UDC 519.50+519.54

MATHEMATICS

M. CHOBAN

ON THE BEHAVIOR OF METRIZABILITY UNDER QUOTIENT \(s\)-MAPPINGS*

(Presented by Academician P. S. Aleksandrov on May 20, 1965)

The present paper is devoted to the study of quotient spaces of metrizable spaces. Its main results are Theorems 1 and 2.

Theorem 1. Let \(f:X\to Y\) be a quotient \(s\)-mapping of a metric space \(X\) onto a regular space \(Y\) with the first axiom of countability. If the space \(Y\) is separable,** then it is also metrizable.

Proof. Let \(\Phi=\{y_1,\ldots,y_n,\ldots\}\) be some countable everywhere dense subset of \(Y\). Put \(X_1=[f^{-1}\Phi]\) and \(f_1=f|X_1\). It is possible to verify that \(fX_1=Y\). By virtue of the separability of the mapping \(f\), the subspace \(X_1\) has a countable base. Let \(\omega=\{U_i\mid i=1,\ldots\}\) be some such base. Suppose that the union of any finite number of elements of \(\omega\) is also contained in \(\omega\). We shall prove that the system

\[ \gamma=\{\operatorname{Int}[f_1U_i]\mid U_i\in\omega\} \]

forms a base in \(Y\). Take an arbitrary point \(y_0\in Y\) and an arbitrary neighborhood \(Oy_0\ni y_0\). Consider a neighborhood \(O_1y_0\) such that \(y_0\in O_1y_0\subset[O_1y_0]\subset Oy_0\). In \(\omega\) there is a sequence \(\{U_{i_k}\mid k=1,\ldots\}\) of sets open in \(X_1\) such that: 1) \(U_{i_k}\subset f_1^{-1}O_1y_0\),

\[ k=1,2,\ldots;\quad 2)\ U_{i_{k+1}}\supset U_{i_k},\ k=1,2,\ldots;\quad 3)\ f_1^{-1}y_0\subset\bigcup_{k=1}^{\infty}U_{i_k}. \]

At the point \(y_0\) take a base \(\{W_k\mid k=1,2,\ldots\}\) such that \(W_k\subset O_1y_0\) and \(W_{k+1}\subset W_k,\ k=1,\ldots\). We shall prove that there is a number \(k_0\) such that \([f_1U_{i_{k_0}}]\supset W_{k_0}\). Suppose the contrary; then \((W_k\setminus [f_1U_{i_k}])\cap\Phi\ne\varnothing\), where \(k=1,2,\ldots\). Construct a sequence \(P=\{y_{n_k}\mid y_{n_k}\in (W_k\setminus [f_1U_{i_k}])\cap\Phi\}\) which converges to the point \(y_0\). From the fact that \(y_{n_k}\in W_k\setminus f_1U_{i_k}\) and \(U_{i_1}\subset\cdots\subset U_{i_k}\subset\cdots\), it follows that each set \(U_{i_k}\) intersects only finitely many sets

\[ f_1^{-1}y_{n_{j_1}},\ldots,f_1^{-1}y_{n_{j_{s(k)}}} \]

from \(f_1^{-1}P\). Hence the set

\[ \widetilde U_{i_k}=U_{i_k}\setminus f_1^{-1}P = U_{i_k}\setminus\bigcup_{j=1}^{s(k)} f_1^{-1}y_{n_{j_i}} \]

is open in \(X_1\). From the fact that \(f_1^{-1}P\cap f_1^{-1}y_0=\varnothing\) and

\[ \bigcup_{k=1}^{\infty}U_{i_k}\supset f_1^{-1}y_0, \]

it follows that

\[ \bigcup_{k=1}^{\infty}\widetilde U_{i_k}\supset f_1^{-1}y_0 \quad\text{and}\quad \bigcup_{k=1}^{\infty}\widetilde U_{i_k}\cap f_1^{-1}P=\varnothing. \]

We have obtained that not a single point of \(f_1^{-1}y_0\) is contained in \([f_1^{-1}P]_{X_1}\). But, since \(X_1=[f^{-1}\Phi]_X\), \(P\subset\Phi\), and by virtue of the quotient character of the mapping \(f\), we obtain that

\[ [f_1^{-1}P]_{X_1}\cap f_1^{-1}y_0\ne\varnothing, \]

a contradiction. Hence there is a number \(k_0\) such that \([f_1U_{i_{k_0}}]\supset W_{k_0}\), and this condition is equivalent to

\[ \operatorname{Int}[f_1U_{i_{k_0}}]\ni y_0. \]

From the fact that \(U_{i_{k_0}}\subset f_1^{-1}O_1y_0\) and

* A mapping \(f:X\to Y\) is called an \(s\)-mapping (or separable) if for every point \(y\in Y\) the set \(f^{-1}y\) is a subspace with a countable base.

** A space \(Y\) is separable if it contains some everywhere dense countable set.

\([O_1y_0]\subset O y_0\), it follows that \(\operatorname{Int}[f_1 U_{i_{k_0}}]\subset O y_0\). This completes the proof of Theorem 1.

Corollary (Stone). Let \(f:X\to Y\) be a quotient mapping of a separable metric space \(X\) onto a regular space \(Y\) with the first axiom of countability. Then \(Y\) is a separable metrizable space.

In Theorem 1 one cannot discard the assumption that the space \(Y\) is separable, since from results of V. Ponomarev it follows that every topological space with a point-countable base (among them there are also nonmetrizable ones) is an open \(s\)-image of some strongly paracompact metric space (see \((^6)\)). Since there exist nonmetrizable spaces that are closed images of metric spaces with a countable base, the requirement of the first axiom of countability is necessary. Thus Theorem 1 may be regarded as a final result. For the same reasons Theorem 2 should also be recognized as final. Theorem 2 is a strengthening of Stone’s well-known theorem on open \(s\)-mappings and of one result of A. Arhangel’skii.

A. Arhangel’skii has recently introduced the notion of a space of point-countable type \((^3)\).

Definition. A space \(X\) is called a space of point-countable type if an arbitrary point \(x\in X\) is contained in some bicompact \(F\subseteq X\) whose character is countable.*

Theorem 2. Let \(f:X\to Y\) be a quotient \(s\)-mapping of a locally separable metric space \(X\) onto a regular weakly paracompact space \(Y\). If \(Y\) is a space of point-countable type, then it is metrizable and satisfies the local second axiom of countability.**

On the way to the proof of Theorem 2 we shall obtain a number of auxiliary assertions which may perhaps be of independent interest. In parallel, a theorem on quotient uniform \(s\)-mappings is proved.

Lemma. (A. Arhangel’skii). Let \(f:X\to Y\) be a quotient mapping of a topological space \(X\) onto a bicompactum \(Y\). For every open cover \(\eta=\{U_\alpha\}\) of the space \(X\) there exists in \(Y\) such a finite set of points
\(\Phi=\{y_1,\ldots,y_n\}\) that, whatever the point \(y\in Y\), for some point \(y'\in\Phi\) and some element \(U_{\alpha'}\in\eta\),
\(f_y^{-1}\cap U_{\alpha'}\ne\varnothing\) and \(f^{-1}y'\cap U_{\alpha'}\ne\varnothing\).

Proof. Suppose the contrary. Let \(\Phi_1=\{y_1\}\), where \(y_1\) is some point of \(Y\). There is a point \(y_2\in Y\) such that the condition
\(\eta f^{-1}\Phi_1\cap f^{-1}y_2=\varnothing\) is satisfied. Put \(\Phi_2=\{y_1,y_2\}\). Suppose the set \(\Phi_k=\{y_1,\ldots,y_k\}\) has been constructed. By the assumption, there is a point \(y_{k+1}\in Y\) such that
\(\eta f^{-1}\Phi\cap f^{-1}y_{k+1}=\varnothing\). Put
\(\Phi_{k+1}=\Phi_k\cup y_{k+1}=\{y_1,\ldots,y_k,y_{k+1}\}\). Let us have constructed the sets \(\Phi_1,\ldots,\Phi_k,\ldots\), \(\Phi_k\subset \Phi_{k+1}\), where \(k=1,2,\ldots\). Let
\(\Phi=\bigcup_{k=1}^{\infty}\Phi_k\). For any two points \(y',y''\in\Phi\) we have

\[ \eta f^{-1}y'\cap f^{-1}y''=\varnothing,\qquad \eta f^{-1}y''\cap f^{-1}y'=\varnothing. \tag{A} \]

* A bicompactum \(F\subset X\) is called a bicompactum of countable character (in \(X\)) if there exists a countable system of open sets \(\varphi=\{U_n\}\) such that for every \(U\supset F\) there is \(U_n\in\varphi\) for which \(F\subset U_n\subseteq U\). The spaces of point-countable type include all metric spaces, all locally bicompact spaces, and even all spaces complete in the sense of Čech.

** Earlier A. Arhangel’skii proved this proposition for the case of a locally compact space \(Y\) under the additional assumption that the mapping \(f\) is pseudo-open.

\(Y\) is bicompact. There is a point \(y_0 \in Y\) such that \(y_0 \in [\Phi]\). Put \(\Phi_0=\Phi \setminus y_0\). \(\Phi_0\) is not closed in \(Y\); therefore, by virtue of the quotient nature of the mapping, the set \(f^{-1}\Phi_0\) is not closed in \(X\), i.e., there is a point \(x_0 \in [f^{-1}\Phi_0]\setminus f^{-1}\Phi_0\). Take some \(U_{\alpha'} \in \eta\) such that \(x_0 \in U_{\alpha'}\). From condition (A) it follows that \(U_{\alpha'}\) meets no more than one \(f^{-1}y_{k_0}\). Hence the open set
\[ \Gamma=U_{\alpha'}\cap (X\setminus f^{-1}y_{k_0}) \]
does not meet \(f^{-1}\Phi_0\), and this means that \(x_0 \notin [f^{-1}\Phi_0]\), a contradiction. Lemma 2 is completely proved.

Theorem 3. Let \(f:X\to Y\) be a quotient uniform \(s\)-mapping of a metrizable space \(X\) onto a bicompact \(Y\). Then the space \(Y\) is metrizable.

Proof. Take the system
\[ \left\{\gamma_n=\{U_\alpha^n\mid \operatorname{diam} U_\alpha^n<\frac1n\}\right\} \]
of locally finite coverings which forms a base in \(X\). Take some \(\gamma_n\). There is a set
\[ \Phi_n=\{y_1^n,\ldots,y_{k(n)}^n\} \]
satisfying the conditions of A. Arhangel′skii’s lemma. Take the system
\[ \bar\gamma_n=\{U_\alpha^n\mid U_\alpha^n\cap f^{-1}\Phi_n\ne \varnothing\}. \]
Since \(f\) is an \(s\)-mapping, the system \(\bar\gamma_n\) is countable. We shall show that the system
\[ \Omega=\bigcup_{n=1}^{\infty} f\bar\gamma_n \]
is a network in the space \(Y\).

Take arbitrary \(y_0\) and \(Oy_0\) from \(Y\). Since \(f\) is uniform, we have
\[ \rho(f^{-1}y_0, X\setminus f^{-1}Oy_0)=r>0 \]
(see (6)). Choose an index \(n_0\) such that \(1/n_0<r/2\). By the condition, there exist \(y'\in \Phi_{n_0}\) and \(U_{\alpha'}^{n_0}\in \bar\gamma_{n_0}\) such that
\[ f^{-1}y_0\cap U_{\alpha'}^{n_0}\ne \varnothing. \]
Since
\[ \operatorname{diam} U_{\alpha'}^{n_0}<1/n_0<r/2, \]
we have
\[ U_{\alpha'}^{n_0}\subset f^{-1}Oy_0. \]
Then
\[ y_0\in fU_{\alpha'}^{n_0}\subset Oy_0, \]
and this means that \(\Omega\) is a countable network in the space \(Y\), and on the basis of Theorem 1 from (2), Theorem 3 is proved.

Lemma 1. Let \(f:X\to Y\) be a quotient \(s\)-mapping of a locally separable metrizable space \(X\) onto a bicompact \(Y\). Then the space \(Y\) is metrizable.

Proof. Since \(X\) locally satisfies the second axiom of countability, it decomposes into some number of discrete sets
\[ \omega=\{M_\lambda\mid \lambda\in \theta\}, \]
where each \(M_\lambda\) has a countable base. The system \(\omega=\{M_\lambda\}\) forms a covering of the space \(X\). On the basis of A. Arhangel′skii’s lemma, there is a finite set of points
\[ \Phi=\{y_1,\ldots,y_n\} \]
such that
\[ f(\omega f^{-1}\Phi)=Y. \]
Denote
\[ X_1=\omega f^{-1}\Phi \]
and \(f_1=f\mid X_1\). Since \(f\) is an \(s\)-mapping, \(X_1\) has a countable base. By the continuity of the mapping \(f_1\), the space \(Y\) has a countable network and, consequently, is metrizable (see Theorem 1 from (2)).

Lemma 2*. If a bicompact \(\Phi\) has countable character in a \(T_2\)-space \(X\), and a point \(x_0\in \Phi\subset X\) has countable character in the subspace \(\Phi\), then the point \(x_0\) has countable character also in the space \(X\).

Proof. Let
\[ \varphi=\{U_n\mid n=1,\ldots\} \]
be a countable system of open sets in \(X\) such that for any open set \(U\supset \Phi\) there is \(U_n\) for which
\[ \Phi\subset U_n\subset U. \]
Let, further,
\[ \pi=\{\Gamma_n\} \]
be a countable system of open sets in \(\Phi\), forming a base at the point \(x_0\). For each point \(x\ne x_0\) of \(\Phi\) there exists a neighborhood \(Ox\) (open in \(X\)) such that
\[ x_0\notin [Ox]. \]
Obviously,
\[ \gamma_n=\{\Gamma_n, Ox\mid x\in \Phi,\ x\ne x_0\} \]
is a covering of the bicompact \(\Phi\). Choose a finite subcovering
\[ \omega_n=\{\Gamma_n, Ox_1^n,\ldots,Ox_{m(n)}^n\}. \]
Denote
\[ M_n=\bigcup_{i=1}^{m(n)} Ox_i^n. \]
We have
\[ x_0\notin [M_n]. \]
Put
\[ V_n=U_n\setminus \bigcup_{i=1}^n [M_i] =U_n\cap\left(X\setminus \bigcup_{i=1}^n [M_i]\right). \]
We shall show that the system
\[ \Omega=\{V_n\mid n=1,\ldots\} \]
forms a base at the point \(x_0\) in the space—

* For the case of completely regular spaces this lemma was first established by A. Arhangel′skii by a method essentially using complete regularity—the passage to bicompact Hausdorff extensions.

space \(X\). Let \(U\) be any open set in \(X\) containing the point \(x_0\). Choose \(\Gamma_{n_0}\) and \(M_{n_0}\) such that \(\Gamma_{n_0} \subset U\) and \(\Phi \subset \Gamma_{n_0} \cup M_{n_0}\). Then \(\Phi \subset U \cup M_{n_0}\). There exists an open set \(U_{n_1}\) \((n_1 \geq n_0)\) such that \(U_{n_1} \subset M_{n_0} \cup U\). Then

\[ V_{n_1}=U_{n_1}\setminus \bigcup_{i=1}^{n_1}[M_i]\subset U_{n_1}\setminus M_{n_0}\subset U. \]

This completes the proof of Lemma 2.

Proof of Theorem 2. Take some point \(y_0 \in Y\). In \(Y\) there is a bicompactum \(\Phi_0\) of countable character such that \(y_0 \in \Phi_0\). On the basis of Lemma 1, the bicompactum \(\Phi_0\) is metrized, and on the basis of Lemma 2 the point \(y_0\) has countable character in the space \(Y\). Hence \(Y\) satisfies the first axiom of countability. Since \(X\) locally satisfies the second axiom of countability, by a known theorem of P. S. Aleksandrov (see \((^1)\)), it decomposes into the sum of a certain number of discrete sets \(\{M_\lambda\}\), where each \(M_\lambda\) has a countable base. Take any point \(y_0 \in Y\). By virtue of the separability of the mapping, the set \(f^{-1}y_0\) intersects no more than a countable number of sets \(M_{\lambda_1},\ldots,M_{\lambda_n},\ldots\), i.e.

\[ f_{y_0}^{-1}\subset \bigcup_{i=1}^{\infty} M_{\lambda_i}. \]

It is known that

\[ \operatorname{Int} f\left(\bigcup_{i=1}^{\infty} M_{\lambda_i}\right)=Y_1 \ni y_0 \]

and \(Y_1\) is separable. Denote \(X_1=f^{-1}Y_1\) and \(f_1=f|X_1\). Since \(f_1\) is a quotient \(s\)-mapping of the metric space \(X_1\) onto the separable space \(Y_1\) with the first axiom of countability, it follows from Theorem 1 that \(Y_1\) has a countable base.

Since \(Y\) is a regular weakly paracompact space with the local second axiom of countability, by another theorem of P. S. Aleksandrov (see Theorem 11 in \((^1)\)), it is metrizable.

From Theorem 2 it follows that

Corollary. Let \(f:X\to Y\) be an almost open \(s\)-mapping \(^*\) of a locally separable metric space \(X\) onto a regular weakly paracompact space \(Y\). Then \(Y\) is metrizable and locally satisfies the second axiom of countability.

Remark. In Theorem 2 the requirement of weak paracompactness can be weakened to the requirement: into every cover one can inscribe a point-countable cover.

The author expresses gratitude to A. V. Arhangel’skii for posing a number of problems and for providing the opportunity to become acquainted with his unpublished work.

Moscow State University
named after M. V. Lomonosov

Received
12 V 1965

REFERENCES

\(^1\) P. S. Aleksandrov, P. S. Urysohn, On compact topological spaces, Tr. po topologii i drugim oblastiam matematiki, 2, 1951.
\(^2\) A. Arhangel’skii, DAN, 126, No. 2 (1959).
\(^3\) A. Arhangel’skii, DAN, 151, No. 4 (1963).
\(^4\) A. Arhangel’skii, DAN, 147, No. 5 (1962).
\(^5\) A. N. Stone, Proc. Am. Math. Soc., 7, 690 (1956).
\(^6\) V. I. Ponomarev, Bull. Polish Acad. Sci., 8, No. 3 (1960).

\(^*\) A mapping \(f\) of a space \(X\) onto \(Y\) is called almost open if for an arbitrary point \(y\in Y\) there is a point \(x\in f^{-1}y\) having a base of open sets each of whose images is open (see \((^4)\)).