UDC 513.83+519.54

M. M. CHOBAN

ON THE THEORY OF \(p\)-SPACES*

(Presented by Academician P. S. Aleksandrov on 6 III 1970)

A. V. Arkhangel’skii introduced and studied a new class of topological spaces, the class of \(p\)-spaces \((^1)\). Subsequently, an important role in the development of the theory of \(p\)-spaces was played by the works of A. V. Arkhangel’skii, V. I. Ponomarev, V. V. Filippov, the author, and others. In the present article mappings of \(p\)-spaces are studied. The work of N. N. Wicke \((^6)\) is also devoted to this question. All spaces are assumed to be completely regular, and mappings—single-valued and continuous, unless the contrary is explicitly stated. Proofs are omitted throughout.

§ 1. A mapping \(f: Z \to X\) of a space \(Z\) onto \(X\) is called inductively open and bicompact if there exists a subset \(Y \subseteq Z\) such that \(fY = X\) and the mapping \(f|Y\) is open and bicompact.

Theorem 1. Let \(f: Z \to X\) be an inductively open bicompact mapping of a paracompact space \(Z\) onto \(X\). Then the space \(X\) is weakly paracompact. If, in addition, \(Z\) is a \(p\)-space, then \(X\) is also a \(p\)-space.

A. V. Arkhangel’skii posed the following question: is every weakly paracompact space an open bicompact image of a paracompact space? This question is extremely difficult, and in such a formulation the existence of a solution seems almost incredible. Nevertheless, in a number of cases one can obtain a positive result.

Theorem 2. Let \(X\) be a regular weakly paracompact \(p\)-space. Then there exists a paracompact \(p\)-space \(Z\) which is mapped onto \(X\) inductively openly and bicompactly. If, in addition, the space \(X\) is complete in the sense of Čech, then \(Z\) is also complete in the sense of Čech.

Theorem 3. Let \(X\) be a weakly paracompact \(p\)-space. If the space \(X\) satisfies one of the following conditions: a) \(X\) is normal; b) \(X\) is locally normal; c) \(X\) is hereditarily weakly paracompact, then there exists a paracompact \(p\)-space \(Z\) which is mapped onto \(X\) openly and bicompactly. If, in addition, the space \(X\) is complete in the sense of Čech, then \(Z\) is also complete in the sense of Čech.

In proving Theorems 2 and 3, an important role is played by a number of special assertions and by the following proposition.

Proposition 1. Every \(F_\sigma\)-subset of a weakly paracompact space is a weakly paracompact subspace.

For locally bicompact spaces, generally speaking, a stronger result than Theorem 2 is valid.

Theorem 4. Every locally bicompact weakly paracompact space \(X\) is an open finite-fold image of some locally bicompact paracompact space \(Z\).

Theorem 1 underlies the proof of the following theorem.

* A \(T_1\)-space \(X\) is called a \(p\)-space if there exists a countable family \(\{\gamma_n \mid n = 1, 2, \ldots\}\) of open covers in \(\omega X\) (\(\omega X\) is the Wallman compactification of the space \(X\)) of the space \(X\) such that
\[ \bigcap_{n=1}^{\infty} \gamma_n x \subseteq X \]
for every point \(x \in X\), where
\[ \gamma_n x = \bigcup \{U \in \gamma_n \mid x \in U\}. \]
If, in addition, for every point \(x \in X\) and every natural number \(n\) there exists a natural number \(m\) such that \([\gamma_m x]_{\omega X} \subseteq \gamma_n x\), then \(X\) is called a strict \(p\)-space.

Theorem 5. The product of a countable number of regular weakly paracompact \(p\)-spaces is a weakly paracompact space.

Corollary 1. The product of a countable number of weakly paracompact Čech-complete spaces is a weakly paracompact Čech-complete space.

§ 2. Theorem 6. Let \(X\) be a \(T_2\)-space. Then the following conditions are equivalent: a) \(X\) is a space of point-countable type*; b) there exist a paracompact \(p\)-space \(Z\) and an open mapping \(f: Z \to X\) such that for every compact set \(F \subseteq X\) of countable character in \(X\) there exists a compact subset \(\Phi \subseteq Z\) for which \(f\Phi \supseteq F\).

We note that in \((^{6})\) it was proved that spaces of point-countable type, and only they, are open images of paracompact \(p\)-spaces.

Theorem 6 is a strengthening of the theorem from \((^{6})\). This strengthening is of important significance. In particular, it allows one to draw the following conclusion.

Corollary 2. Spaces of countable type, and only they, are open \(k\)-covering** images of paracompact \(p\)-spaces.

Theorem 7. Every Čech-complete space \(X\) is an open image of some paracompact Čech-complete space \(Z\).

§ 3. A mapping \(f: X \to Y\) is called regularly complete if there exists a Čech-complete space \(\widetilde X \supseteq X\) such that for every \(y \in Y\) the set \(f^{-1}y\) is closed in \(\widetilde X\).

Examples of regularly complete mappings: 1) compact mappings. 2) Continuous mappings of complete spaces. 3) Let \(f: X \to Y\) be a mapping of a metric space \((X,\rho)\) onto \(Y\). If for every point \(y \in Y\) the set \(f^{-1}y\) is complete with respect to the metric \(\rho\), then the mapping \(f\) is regularly complete.

Theorem 8. For every regular topological space \(X\) the following conditions are equivalent: a) \(X\) is an open regularly complete image of some \(p\)-space; b) \(X\) is an open regularly complete image of some paracompact \(p\)-space \(Z\).

Corollary 3. Every \(p\)-space \(X\) is an open regularly complete image of some paracompact \(p\)-space \(Z\).

It was proved in \((^{7})\) that open regularly complete mappings of a \(p\)-space are \(k\)-covering.

Theorem 9. Let \(f: X \to Y\) be an open regularly complete mapping of a \(p\)-space \(X\) onto a normal weakly paracompact space \(Y\). Then there exist a paracompact \(p\)-space \(Z\) and a mapping \(g: Z \to X\) of the space \(Z\) into \(X\) such that: 1) \(f(gZ) = Y\); 2) the mapping \(\varphi: Z \to Y\), where \(\varphi = f \circ g\), is open and compact; 3) the mapping \(g: Z \to gZ\) is open and compact.

Corollary 4. Let \(f: X \to Y\) be an open regularly complete mapping of a \(p\)-space \(X\) onto a normal weakly paracompact space \(Y\). Then there exists a weakly paracompact \(p\)-space \(S \subseteq X\) such that \(fS = Y\) and the mapping \(f|S\) is open and compact.

Remark 1. In all the theorems indicated it is possible to add the further assertion that the constructed paracompact \(p\)-space \(Z\) satisfies the following conditions: 1) the weight of the space \(Z\) is equal to the weight of the space \(X\); 2) the space \(Z\) is perfectly mapped onto some zero-dimensional (in the sense of \(\dim\)) metric space; 3) \(\dim Z \leq \max\{\dim F \mid F \text{ is a compact subset of the space } X\}\).

* A space \(X\) is called a space of point-countable (respectively, countable) type if for every point \(x \in X\) (respectively, compact set \(F \subseteq X\)) there exists a compact set \(\Phi\) of countable character in \(X\) such that \(x \in \Phi\) (respectively, \(F \subseteq \Phi\)).

** A mapping \(f: X \to Y\) is called \(k\)-covering if for every compact set \(F \subseteq Y\) there is a compact set \(\Phi \subseteq X\) for which \(f\Phi \supseteq F\).

§ 4. A mapping \(f:X\to Y\) is called uniformly complete,* if there exists a countable family \(\{\gamma_n\mid n=1,2,\ldots\}\) of open covers of the space \(X\) such that for every monotone sequence
\[ \{F_n\subset \Gamma\in\gamma_n\mid n=1,2,\ldots\} \]
we have
\[ \bigcap_{n=1}^{\infty}[F_n]\ne\varnothing, \]
as soon as \(F_1\cap f^{-1}y\ne\varnothing,\ldots, F_n\cap f^{-1}y=\varnothing,\ldots\) for some point \(y\in Y\). We note that every regular complete mapping is uniformly complete.

Theorem 10. Let \(\varphi:Y\overset{\text{onto}}{\longrightarrow}S\) and \(\psi:S\overset{\text{onto}}{\longrightarrow}X\) be open uniformly complete mappings. If \(Y\) is a \(p\)-space, then there exists a paracompact \(p\)-space \(Z\) that is mapped openly and regularly completely onto \(X\).

Theorem 11. Let \(f:X\to Y\) be an open uniformly complete mapping of a \(p\)-space \(X\) onto a weakly paracompact space \(Y\). Then \(Y\) is a \(p\)-space.

Remark 2. Theorem 12 is valid for \(Wp\)-spaces.

§ 5. Denote by \(A(X)\) the collection of all nonempty subsets of the space \(X\). Put \(F(X)=\{L\in A(X)\mid L\text{ is closed in }X\}\). A single-valued mapping \(\theta:X\to A(Y)\) is called a set-valued mapping of the space \(X\) into \(Y\). In what follows we shall consider only set-valued mappings.

A mapping \(\theta:X\to A(Y)\) of a space \(X\) into a metric space \((Y,\rho)\) is called a \(\pi\)-\(Y\)-mapping if, for every point \(x\in X\) and every open set \(U\supset \theta^{-1}(\theta x)\) in \(X\), we have
\[ \rho(\theta x,\,Y\setminus \theta^{\#}U)>0, \]
where
\[ \theta^{\#}U=\{y\in Y\mid \theta^{-1}y\subset U\}. \]
When the mapping \(\theta^{-1}\) is single-valued, we obtain the definition of V. I. Ponomarev (see \((^4)\)).

A system \(\mathcal L\) of subsets of a space \(X\) is called an \(N\)-system if, for every open set \(U\supset L\) in \(X\), where \(L\in\mathcal L\), there is an open set \(V\) in \(X\) such that
\[ L\subset V\subset [V]\subset U. \]

Theorem 12. Let \(\theta:X\to A(Y)\) be a lower semicontinuous \(\pi\)-\(Y\)-mapping of a \(T_1\)-space \(X\) into a metric space \((Y,\rho)\). Suppose, further, that there exists a space \(Z\) such that \(X\subset Z\subset \omega X\)* and, for every point \(x\in X\), the set \(\theta^{-1}\theta x\) is closed in \(Z\). If the family \(\{\theta^{-1}\theta x\mid x\in X\}\) is an \(N\)-system, then there exists a countable family \(\{\gamma_n\mid n=1,2,\ldots\}\) of open covers in \(Z\) of the space \(X\) such that
\[ \bigcap_{n=1}^{\infty}\gamma_n x\subset X \]
for every point \(x\in X\).

A mapping \(\theta:X\to A(Y)\) is called \(XY\)-bicompact if, for every point \(x\in X\), the set \(\theta^{-1}(\theta x)\) is bicompact.

Theorem 13. For every regular space \(X\) the following conditions are equivalent: 1) \(X\) is a strict \(p\)-space; 2) there exists an \(XY\)-bicompact lower semicontinuous \(\pi\)-\(Y\)-mapping \(\theta:X\to A(Y)\) into some metric space \((Y,\rho)\); 3) there exists an \(XY\)-bicompact lower semicontinuous \(\pi\)-\(Y\)-mapping \(\theta:X\to F(Y)\) into some complete zero-dimensional metric space \((Y,\rho)\).

§ 6. Propositions 1 and 2 of \((^8)\) establish a connection between single-valued and set-valued mappings. It is precisely these propositions that make it possible to translate a number of theorems for single-valued mappings into the language of set-valued mappings.

A mapping \(\theta:Y\to A(X)\) is called \(Y\)-perfect if the mapping \(\theta\) is upper semicontinuous and \(\theta y\) is bicompact for every point \(y\in Y\).

The theorem from \((^6)\) is equivalent to the following theorem.

* Uniformly complete mappings were studied in \((^5)\).

** A mapping \(\theta:X\to A(Y)\) is lower (upper) semicontinuous if, for every open (closed) set \(A\) in \(Y\), the set \(A^{-1}\) is open (closed) in \(X\), respectively.

*** The space \(\omega X\) may be replaced by any regular bicompact extension of the space \(X\).

Theorem 14. A \(T_2\)-space \(X\) is a space of point-countable type if and only if \(X\) is an open \(Y\)-complete image of some zero-dimensional metrizable space \(Y\).

A mapping \(\theta: Y \to A(X)\) is called \(k\)-covering if for every bicompactum \(\Phi \subset X\) there exists a bicompactum \(F \subseteq Y\) such that \(\theta F \supset \Phi\).

Corollary 2 makes it possible to draw the following conclusion.

Theorem 15. A \(T_2\)-space \(X\) is a space of countable type if and only if \(X\) is an open \(Y\)-complete \(k\)-covering image of some metrizable space \(Y\).

A mapping \(\theta: Y \to A(X)\) is \(X\)-bicompact if \(\theta^{-1}x\) is a bicompactum for every point \(x \in X\).

From Theorems 3 and 7, respectively, it follows:

Theorem 16. A normal space \(X\) is a weakly paracompact \(p\)-space if and only if \(X\) is an open \(Y\)-complete \(X\)-bicompact image of some (zero-dimensional) metrizable space \(Y\).

Theorem 17. Every Čech-complete space \(X\) is an open \(Y\)-complete image of some complete zero-dimensional metrizable space \(Y\).

Mechanical-Mathematical Faculty
of M. V. Lomonosov Moscow State University

Received
25 II 1970

REFERENCES

1. A. V. Arkhangel’skii, Matem. sborn., 67 (109), No. 1, 55 (1965).
2. A. V. Arkhangel’skii, Tr. Moskovsk. matem. obshch., 13, 3 (1965).
3. A. V. Arkhangel’skii, UMN, 21, no. 4, 133 (1966).
4. V. I. Ponomarev, Byull. Pol’sk. Akad. nauk, 8, No. 3, 127 (1960).
5. H. H. Wicke, J. M. Worrell, Duke Math. J., 34, No. 2, 255 (1967).
6. H. H. Wicke, Proc. Am. Math. Soc., 22, No. 1, 136 (1969).
7. M. M. Choban, DAN, 190, No. 2 (1970).
8. M. M. Choban, DAN, 182, No. 3, 514 (1968).