UDC 513.831

MATHEMATICS

B. A. PASYNKOV

ON SECTIONS OVER ZERO-DIMENSIONAL SUBSETS OF QUOTIENT SPACES OF LOCALLY BICOMPACT GROUPS

(Presented by Academician P. S. Aleksandrov, 13 IV 1967)

Reiter in (¹) proved that if in the quotient group \(G/H\) of a locally bicompact group \(G\) with the first axiom of countability one takes an arbitrary derived (i.e., containing no nonempty perfect subset) closed set \(A'\), then in \(G\) there exists a closed set \(A\) which is mapped onto \(A'\) homeomorphically by means of the natural mapping \(G \to G/H\). In the same paper (¹) Reiter posed the question of extending his result to arbitrary locally bicompact groups. It turns out that such an extension is possible.

Theorem 1. If \(G\) is a locally bicompact group and \(H\) is its closed subgroup, then for every paracompact zero-dimensional* set \(A' \subseteq G/H = X\) there is a set \(A \subseteq G\) which is mapped onto \(A'\) homeomorphically by means of the natural mapping \(p : G \to X\).

Remark. As \(A'\) one may take, for example, any countable subset of \(X\) or any zero-dimensional closed subset of \(X\).

If in the theorem \(\dim X < \infty\), then, by Mostert’s theorem (²) (see also (³)), the mapping \(p : G \to X\) is a locally trivial fibration. In view of the paracompactness and zero-dimensionality of \(A'\), the mapping \(p : p^{-1}(A') \to A'\) will already be a trivial fibration, whence the existence of the set \(A\) follows.

In the general case we shall need several lemmas.

Lemma 1. Let \(X\) be the limit of such an inverse spectrum

\[ S=\{X_{\alpha}, \mathfrak{F}^{\beta}_{\alpha}\},\quad \alpha \in \mathfrak{A}, \]

that: 1) the indices \(\alpha\) are all ordinal numbers less than a certain limit number \(\theta\); 2) the projections \(\mathfrak{F}^{\alpha+1}_{\alpha}\) are locally trivial fibrations; 3) for every limit number \(\beta\), the element of the spectrum \(X_{\beta}\) is the limit of the spectrum

\[ S_{\beta}=\{X_{\alpha}, \mathfrak{F}^{\alpha'}_{\alpha}\},\quad \alpha<\beta . \]

Then for every paracompact zero-dimensional set \(A' \subseteq X_{1}\) there exists a set \(A\) in \(X=X_{0}\) which is mapped onto \(A'\) homeomorphically by means of the projection \(\mathfrak{F}_{1}:X \to X_{1}\).

Proof. The set \(A \subseteq X\) is obtained as the limit of a spectrum \(\{A_{\alpha}, \mathfrak{F}^{\beta}_{\alpha}\}\), \(\alpha \in \mathfrak{A}\), for which \(A_{1}=A'\), \(A_{\alpha}\subseteq X_{\alpha}\), and the mappings \(\mathfrak{F}^{\beta}_{\alpha}: A_{\beta}\to A_{\alpha}\) are homeomorphisms. The sets \(A_{\alpha}\) are constructed by transfinite induction. (The construction of \(A_{\alpha+1}\) from \(A_{\alpha}\) uses condition 2) and the fact that the fibration \(\mathfrak{F}^{\alpha+1}_{\alpha} : (\mathfrak{F}^{\alpha+1}_{\alpha})^{-1}A_{\alpha}\to A_{\alpha}\) is trivial. The construction of \(A_{\beta}\) for a limit \(\beta\) uses condition 3).)

Lemma 2. If \(G\) is a locally bicompact projective-Lie (⁴) group and \(H\) is its closed subgroup, then the space of the group \(G\)

* By dimension is meant dimension defined by means of coverings.

is the limit of the spectrum \(S=\{X_\alpha,\delta_\alpha^\beta\}\), \(\alpha\in \mathfrak A\), satisfying the conditions of Lemma 1, and \(X_1=G/H\).

Proof. We construct for \(G\) an analogue of the Pontryagin–Lee series, beginning with the space \(G/H\). In \(G\) one can \((^3)\) choose a system of bicompact normal divisors \(G_\alpha\), indexed by all ordinal numbers \(\alpha\), \(2\leq \alpha\leq \omega_\tau\), where \(\tau\) is the weight of \(G\) at a point, such that: 1) \(G_{\alpha+1}\subset G_\alpha\), 2) \(G_\beta=\bigcap_{\alpha<\beta}G_\alpha\) for limit numbers \(\beta\); 3) every neighborhood of the identity contains at least one of the normal divisors \(G_\alpha\), in particular \(\bigcap_\alpha G_\alpha=e\); 4) the quotient groups \(G/G_2\) and \(G_\alpha/G_{\alpha+1}\) are Lee groups.

Denote the subgroups \(H\cap G_\alpha\) by \(H_\alpha\), and \(H\) by \(H_1\). Since for \(\beta>\alpha\) we have the inclusion \(H_\beta\subseteq H_\alpha\), the natural projection \(\delta_\alpha^\beta\) of the quotient space \(X_\beta=G/H_\beta\) onto the quotient space \(G/H_\alpha=X_\alpha\) is defined; moreover, for all \(\alpha\) the natural projections \(\delta_\alpha:G\to X_\alpha\) are defined. Since all subgroups \(H_\alpha\) for \(\alpha\geq 2\) are bicompact, all projections \(\delta_\alpha\) and \(\delta_\alpha^\beta\) for \(\alpha\geq 2\) are perfect.* Thus the spectrum \(S=\{X_\alpha,\delta_\alpha^\beta\}\), \(1\leq \alpha<\omega_\tau\), is defined.

Consider a limit number \(\beta\leq \omega_\tau\) (putting \(G_{\omega_\tau}=e\) and \(X_{\omega_\tau}=G\)). The space \(X_\beta\), for each \(\alpha\), \(1\leq \alpha<\beta\), has a continuous mapping \(\delta_\alpha^\beta\) onto the space \(X_\alpha\), and \(\delta_\alpha^{\alpha'}\cdot \delta_{\alpha'}^\beta=\delta_\alpha^\beta\) for \(\alpha'>\alpha\), i.e. there is defined a continuous mapping \(f_\beta\) of the space \(X_\beta\) onto the everywhere dense subset \(\bar X_\beta\) of the limit of the spectrum \(S_\beta=\{X_\alpha,\delta_\alpha^{\alpha'}\}\), \(\alpha<\beta\), satisfying the relations \(\delta_\alpha^\beta=\pi_\alpha\cdot f_\beta\) (where \(\pi_\alpha\) denotes the projection of the limit of the spectrum \(S_\beta\) onto the element of the spectrum \(X_\alpha\)). Since the mappings \(\delta_\alpha^\beta\) are bicompact mappings “onto” and \(\delta_\alpha^{\alpha'}\cdot\delta_{\alpha'}^\beta=\delta_\alpha^\beta\) for \(\alpha'>\alpha\), the mapping \(f_\beta\) is also a bicompact mapping “onto” (if \(\delta_\alpha^{\alpha'}(x_{\alpha'})=x_\alpha\), then \((\delta_{\alpha'}^\beta)^{-1}x_{\alpha'}\subseteq(\delta_\alpha^\beta)^{-1}x_\alpha\)). Since \(\delta_\alpha^\beta=\pi_\alpha\cdot f_\beta\) and the mapping \(\delta_\alpha^\beta\) is perfect, \(f_\beta\) will also be perfect. Finally, let \(x_1\ne x_2\in X_\beta\), i.e. \(x_1=g_1H_\beta\ne x_2=g_2H_\beta\). If \(\delta_\alpha^\beta(x_1)=\delta_\alpha^\beta(x_2)\), then \(g_1H_\alpha=g_2H_\alpha\), whence follows the existence of an index \(\alpha_0<\beta\) such that \(g_1H_{\alpha_0}\ne g_2H_{\alpha_0}\). (If this were not so, then
\[ g_1H_\beta = g_1\left(\bigcap_{\alpha<\beta}H_\alpha\right) = \bigcap_{\alpha<\beta}g_1H_\alpha = \bigcap_{\alpha<\beta}g_2H_\alpha = g_2\left(\bigcap_{\alpha<\beta}H_\alpha\right) = g_2H_\beta, \]
which contradicts the choice of \(x_1\) and \(x_2\).) Thus there exists an index \(\alpha_0\) for which \(\delta_{\alpha_0}^\beta(x_1)\ne \delta_{\alpha_0}^\beta(x_2)\). One-to-one-ness, and hence homeomorphism, of \(f_\beta\) is proved.

Finally, let us show that each projection \(\delta_\alpha^{\alpha+1}:X_{\alpha+1}\to X_\alpha\) is a locally trivial fibration; for this it suffices to show that any quotient group \(H_{\alpha+1}/H_\alpha\) is a manifold (\((^3)\), Theorem 14).

The quotient group \(H_1/H_2=H/H\cap G_2\) is homeomorphic to the image of the group \(H\) in the quotient group \(G/G_2\), i.e. to a closed subgroup of a Lee group, hence to a Lee group. Indeed, let us put in correspondence to the class \(h\cdot H_2=h(H\cap G_2)\in H_1/H_2\), \(h\in H=H_1\), the class \(f(hH_2)=hG_2=h\cdot H_2G_2\in G/G_2\). If \(h'\cdot H_2\ne h''\cdot H_2\), then \(h'\cdot G_2\ne h''\cdot G_2\), for otherwise \(h'g'=h''g''\), \(g'\) and \(g''\in G_2\), i.e. \((h'')^{-1}h'=g''\cdot(g')^{-1}\in H\cap G_2=H_2\), whence \(h'=h''(g''(g')^{-1})\in h''\cdot H_2\). Thus the mapping \(f\) is one-to-one. It is obviously a mapping “onto.” If the set \(F\) is closed in \(H_1/H_2\), then its inverse image \(\Phi=\Phi\cdot H_2=\Phi(H\cap G_2)\) is closed in \(H_1\), and then the set \(\Phi\cdot G_2\) is closed in \(G\) (by virtue of bicompactness of \(G_2\)), i.e. so is its image in \(G/G_2\), equal to \(f(F)\). The closedness of \(f\) is proved.

Let us prove the continuity of \(f\). Let the set \(F'\) be closed in the image of the subgroup \(H\) in the quotient group \(G/G_2\). Then the inverse image \(\widetilde\Phi\) of the set is closed in \(G\)

* That is, they are closed (the image of a closed set is closed) and bicompact (the inverse image of a point is bicompact) mappings.

\(F'\), and hence the set \(\Phi'=\widetilde{\Phi}\cap H\) is closed in \(H\). The set \(\Phi'\) coincides with the set \(\Phi'\cdot H_2\), since
\[ \Phi'=\widetilde{\Phi}\cap H\subseteq (\widetilde{\Phi}\cap H)\cdot H_2 =\Phi'\cdot H_2=(\widetilde{\Phi}\cap H)\cdot (G_2\cap H)\subseteq (\widetilde{\Phi}\cdot G_2)\cap H=\widetilde{\Phi}\cap H=\Phi' . \]
The image of the set \(\Phi'\) in \(H_1/H_2\) is closed and, by the one-to-one character of \(f\), coincides with the set \(f^{-1}(F')\). The homeomorphism \(f\) is proved.

The space of the factor group \(H_\alpha/H_{\alpha+1}\) for \(\alpha>1\) is bicompact and homeomorphic to the image of the group \(H_\alpha\) in the Lie group \(G_\alpha/G_{\alpha+1}\), i.e. the spaces \(H_\alpha/H_{\alpha+1}\) for \(\alpha>1\) are also manifolds. Thus the local triviality of the decompositions
\[ \omega_\alpha^{\alpha+1}:X_{\alpha+1}\to X_\alpha \]
in the case of a projective-Lie group \(G\) is proved, i.e. the lemma is also proved.

From Lemmas 1 and 2 the validity of the theorem for projective-Lie groups follows immediately.

If now the group \(G\) is arbitrary, then it always contains an open projective-Lie subgroup \(\Gamma\) \((^5)\), and the factor space \(X=G/H\) decomposes into a sum of pairwise disjoint open sets \(X_\nu\), whose inverse images \(p^{-1}(X_\nu)\) in \(G\) under the natural mapping \(p:G\to X\) have, for some \(a_\nu\in G\), the form \(\Gamma\cdot a_\nu\cdot H\) (see, for example, \((^3)\), Lemma 1). The mapping \(p\) of the set \(\Gamma\cdot a_\nu\subseteq \Gamma\cdot a_\nu\cdot H\) onto \(X_\nu\) can be identified \((^3)\), Lemma 1, with the natural mapping of the projective-Lie group \(\Gamma\) onto the factor space
\[ \Gamma/(\Gamma\cap a_\nu\cdot H\cdot a_\nu^{-1}), \]
i.e. everything has been reduced to the case of a projective-Lie group. The theorem is completely proved.

In the case of countable sets, Theorem 1 can be proved under more general assumptions. (All spaces considered are assumed to be completely regular.)

Definition. A space \(X\) has the almost everywhere first axiom of countability if \(X\) contains an everywhere dense subset of points of countable character in \(X\).

Theorem 2. Let an open mapping \(f:X\to Y\) be given such that every set \(f^{-1}(y)\), \(y\in Y\), has the almost everywhere first axiom of countability. Then for every \(\sigma\)-discrete (in itself) paracompact set \(N\subseteq Y\) there is a set \(M\subseteq X\), mapped onto \(N\) by \(f\) homeomorphically (in short, in \(X\) over \(N\) there exists a section).

Corollary 1. If an open mapping \(f:X\to Y\) of a paracompact \((^6)\) space \(X\) with the first axiom of countability (in particular, complete with the first axiom of countability or perfectly mapped onto a metrizable space with the first axiom of countability) is given, then for every \(\sigma\)-discrete (in itself) paracompact (for example, countable) set \(N\subseteq Y\), in \(X\) over \(N\) there exists a section.

It is useful to compare this corollary with Theorem 1 from \((^1)\).

Corollary 2. If \(G\) is a topological group, \(H'\) and \(H''\) are its closed subgroups, \(H'\supseteq H''\), and the factor spaces \(X'=G/H'\) and \(X''=G/H''\) are metrizable, then in \(X''\), over any \(\sigma\)-discrete set \(N\subseteq X'\), with respect to the natural projection \(p:X''\to X'\), there exists a section.

Corollary 3. If a group \(G\) is almost metrizable \((^7)\), and \(H'\) and \(H''\) are its closed subgroups, \(H'\supseteq H''\), and the factor space \(H'/H''\) is metrizable, then with respect to the natural mapping \(p:G/H''\to G/H'\), for every \(\sigma\)-discrete set closed in \(G/H'\) there exists a section in \(G/H''\).

Theorem 3. If a group \(G\) is almost metrizable and \(H\) is its closed subgroup, then with respect to the natural projection \(p:G\to G/H\), for every \(\sigma\)-discrete set \(N\) closed in \(G/H\), there exists a section in \(G\).

Instead of closedness of \(N\) one may require that \(N\) be a paracompact subset of type \(G_\delta\) in \(G/H\).

The proof of Theorem 3 is analogous to the proof of Theorem 1.

From Theorem 3 follows the validity of Theorem 1, however, only for a \(\sigma\)-discrete set \(A'\) closed in \(X\).

Let us also note that spaces \(X\) of point-countable type, all points of which are sets of type \(G_\delta\) in \(X\), satisfy the first axiom of countability; whence it follows

Theorem 4. The quotient space \(G/H\) of an almost metrizable group \(G\) is metrizable if and only if at least one of its points has type \(G_\delta\) in \(G/H\).

Moscow State University
named after M. V. Lomonosov

Received
22 III 1967

REFERENCES

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