UDC 513.831

MATHEMATICS

V. I. PONOMAREV

ON SPACES COABSOLUTE WITH METRIC SPACES

(Presented by Academician P. S. Aleksandrov on 14 V 1965)

Two topological spaces will be called coabsolute if their absolutes (see \((^{14,\,8-10,\,13})\)) are homeomorphic to one another. As was proved in \((^{14,\,8-10,\,13})\), the coabsoluteness of two spaces is necessary and sufficient for the existence of an irreducible, perfect (in general, many-valued) mapping of one of these spaces onto the other.

Definition 1. A system \(\Sigma\) of open sets of a space \(X\) is called dense in this space if every open \(H \subseteq X\) contains some \(U \in \Sigma\). The least cardinality of a dense system in \(X\) will be called the \(\pi\)-weight of the space \(X\).

Main theorem. In order that a space \(X\) admit a perfect irreducible single-valued mapping onto some metrizable space, it is necessary and sufficient that the space \(X\) be a paracompact \(p\)-space * and that in \(X\) there exist a dense system \(\Sigma\) of open sets decomposing into a countable sum*

\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i \]

of disjoint systems \(\Sigma_i\), each of which is locally finite in \(X\). If a paracompact \(p\)-space \(X\) admits a many-valued perfect irreducible mapping onto some metrizable space, then there exists a metrizable space \(Y\) (in general, another one) and a single-valued perfect irreducible mapping \(f:X\to Y\).

For the proof of this theorem we shall need some auxiliary notions and propositions, which are also of independent interest.

1. Product of mappings. Suppose mappings \(f_\alpha:X\to Y_\alpha\) are given from one and the same space \(X\) onto spaces \(Y_\alpha\). The mapping

\[ f:X\to Y\subseteq \prod_\alpha Y_\alpha, \]

which assigns to each point \(x\in X\) the point \(\{f_\alpha x\}\) of the product \(\prod_\alpha Y_\alpha\), is called the product of the mappings \(f_\alpha:X\to Y_\alpha\). By \(P_f\) we shall denote the collection of all distinguished subsets ** of the space \(X\) under the mapping \(f:X\to Y\). The following assertions hold:

Lemma 1. If \(f:X\to Y\) is the product of mappings \(f_\alpha:X\to Y_\alpha\), then \(f^{-1}fx=\bigcap_\alpha f_\alpha^{-1}f_\alpha x\) for all \(x\in X\) and \(P_f\supseteq \bigcup_\alpha P_{f_\alpha}\).

Lemma 2. Let \(\omega\) be some open covering of the space \(X\),

* Paracompacts that admit a perfect mapping onto a metrizable space were studied in detail by A. V. Arhangel’skii \((^{1,\,2})\) (see also \((^{11})\)). We shall call these spaces paracompact \(p\)-spaces. In particular, in \((^{1,\,2,\,11})\) intrinsic criteria are given for a space \(X\) to be a paracompact \(p\)-space. In the paper \((^{4})\) it is proved (in our terminology) that a paracompact space complete in the sense of Čech is a paracompact \(p\)-space.

** A set \(A\subseteq X\) is called distinguished under a mapping \(f:X\to Y\) if \(A=f^{-1}fA\).

and among the mappings \(f_\alpha:X\to Y_\alpha\) there is at least one \(\omega\)-mapping with respect to this \(\omega\). Then the product \(f:X\to Y\subseteq \prod_\alpha Y_\alpha\) will necessarily be an \(\omega\)-mapping with respect to the covering \(\omega\).

Lemma 3. Suppose that among the mappings \(f_\alpha:X\to Y_\alpha\) there is at least one perfect mapping. Then the product \(f:X\to Y\) of the mappings \(f_\alpha\) is also perfect.

Lemma 4*. Let \(X\) be a paracompact \(p\)-space, and let \(\omega\) be an arbitrary open covering of it. Then there exists a metrizable space \(Y_\omega\) and a perfect \(\omega\)-mapping \(f_\omega:X\to Y_\omega\).

Lemma 5. Let \(X\) be a paracompact \(p\)-space, and let \(U\subseteq X\) be an open subset of type \(F_\sigma\). Then there exists a metrizable space \(Y_U\) and a perfect mapping \(f_U:X\to Y_U\), under which the set \(U\) is distinguished, i.e. \(U=f^{-1}fU\).

n. 2. Dense systems of open sets.

Lemma 6. Let \(f:X\to Y\) be a one-to-one perfect irreducible mapping, and let \(\Sigma=\{U\}\) be a system of open sets dense in \(X\). Then the system** \(f^\#\Sigma=\{f^\#U\}\) of open sets in \(Y\) is dense in \(Y\).

Lemma 7. Let \(f:X\to Y\) be a one-to-one perfect irreducible mapping, and let \(\Sigma=\{V\}\) be a system of open sets dense in \(Y\). Then the system \(f^{-1}\Sigma=\{f^{-1}V\}\) is dense in \(X\).

Lemma 8. Let \(f:X\to Y\) be a many-valued (in particular, one-valued) perfect irreducible mapping (see (9)). Then the \(\pi\)-weight of the space \(X\) is equal to the \(\pi\)-weight of the space \(Y\).

n. 3. Locally finite systems.

Lemma 9. Let \(\Sigma=\{U_\alpha\}\) be a locally finite in \(X\) system of open sets of type \(F_\sigma\). Then the open set
\[ \widetilde{\Sigma}=\bigcup_\alpha U_\alpha \]
—the body of the system \(\Sigma\)—also has type \(F_\sigma\).

Lemma 10. Let \(\Sigma\) be a system of open sets dense in \(X\) and \(\sigma\)-locally finite*** (in all of \(X\)). Then for every open set \(U\subseteq X\) there is an open set \(U'\subseteq U\), everywhere dense in \(U\), of type \(F_\sigma\).

Lemma 11. If in the space \(X\) there is a system \(\Sigma\) of open sets that is dense and \(\sigma\)-locally finite (in all of \(X\)), then there is a dense \(\sigma\)-locally finite (also in all of \(X\)) system \(\Sigma'\) of open sets of type \(F_\sigma\).

Lemma 12. Let \(f:X\to Y\) be a one-to-one perfect irreducible mapping, and let \(\Sigma=\{U\}\) be a locally finite in \(X\) system of open sets; then the system \(f^\#\Sigma=\{f^\#U\}\) of open nonempty sets is locally finite in \(Y\).

n. 4. Construction of mappings.

Basic Lemma 13. Let \(\Sigma=\{U_\alpha\}\) be a disjoint locally finite in \(X\) system of open sets of type \(F_\sigma\). Then there exists a metrizable space \(Y_\Sigma\) and a perfect mapping \(f_\Sigma:X\to Y_\Sigma\), under which all sets \(U_\alpha\in\Sigma\) are distinguished.

Proof. By Lemma 9 the open set \(\widetilde{\Sigma}\) has type \(F_\sigma\), and the closed set \(\Phi=X\setminus\widetilde{\Sigma}\) has type \(G_\delta\). Let
\[ \Phi=\bigcap_{i=1}^{\infty}\Gamma_i . \]
By \(\Omega_i\) denote the covering of the whole space \(X\), consisting of all

* The proof of this lemma is contained in (³).

** In order that a one-to-one mapping \(f:X\to Y\) be simultaneously closed and irreducible, it is necessary and sufficient that for every open \(U\subseteq X\) the set
\[ f^\#U=\mathscr{E}(y\in Y,\ f^{-1}y\subseteq U) \]
be nonempty and open in \(Y\) (proved in (⁹, ¹²)).

*** A system \(\Sigma\) is called \(\sigma\)-locally finite in \(X\) if
\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i, \]
where the \(\Sigma_i\) are locally finite in \(X\). The systems \(\Sigma_i\) need not be coverings of the whole space \(X\).

sets \(U_\alpha \in \Sigma\) and also the sets \(\Gamma_i\). By Lemma 4 there exists a metrizable space \(Y_i\) and a perfect \(\Omega_i\)-map \(f_i: X \to Y_i\).

Consider the product \(f: X \to Y \subseteq \prod_{i=1}^{\infty} Y_i\) of the mappings \(f_i: X \to Y_i\). By Lemma 2 the mapping \(f\) is an \(\Omega_i\)-map for every \(i\), and by Lemma 3 it is perfect. It remains now to prove that every \(U_\alpha \in \Sigma\) is distinguished for the mapping \(f\). Let \(x_0 \in U_\alpha\) be arbitrary. We must prove that \(f^{-1} f x_0 \subseteq U_\alpha\). Choose \(i_0\) such that \(x_0 \in \Gamma_{i_0}\). Since \(f\) is an \(\Omega_{i_0}\)-map, there exists \(U \in \Omega_{i_0}\) such that \(f^{-1} f x_0 \subseteq U\), and this \(U\) is necessarily one of the sets \(U_\alpha \in \Sigma\) (here the disjointness of the system \(\Sigma\) is essential). We have \(f^{-1} f x_0 \subseteq U_\alpha\), as was required to prove.

Main Lemma 14. Let in a paracompact \(p\)-space \(X\) there be a system \(\Sigma\) of open sets of type \(F_\sigma\), decomposing into a countable sum
\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i \]
of disjoint systems \(\Sigma_i\), each of which is locally finite in \(X\). Then there exists a metrizable space \(Y_\Sigma\) and a perfect mapping \(f_\Sigma: X \to Y_\Sigma\) under which all sets \(U \in \Sigma\) are distinguished.

Proof. By Lemma 13, for each \(\Sigma_i\) there exists a metrizable space \(Y_i\) and a perfect mapping \(f_i: X \to Y_i\), under which every set \(U \in \Sigma_i\) is distinguished. Consider the product \(f: X \to Y\) of the mappings \(f_i: X \to Y_i\). The space \(Y\) is metrizable, and the mapping \(f\) is perfect by Lemma 3. Further, by Lemma 1, all sets \(U \in \Sigma\) are distinguished. The lemma is proved.

5. Proof of the Main Theorem. a) Let \(X\) be a paracompact \(p\)-space, and let \(\Sigma=\{U_\alpha\}\) be a dense system in \(X\) of open sets decomposing into a countable sum
\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i \]
of disjoint systems \(\Sigma_i\) locally finite in \(X\). By Lemma 10, in \(X\) there exists a dense system \(\Sigma'=\{U'_\alpha\}\) of open sets \(U'_\alpha\) of type \(F_\sigma\), with each \(U'_\alpha\) everywhere dense in \(U_\alpha\) and \(U'_\alpha \subseteq U_\alpha\). Denote by \(\Sigma'_i\) for each \(\Sigma_i=\{U^i_\lambda\}\subseteq\Sigma\) the collection of all \(U^{\prime i}_\lambda\). We obtain
\[ \Sigma'=\bigcup_{i=1}^{\infty}\Sigma'_i, \]
and each system \(\Sigma'_i\) is disjoint and locally finite in \(X\). By Lemma 14, consider a metrizable space \(Y\) and a perfect mapping \(f: X \to Y\), under which all \(U^{\prime i}_\alpha \in \Sigma'\) are distinguished. We shall prove that the mapping \(f\) is irreducible. Indeed, it is enough to prove that in every open set \(\Gamma \subseteq X\) there is contained the complete preimage \(f^{-1}y\) of some point \(y \in Y\). But the system \(\Sigma'\) is dense in \(X\). Therefore there exists \(U'_\alpha \in \Sigma'\) such that \(U'_\alpha \subseteq \Gamma\), and the set \(U'_\alpha\) is distinguished for the mapping \(f\).

b) Let the space \(X\) admit a one-to-one perfect mapping \(f: X \to Y\) onto some metrizable space \(Y\). By Bing’s metrization criterion (7), in \(Y\) there exists a \(\sigma\)-discrete base \(\Sigma\),
\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i, \]
where the \(\Sigma_i\) are discrete in \(Y\). Every base of a space is necessarily a dense system. Therefore, by Lemma 7, the system \(f^{-1}\Sigma\) is dense in \(X\); moreover, the systems \(f^{-1}\Sigma_i\) are disjoint and locally finite in \(X\), and all \(U \in f^{-1}\Sigma\) even have type \(F_\sigma\).

c) Let the paracompact \(p\)-space \(X\) admit a many-to-one perfect irreducible mapping \(f: X \to Y\) onto some metrizable space \(Y\). Then there exist a space \(Z\) and one-to-one perfect irreducible mappings \(p_X: Z \to X\) and \(p_Y: Z \to Y\) such that
\[ f=p_Y^{\#}p_X^{-1} \]
(see \((8^\circ)\)). Consider in \(Y\) a \(\sigma\)-discrete base \(\Sigma=\{V\}\). Then, by Lemmas 6 and 7, the system \(p_X^{\#}p_Y^{-1}\Sigma\) will be a dense system of open sets of the space \(X\); moreover, if
\[ \Sigma=\bigcup_{i=1}^{\infty}\Sigma_i, \]

\(\Sigma_i\) are discrete, then \(p_X^{\#}p_Y^{-1}\Sigma=\bigcup_{i=1}^{\infty}p_X^{\#}p_Y^{-1}\Sigma_i\), and the \(p_X^{\#}p_Y^{-1}\Sigma_i\) are disjoint and locally finite in \(X\) (by Lemma 12). Thus, in the space \(X\) there exists a dense system \(p_X^{\#}p_Y^{-1}\Sigma\), decomposing into a countable sum of disjoint systems locally finite in \(X\). And then, as has already been proved, there exists a metrizable space \(Y'\) and a one-to-one perfect irreducible mapping \(g:X\to Y'\). The theorem is completely proved.

§ 6. Consequences.

Theorem 2. Let \(X\) be a bicompactum. Then, in order that \(X\) admit a one-to-one irreducible mapping onto some compactum, it is necessary and sufficient that the \(\pi\)-weight of this bicompactum be countable. If a bicompactum \(X\) admits a many-valued irreducible mapping onto some compactum, then it also admits a one-to-one irreducible mapping onto some (generally speaking, different) compactum.

Theorem 3. In order that a completely normal bicompactum \(X\) admit an irreducible mapping onto a compactum, it is necessary and sufficient that it contain an everywhere dense countable set.

Theorem 4. In order that a bicompactum be coabsolute with the Cantor perfect set, it is necessary and sufficient that it have no isolated points and have countable \(\pi\)-weight.

Theorem 5. In order that a space \(X\) admit a one-to-one perfect irreducible mapping onto some metric space with complete metric, it is necessary and sufficient that \(X\) be a Čech-complete paracompact space and have a dense system \(\Sigma\) of open sets decomposing into a countable sum of disjoint and locally finite systems.

If a space \(X\) admits a many-valued perfect irreducible mapping onto some metrizable space with complete metric, then \(X\) is a Čech-complete paracompact space and there exists a one-to-one perfect irreducible mapping of it onto some metrizable space with complete metric.

R. Engelking and A. Pełczyński proved \((^6)\) that a dyadic bicompactum admitting a one-to-one irreducible mapping onto some compactum is metrizable. We shall supplement this result with the following propositions, which follow easily from our main theorem:

Theorem 6. If a dyadic bicompactum is coabsolute with some compactum, then it is necessarily metrizable.

Theorem 7. Let \(X\) be a dyadic bicompactum, and let \(X_0\) be its everywhere dense subset coabsolute with some metric space \(Y_0\). Then \(Y_0\) and \(X\) (and, of course, \(X_0\subseteq X\)) have a countable base.

From this there immediately follows a theorem of B. Efimov \((^5)\): all dyadic bicompact extensions of a space with a countable base are compacta.

Moscow State University
named after M. V. Lomonosov

Received
21 I 1965

CITED LITERATURE

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