Reports of the Academy of Sciences of the USSR
1969. Volume 186, No. 5

UDC 513.831

MATHEMATICS

V. I. PONOMAREV

ON STAR-FINITE COVERINGS

AND OPEN-CLOSED SETS

(Presented by Academician P. S. Aleksandrov on 29 X 1968)

§ 1. The following results are known.

1. The property of strong paracompactness, generally speaking, is not preserved under perfect (even irreducible) mappings (see \((^{10})\)).

2. The property of strong paracompactness is preserved: a) under open perfect mappings (see \((^{2})\)); b) under quotient quasi-monotone mappings (see \((^{3})\)).

3. A space \(Y\) with the first axiom of countability which is a quotient quasi-monotone image of a strongly metrizable space is itself strongly metrizable \((^{8})\).

It turns out that in the results listed above what is important is not that the mapping \(f\) is open or quasi-monotone, but that it carries every open-closed set into an open-closed set. For us the following will be fundamental.

Definition. A mapping \(f : X \to Y\) will be called a \(\Lambda\)-mapping*, if two conditions are simultaneously fulfilled: a) the image of every open-closed set is an open-closed set; b) for each point \(y \in Y\) and every covering \(\omega_y\) of the set \(f^{-1}y\) by open-closed sets in \(X\), there exists a finite subcovering.

Main Theorem 1. The image of a strongly paracompact (fully paracompact) space \(X\) under a \(\Lambda\)-mapping \(f : X \to Y\) is again a strongly paracompact (fully paracompact) space.

Main Theorem 2. Let \(f : X \to Y\) be a quotient \(\Lambda\)-mapping of a strongly metrizable space \(X\) onto a space \(Y\) with the first axiom of countability. Then \(Y\) is also strongly metrizable.

§ 2. For the definition of a component of a star-finite covering see \((^{3–5})\).

If \(\eta\) is some system of sets, then by \(\widetilde{\eta}\) we shall denote the body of this system, i.e. the union of all sets in \(\eta\). The bodies of different components of a star-countable covering \(\omega\) do not intersect and are open-closed sets of the space. If \(\tau_1, \ldots, \tau_s\) are any coverings of a space \(X\), then by \(\tau_{12\ldots s} = \tau_1 \wedge \cdots \wedge \tau_s\) we shall denote the covering consisting of sets that are intersections of one set from each of the coverings \(\tau_1, \tau_2, \ldots, \tau_s\). All mappings considered by us are continuous, and all spaces are regular. If \(f : X \to Y\) is a mapping and \(U \subseteq X\), then
\[ f^\# U = \mathscr{E}(y \in Y,\ f^{-1}y \subseteq U)=Y\setminus f(X\setminus U). \]

§ 3. Proposition 1. The following properties of a space \(X\) are equivalent: 1) \(X\) is strongly paracompact. 2) For every open covering \(\omega\) there exists a disjoint open-closed covering \(\Omega=\{\Gamma_\lambda\}\), such that there exist countable closed systems \(\xi(\Gamma_\lambda)\), \(\Gamma_\lambda=\widetilde{\xi}(\Gamma_\lambda)\), such that \(\xi=\bigcup_\lambda \xi(\Gamma_\lambda)\) is inscribed in \(\omega\).

* Every open perfect mapping, as well as every quotient quasi-monotone mapping, is a \(\Lambda\)-mapping. Therefore the results listed above follow automatically from our theorems.

** Theorem 2 is true if one requires only that \(Y\) be of point-countable type.

Proof of Main Theorem 1. Let \(\omega\) be an arbitrary open cover of the space \(Y\). Inscribe in it a cover \(\omega'\) so that the cover consisting of the closures of its elements is also inscribed in \(\omega\). Consider the cover
\[ f^{-1}\omega'=\{f^{-1}V,\ V\in\omega'\} \]
and inscribe in it a star-finite cover \(\gamma\). Consider the components \(\gamma_\alpha\) of the cover, the bodies \(\widetilde{\gamma}_\alpha\) of these components, and the disjoint open-and-closed cover
\[ \delta=\{\widetilde{\gamma}_\alpha\} \]
of the space. The cover
\[ f\delta=\{f\widetilde{\gamma}_\alpha,\ \widetilde{\gamma}_\alpha\in\delta\} \]
of the space \(Y\), generally speaking, is not disjoint, nor is it necessarily open-and-closed in \(Y\), because \(f\) is a \(\Lambda\)-mapping. For each \(y\in Y\) consider the system
\[ \delta y=\mathcal{E}(\widetilde{\gamma}_\alpha\in\delta,\ \widetilde{\gamma}_\alpha\cap f^{-1}y\ne\Lambda). \]
The system \(\delta y\) is finite and consists of open-and-closed sets. Therefore, from the fact that \(f\) is a \(\Lambda\)-mapping it follows that the set
\[ Wy=f^\#\delta y\cap\bigcap_{\widetilde{\gamma}_\alpha\in\delta y}\widetilde{\gamma}_\alpha \]
is open and closed in \(Y\). Further, it is easily proved that the open-and-closed cover
\[ \tau=\{Wy,\ y\in Y\} \]
of the space \(Y\) is disjoint and is inscribed in \(f\delta\). In addition, the equality
\[ f\widetilde{\gamma}_\alpha=\bigcup_{U\in\gamma_\alpha} fU = \bigcup_{U\in\gamma_\alpha}[fU]_Y \]
is true. Associate with each element \(W\in\tau\) some one element \(f\widetilde{\gamma}_{\alpha(W)}\in f\delta\) in which \(W\) is contained. Denote
\[ \xi(W)=\{\,W\cap [fU]_Y,\ U\in\gamma_{\alpha(W)}\,\},\qquad W\in\tau. \]
The system \(\xi(W)\) is countable, consists of closed sets, and \(W=\xi(W)\). Put
\[ \xi=\bigcup_{\in\tau}\xi(W). \]
The system \(\xi\) is inscribed in the cover \(f\gamma\), and hence also in \(\omega\), and is a closed star-countable cover. Moreover, it has the property that for every \(W\) from the disjoint open-and-closed cover \(\tau\) one necessarily has
\[ W=\widetilde{\xi}(W), \]
where \(\xi(W)\subset\xi\) is a countable subsystem. Thus condition 2 of Proposition 1 is fulfilled, and from it follows the strong paracompactness of the space \(Y\).

p. 4. Lemma 1 (Stone (9)). Let \(f:X\to Y\) be a quotient mapping and \(y_0\in Y\). Suppose that the first axiom of countability is satisfied at the point \(y_0\). Let \(\{\Gamma_n\}\), \(n=1,2,\ldots\), be a countable base at the point \(y_0\in Y\); let \(\{V_n\}\), \(n=1,2,\ldots\), be a countable system of open subsets of \(X\). Suppose the following conditions are satisfied:
\[ [\Gamma_{n+1}]\subseteq \Gamma_n,\quad n=1,2,\ldots;\qquad V_n\subseteq V_{n+1},\quad n=1,2,\ldots, \]
\[ \ldots,\quad f^{-1}y_0\subseteq \bigcup_{n=1}^{\infty}V_n. \]
Then there is an index \(n_0\) such that
\[ \Gamma_{n_0}\subseteq [fV_{n_0}]_Y. \]

From this lemma follows

Lemma 2. Let \(f:X\to Y\) be a quotient mapping onto a space with the first axiom of countability. Suppose that for each point \(y\in Y\) there is defined a countable finitely additive* system \(\Sigma(y)\) of open subsets of \(X\), forming a base in the subspace \(f^{-1}y\subseteq X\). Then the system
\[ I[f\Sigma(y)]=\{\,I[fU]_Y,\ U\in\Sigma(y)\,\} \]
forms a base at the point \(y\), and the system
\[ \bigcup_{y\in Y} I[f\Sigma(y)] \]
is a base in the whole space.

Proposition 2. For the strong metrizability of a space \(X\), it is sufficient and necessary that there exist a \(\sigma\)-star-countable base.

p. 5. Proof of Main Theorem 2. Let
\[ \sigma=\bigcup_{i=1}^{\infty}\omega_i \]
be a base of the strongly metrizable space \(X\), decomposing into a countable number of star-finite covers \(\omega_i\). Consider the components \(\omega_{i\alpha}\) of the covers \(\omega_i\), the bodies \(\widetilde{\omega}_{i\alpha}\) of these components; the disjoint open-and-closed covers
\[ \delta_i=\{\widetilde{\omega}_{i\alpha}\} \]
of the space \(X\), and the open-and-closed (because \(f\) is a \(\Lambda\)-mapping) covers
\[ f\delta_i=\{f\widetilde{\omega}_{i\alpha},\ \widetilde{\omega}_{i\alpha}\in\delta_i\} \]
of the space \(Y\). Also consider the finite (because \(f\) is a \(\Lambda\)-mapping) systems
\[ \delta_i y=\{\widetilde{\omega}_{i\alpha}\in\delta_i,\ f^{-1}y\cap\widetilde{\omega}_{i\alpha}\ne\Lambda\} \]
and the open-and-closed in \(Y\)

\[ \text{* The system }\Sigma(y)\text{ is finitely additive if, together with a finite collection }U_1,\ldots,U_s \]
\[ \text{of its elements, it also contains the set }\bigcup_i U_i\text{ (the union of these elements).} \]

(by virtue of the fact that \(f\) is a \(\Lambda\)-mapping) the sets

\[ W_y^i=f^\#\delta_i y\cap \bigcap_{\widetilde{\omega}_{ia}\in\delta_i y} f\omega_{ia}. \tag{1} \]

Just as in Theorem 1, we assert that \(\tau_i=\{W_y^i,\ y\in Y\}\) is a disjoint open-and-closed cover, inscribed in the open-and-closed cover \(f\delta_i\). Moreover, note that

\[ W_{y_1}^i=W_{y_2}^i,\quad \text{if } y_1\in W_{y_2}^i \text{ or } y_2\in W_{y_1}^i . \tag{2} \]

Now consider an arbitrary \(W_{y_0}^i\in\tau_i\) and

\[ \omega_i y_0=\bigcup_{\widetilde{\omega}_{ia}\in\delta_i y_0}\omega_{ia}. \tag{3} \]

Also consider the system \(\Sigma(\omega_i y_0)\), consisting of sets that are finite unions of sets from \(\omega_i y_0\). Note that \(\omega_i y_0\) is countable; hence the system \(\Sigma(\omega_i y_0)\) will also be countable. Finally, consider

\[ \xi(W_{y_0}^i)=\{W_{y_0}^i\cap I[fV],\ V\in\Sigma(\omega_i y_0)\}. \tag{4} \]

The system \(\xi(W_{y_0}^i)\) is also countable. From (2) and Lemma 1 there follows the equality

\[ \widetilde{\xi}(W_{y_0}^i)=W_{y_0}^i. \tag{5} \]

Now consider the systems

\[ \xi_i=\bigcup_{W^i\in\tau_i}\xi(W^i). \tag{6} \]

From equality (5), the disjointness of the open-and-closed cover \(\tau_i\), and the countability of \(\xi(W^i)\), it follows that \(\xi_i\) is a star-countable open cover of the space \(Y\). Let now \(i_1 i_2,\ldots,i_k\) be some finite sequence of natural numbers. For each point \(y_0\in Y\), consider the systems \((\omega_{i_1}y_0,\ldots,\omega_{i_k}y_0)\) and the system \(\Sigma(\omega_{i_1}y_0\cup\ldots\cup\omega_{i_k}y_0)\), consisting of all possible finite unions of sets of the system \(\omega_{i_1}y_0\cup\omega_{i_2}y_0\cup\ldots\cup\omega_{i_k}y_0\). Also consider the sets \(W_{y_0}^{i_1},\ldots,W_{y_0}^{i_k}\). Denote

\[ \xi(W^{i_1}y_0,\ldots,W^{i_k}y_0)=\{W^{i_1}y_0\cap\ldots\cap W^{i_k}y_0\cap I[fV],\ V\in\Sigma(\omega_{i_1}y_0\cup\ldots\cup\omega_{i_k}y_0)\}. \tag{7} \]

From equality (5) and the inclusion

\[ I[f(V^{i_1}\cup\ldots\cup V^{i_k})]\supset I[fV^{i_1}]\cap\ldots\cap I[fV^{i_k}] \tag{8} \]

there follows the equality

\[ \widetilde{\xi}(W^{i_1}y_0,\ldots,W^{i_k}y_0)=W^{i_1}y_0\cap\ldots\cap W^{i_k}y_0. \tag{9} \]

Denote

\[ \xi_{i_1 i_2\ldots i_k}=\bigcup\{\xi(W^{i_1},\ldots,W^{i_k}),\ W^{i_1}\in\tau_{i_1},\ldots,W^{i_k}\in\tau_{i_k}\}. \tag{10} \]

From equality (9), the countability of the systems \(\xi(W^{i_1},\ldots,W^{i_k})\), \(W^{i_1}\in\tau_{i_1},\ldots,W^{i_k}\in\tau_{i_k}\), and the disjointness of the cover \(\tau_{i_1 i_2\ldots i_k}=\tau_{i_1}\wedge\ldots\wedge\tau_{i_k}\), there follows the star-countability of the system \(\xi_{i_1 i_2\ldots i_k}\).

Consider

\[ \xi=\bigcup_{i_1 i_2\ldots i_k}\xi_{i_1 i_2\ldots i_k}. \tag{11} \]

The system \(\xi\) consists of open sets and is the union of a countable number of star-countable systems \(\xi_{i_1,i_2\ldots i_k}\). We shall prove that \(\xi\) is a base of the space \(Y\). Let \(y_0\in Y\) and let the neighborhood \(Oy_0\) be arbitrary. Consider

a countable system

\[ \omega y_0=\bigcup_{i=1}^{\infty}\omega_i y_0 \tag{12} \]

and the countable system \(\Sigma(\omega y_0)\), consisting of all possible unions of a finite number of elements from \(\omega y_0\). By Lemma 2, the system

\[ I[f\Sigma(\omega y_0)]=\{I[fV],\ V\in\Sigma(\omega y_0)\} \tag{13} \]

forms a base at the point \(y_0\). Therefore there will be such a \(V^*\in\Sigma(\omega y_0)\) that

\[ y_0\in I[fV^*]\subseteq Oy_0. \tag{14} \]

But the equality

\[ \Sigma(\omega y_0)=\bigcup_{i_1 i_2\ldots i_k}\Sigma(\omega_{i_1}y_0\cup\omega_{i_2}y_0\cup\ldots\cup\omega_{i_k}y_0) \tag{15} \]

is valid.

Consequently, there exists a sequence \(i_1,\ldots,i_s\) such that

\[ V^*\in\Sigma(\omega_{i_1}y_0\cup\omega_{i_2}y_0\cup\ldots\cup\omega_{i_s}y_0). \tag{16} \]

Then \(y_0\in W^{i_1}y_0\cap\ldots\cap W^{i_s}y_0\cap I[fV^*]=O^*y_0\ne\Lambda\) and \(O^*y_0\subseteq Oy_0\) (14). Moreover, by the definition of the system \(\xi(W^{i_1}y_0,\ldots,W^{i_s}y_0)\) (7), necessarily
\(O^*y_0\in\xi(W^{i_1}y_0,\ldots,W^{i_s}y_0)\subseteq\xi_{i_1 i_2\ldots i_s}\subseteq\xi\), whereby it has been proved that \(\xi\) is a base of the space \(Y\); here \(\sigma\) is star-countable, whence (by virtue of Proposition 2) the strong metrizability of the space \(Y\) follows. Everything is proved.

Faculty of Mechanics and Mathematics
Moscow State University
named after M. V. Lomonosov

Received
1 X 1968

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