\right| \cdot \sup \left\{ {\chi \left( {G}_{i}\right) :i \in I}\right\} \;\text{ and }\;w\left( G\r... | Proof. For every \( i \in I \), let \( {\mathcal{B}}_{i} \) be a base of \( {\mathcal{V}}_{{G}_{i}}\left( {e}_{{G}_{i}}\right) \) with \( \left| {\mathcal{B}}_{i}\right| = \chi \left( {G}_{i}\right) \) . For any finite subset \( J \subseteq I \) and for \( {U}_{i} \in {\mathcal{B}}_{i} \) when \( i \in J \), let\n\n\[ ... | Yes |
Let \( {\ell }_{2} \) denote the set of all sequences \( x = {\left\{ {x}_{n}\right\} }_{n \in \mathbb{N}} \) of real numbers such that the series \( \mathop{\sum }\limits_{{n \in \mathbb{N}}}{x}_{n}^{2} \) converges. Then \( {\ell }_{2} \) has a natural structure of vector space (induced by the Cartesian product \( {\... | This defines a norm on the abelian group \( \left( {{\ell }_{2}, + }\right) \) , that provides an invariant metric on \( {\ell }_{2} \) making it a metric space and a topological group. | Yes |
For \( n \in {\mathbb{N}}_{ + }, q \in \lbrack 1,\infty ) \) and \( x = \left( {{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{R}}^{n} \), let\n\n\[ \parallel x{\parallel }_{q} = {\left( \mathop{\sum }\limits_{{i = 1}}^{n}{\left| {x}_{i}\right| }^{q}\right) }^{\frac{1}{q}} \]\n\n(i) For \( q = 2 \), the norm \( \parallel... | See Example B.3.6. | No |
Let \( G \) be a finitely generated group and let \( S = {S}^{-1} \) be a finite set of generators of \( G \) . Given \( g \in G \smallsetminus \left\{ {e}_{G}\right\} \), its word norm \( {\ell }_{S}\left( g\right) \) with respect to \( S \) is the shortest length of a word in the alphabet \( S \) whose evaluation is ... | \[ {\ell }_{S}\left( g\right) = \min \left\{ {n \in {\mathbb{N}}_{ + } : g = {s}_{1}\cdots {s}_{n},{s}_{1},\ldots ,{s}_{n} \in S}\right\} . \] Moreover, let \( {\ell }_{S}\left( {e}_{G}\right) = 0 \) . It is easy to check that the function \( {\ell }_{S} \) satisfies the conditions that determine a word norm: - \( \;{\... | Yes |
Theorem 5.2.14. Every Hausdorff group \( G \) is a Tichonov space. | Proof. Let \( \varnothing \neq F \) be a closed set of \( G \) with \( a \notin F \) . By the homogeneity of \( G \), we may assume that \( a = {e}_{G} \) . Then we can find a chain \( \left\{ {{U}_{n} : n \in \mathbb{N}}\right\} \) as in (5.2) of symmetric open neighborhoods of \( {e}_{G} \) with \( {U}_{n + 1}{U}_{n ... | Yes |
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. | If \( {G}_{i} \) is separable for every \( i \in I \), in view of the equality \( w\left( G\right) = \chi \left( G\right) \cdot d\left( G\right) \) from Lemma 5.1.7 and the inequality \( d\left( G\right) \leq \left| I\right| \) from Example 5.1.12, we conclude that \( w\left( G\right) = \) \( \chi \left( G\right) = \le... | Yes |
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 ... | Yes |
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... | This norm gives rise to the invariant metric \( d = {d}_{\parallel - \parallel } \) described by\n\n\[ \nd\left( {f, g}\right) = \parallel f - g\parallel = \sup \{ \left| {f\left( x\right) - g\left( x\right) }\right| : x \in X\} \;\text{ for every }f, g \in {C}^{ * }\left( X\right) .\n\]\n\nThe metric topology related ... | 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. | Yes |
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 |
(b) 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 \( {xQ}\left( G\right) = \) \( Q\left( G\right) x \). | Proof. 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 \( {e}_{G} \), henc... | 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 |
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