MATHEMATICS

V. PONOMAREV

ON STRONGLY PARACOMPACT SPACES*

(Presented by Academician P. S. Aleksandrov, 23 XI 1961)

Let \(X\) be a normal space, and let \(\overline X\) be some extension of it. We shall say that \(X\) is paracompactly situated in \(\overline X\) if, whatever the neighborhood \(VX\) of the set \(X\) in the space \(\overline X\), there exists a smaller neighborhood \(V'X \subseteq VX\) which is a paracompactum.

Main theorem. In order that a normal space \(X\) be strongly paracompact, it is necessary and sufficient that it be paracompactly situated in its maximal bicompact extension \(\beta X\) (the Stone–Čech extension).

Preliminary remarks. \(1^\circ\). If \(\gamma=\{H_\alpha\}\) is some covering of the given space \(X\), then by \(\overline\gamma=\{[H_\alpha]\}\) we shall always denote the covering (of the same space \(X\)) whose elements are the closures \([H_\alpha]\) of the elements of the covering \(\gamma\).

\(2^\circ\). Let \(H\) be an arbitrary open set of the space \(X\); by \(OH\) we denote the largest open set in \(\beta X\) that gives, in intersection with \(X\), the set \(H\). As is known, sets of the form \(OH\) form a base of the space \(\beta X\).

Hence (and from the regularity of the space \(\beta X\)) it follows at once:

Lemma 1. If \(VX\) is an arbitrary neighborhood of the set \(X\) in the space \(\beta X\), then there exists a covering \(\gamma=\{\Gamma\}\) such that:

\(1^\circ\). \([\Gamma]_{\beta X} \subseteq VX\) for every \(\Gamma \in \gamma\).

\(2^\circ\). \(\Gamma = O(X \cap \Gamma)\).

The theorem formulated above is contained in the following two propositions:

I. If a completely regular space \(X\) is paracompactly situated in some of its bicompact extensions \(bX\), then \(X\) is strongly paracompact.

II. If \(X\) is a strongly paracompact space, then every neighborhood \(VX\) of the set \(X\) in the space \(\beta X\) contains a smaller neighborhood \(V'X \subseteq VX\) which is a strongly paracompact space**.

Proof of Proposition I. Let \(\omega=\{H\}\) be an arbitrary covering of the space \(X\). Put \(\Omega=\{OH\}\),

\[ VX=\bigcup_{H\in\omega} OH. \]

By hypothesis, there exists a paracompact neighborhood \(V'X \subseteq VX\). For each point \(x\in V'X\) we take such a neighborhood \(Ox\) that its closure \([Ox]_{\beta X}\) is contained in \(V'X\) and in some \(OH\in\Omega\). Into the covering formed by these neighborhoods we inscribe a locally finite covering \(\Omega_1\), and we take a covering \(\Omega'=\{U_\alpha\}\) of the same space \(V'X\) such that \(\overline{\Omega'}\) is

* As is known, a space \(X\) is called strongly paracompact (or star-paracompact) if into every one of its (open) coverings \(\omega\) one can inscribe a star-finite covering \(\omega'\). Yu. M. Smirnov proved \((^2)\) that in this definition star-finiteness can be replaced by star-countability.

** It may be noted (this will be proved later) that every paracompactum which is an open set in a bicompactum will be strongly paracompact.

combinatorially inscribed in \(\Omega_1\). Then \(\overline{\Omega}'\) is a locally finite system of bicompacts (inscribed in \(\Omega\) and covering \(V'X\)); this system is necessarily star-finite. Its elements, when intersected with \(X\), give a star-finite covering of the space \(X\), inscribed in \(\omega\), as was required to prove.

Proof of Proposition II.
The following is known.

Lemma 2. Every regular space \(R\) that is the body of a star-finite system of bicompacts \(\sigma=\{\Phi_\alpha\}\) (strongly paracompact spaces) is necessarily strongly paracompact.

This lemma may be proved, for example, as follows. Following P. S. Aleksandrov (see, for example, (1), Ch. 5, Sec. 10, or (2), Lemma 2), call a chain of elements of the given system of sets \(\sigma\) any finite sequence of the form \(\Phi_{\alpha_1},\ldots,\Phi_{\alpha_s}\), where \(\Phi_{\alpha_i}\cap\Phi_{\alpha_{i+1}}\ne\Lambda\). A system of sets is called connected if any two of its elements can be joined by a chain. Every system of sets decomposes into “components,” i.e. into maximal connected subsystems. If the system is star-finite, then its components are finite or countable subsystems. The system \(\sigma\) is a star-finite system of bicompacts; the components \(\sigma_\nu\) are countable subsystems whose bodies are disjoint open-and-closed sets \(R_\nu=\widetilde{\sigma}_\nu\), the body \(R\) of the whole system \(\sigma\). Let \(\gamma=\{H\}\) be an arbitrary open covering of the space \(R\); without loss of generality it may be assumed to be inscribed in the disjoint covering \(\Xi=\{R_\nu\}\) of the space \(R\). Then
\[ \gamma=\bigcup_\nu \gamma_\nu, \]
where \(\gamma_\nu\) consists of all \(H\in\gamma\) that intersect \(R_\nu\) (and, consequently, do not intersect any \(R_{\nu'}\), \(\nu'\ne\nu\)). We have
\[ R_\nu=\widetilde{\sigma}_\nu=\bigcup_{i=1}^{\infty}\Phi_i^\nu, \]
where the \(\Phi_i^\nu\) are all elements of the system \(\sigma\) forming the subsystem \(\sigma_\nu\).

For each \(\Phi_i^\nu\) choose a finite subsystem \(\gamma_i^\nu\) of the system \(\gamma_\nu\) covering the bicompact \(\Phi_i^\nu\); we obtain a countable covering
\[ \gamma_\nu'=\bigcup_i \gamma_i^\nu \]
of the set \(R_\nu\).

Since two elements belonging respectively to the systems \(\gamma_\nu'\) and \(\gamma_{\nu'}'\), with \(\nu'\ne\nu\), do not intersect, it follows that
\[ \gamma'=\bigcup_\nu \gamma_\nu' \]
is a star-countable subsystem of the system \(\gamma\), which is a covering of the whole space \(R\). Thus every open covering of the space \(R\) contains a star-countable one, which, by the already mentioned theorem of Yu. M. Smirnov, means that \(R\) is strongly paracompact.

We turn to the proof of Proposition II. Let \(X\) be strongly paracompact, and let \(VX\) be an arbitrary neighborhood of the set \(X\) in the space \(\beta X\). In view of Lemma 2 it is enough to construct a neighborhood \(V_1X\subseteq VX\) that is the body of a star-finite system of bicompacts. According to Lemma 1, take a covering \(\gamma=\{\Gamma_\nu\}\) of the set \(VX\) whose elements satisfy the conditions \(\Gamma_\nu=O(X\cap\Gamma_\nu)\), \([\Gamma_\nu]_{\beta X}\subseteq VX\). Take the covering \(X\gamma=\{H_\nu\}\) of the space \(X\), consisting of all elements \(H_\nu=\Gamma_\nu\cap X\). As a consequence of the strong paracompactness of the space \(X\), there exists a covering \(\omega=\{U_\alpha\}\) of the space \(X\) such that \(\omega=\{[U_\alpha]_X\}\) is star-finite and inscribed in \(X\gamma\). Take a covering \(\omega'=\{U'_\alpha\}\) of the space \(X\) such that \(\omega'=\{[U'_\alpha]_X\}\) is combinatorially inscribed in \(\omega\), so that for every \(\alpha\) we have
\[ U'_\alpha\subseteq[U'_\alpha]_X\subseteq U_\alpha\subseteq[U_\alpha]_X\subseteq H_\nu=X\cap\Gamma_\nu, \tag{1} \]
where \(\nu=\nu(\alpha)\).

Obviously, \(\overline{\omega}'\) is star-finite. By virtue of inclusion (1) and of the basic properties of the space \(\beta X\), we have
\[ [OU'_\alpha]_{\beta X}\subseteq OU_\alpha\subseteq[OU_\alpha]_{\beta X}\subseteq OH_\nu=\Gamma_\nu. \tag{2} \]

Put

\[ V_1X=\bigcup_\alpha OU_\alpha; \]

from the last inclusion (2) it follows that \(V_1X\subseteq VX\). Since for any two open subsets \(H_1\) and \(H_2\) of \(X\) one always has \(OH_1\cap OH_2=O(H_1\cap H_2)\), i.e., the operator \(O\) preserves the intersection scheme, the systems \(\Omega=\{OU_\alpha\}\), \(\Omega'=\{OU'_\alpha\}\) are star-finite. By virtue of (2), the system of bicompacts \(\overline{\Omega}'=\{[OU'_\alpha]_{\beta X}\}\) is also star-finite. Since \(X\) is everywhere dense in \(\beta X\), we have

\[ V_1X=\left[\bigcup_\alpha U'_\alpha\right]_{V_1X} \subseteq \left[\bigcup_\alpha OU'_\alpha\right]_{V_1X}. \]

The system \(\Omega\) is star-finite, and hence a fortiori locally finite in its body \(V_1X\); since the system \(\overline{\Omega}'\) is combinatorially inscribed in \(V_1X\), it too is locally finite in \(V_1X\); therefore

\[ \left[\bigcup_\alpha OU'_\alpha\right]_{V_1X} = \bigcup_\alpha [OU'_\alpha]_{V_1X}. \tag{3} \]

But (in view of (2) and the definition of the set \(V_1X\)) we have \([OU'_\alpha]_{V_1X}=[OU]_{\beta X}\), so that (3) is rewritten in the form

\[ V_1X=\bigcup_\alpha [OU'_\alpha]_{\beta X}, \]

which proves everything.

Remark. The following question remains open: can the maximal bicompact extension \(\beta X\) in the formulation of Theorem 1 be replaced by an arbitrary bicompact extension \(bX\) of the space \(X\)?

Moscow State University
named after M. V. Lomonosov

Received
12 XI 1961

CITED LITERATURE

¹ P. S. Aleksandrov, P. S. Uryson, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 31 (1950). ² Yu. M. Smirnov, Izv. AN SSSR, ser. matem., 20, 253 (1956).