UDC 519.50+519.54

MATHEMATICS

P. KENDEROV

ON \(Q\)-SPACES

(Presented by Academician P. S. Aleksandrov on 4 X 1966)

E. Hewitt, in the paper \((^{1})\), defined the class of \(Q\)-spaces. Most often these spaces are called functionally closed.

Definition. A completely regular space \(X\) is called a \(Q\)-space (functionally closed) if every maximal countably centered system of zero-sets has a nonempty intersection. Here a system \(\xi=\{Z_\alpha\}\) is called countably centered if every countable subsystem of this system of sets has a nonempty intersection. A set \(A\subseteq X\) is called a zero-set if on the space \(X\) there exists a continuous function such that the set \(A\) is the complete preimage of zero.

Theorem 1. A perfect image of a normal countably paracompact \(Q\)-space \(X\) is always normal, countably paracompact, and is a \(Q\)-space.

Proof. Let \(f:X\to Y\) be a perfect mapping, and let \(X\) be a normal countably paracompact \(Q\)-space. Then the space \(Y\) is normal and countably paracompact (for this it is sufficient only that the mapping \(f\) be closed). We shall prove that \(Y\) is a \(Q\)-space. Let \(\xi=\{Z_\alpha\}\) be an arbitrary maximal countably centered system of zero-sets of the space \(Y\). This means that in the space \(Y\) there exist continuous functions \(\varphi_\alpha(y)\) such that \(Z_\alpha=\varphi_\alpha^{-1}(0)\). Then \(\psi_\alpha(x)=\varphi_\alpha(fx)\) is a continuous function on the space \(X\), and
\[ \psi_\alpha^{-1}(0)=f^{-1}(\varphi_\alpha^{-1}(0))=f^{-1}Z_\alpha, \]
therefore the system
\[ \eta=f^{-1}\xi=\{f^{-1}Z_\alpha\} \]
is countably centered and consists of zero-sets of the space \(X\). The system \(\eta\), generally speaking, is not a maximal countably centered system of zero-sets. Complete it to a maximal countably centered system \(\hat{\eta}\) of zero-sets of the space \(X\). We shall prove that in our case the system obtained, \(\eta\), will be countably centered.

Consider an arbitrary countable subsystem \(\hat{\eta}_0=\{\Phi_{\lambda_i}\}\subseteq\hat{\eta}\) of the system \(\hat{\eta}\). Without loss of generality we may assume that
\[ \Phi_{\lambda_1}\supseteq \Phi_{\lambda_2}\supseteq \cdots \supseteq \Phi_{\lambda_i}\supseteq \cdots, \]
since the system \(\hat{\eta}\), together with any two of its elements, also contains their intersection. Consider the system
\[ f\hat{\eta}_0=\{f\Phi_{\lambda_i}\} \]
of subsets closed in \(Y\). Since for arbitrary \(i,\alpha\) one has
\[ \Phi_{\lambda_i}\cap f^{-1}Z_\alpha\ne\Lambda, \]
it follows that
\[ f\Phi_{\lambda_i}\cap Z_\alpha\ne\Lambda \]
for all \(i,\alpha\). We shall now prove that
\[ \bigcap_{i=1}^{\infty} f\Phi_{\lambda_i}\ne\Lambda. \]
Suppose the contrary; let
\[ \bigcap_{i=1}^{\infty} f\Phi_{\lambda_i}=\Lambda. \]
Then, by virtue of the normality and countable paracompactness of the space \(Y\) (see \((^{2})\)), there exist neighborhoods \(Of\Phi_{\lambda_i}\) of the sets \(f\Phi_{\lambda_i}\) for which
\[ \bigcap_{i=1}^{\infty} Of\Phi_{\lambda_i}=\Lambda. \]
Further, by Urysohn’s lemma, there exist zero-sets \(Z_i\) of the space \(Y\) such that
\[ f\Phi_{\lambda_i}\subseteq Z_i\subseteq Of\Phi_{\lambda_i}. \]
Since
\[ f\Phi_{\lambda_i}\cap Z_\alpha\ne\Lambda \]
for all \(i,\alpha\), the zero-set \(Z_i\supseteq f\Phi_{\lambda_i}\),

system \(\xi=\{Z_\alpha\}\) is a maximal countably centered system of zero-sets, then \(Z_i\in\xi\), \(i=1,2,\ldots\). Since \(Z_i\subset Of\Phi_{\lambda_i}\), \(\bigcap_{i=1}^{\infty} Of\Phi_{\lambda_i}=\Lambda\), it follows that \(\bigcap_{i=1}^{\infty} Z_i=\Lambda\), and we have arrived at a contradiction with the countable centeredness of the system \(\xi\). Thus, necessarily \(\bigcap_{i=1}^{\infty} f\Phi_{\lambda_i}\ne\Lambda\). Let \(y_0\in \bigcap_{i=1}^{\infty} f\Phi_{\lambda_i}\) be an arbitrary point.

Consider the countable system \(\{f^{-1}y_0\cap \Phi_{\lambda_i}\}_{i=1}^{\infty}\) of nonempty closed subsets of the bicompactum \(f^{-1}y_0\). Since \(\Phi_{\lambda_1}\supseteq \Phi_{\lambda_2}\supseteq\cdots\supseteq\Phi_{\lambda_i}\supseteq\cdots\), we have
\[ \bigcap_{i=1}^{\infty}(f^{-1}y_0\cap \Phi_{\lambda_i})\ne \Lambda, \]
and hence the system \(\hat\eta\) is countably centered. Thus, we have extended our countably centered system \(\eta=f^{-1}\xi=\{f^{-1}Z_\alpha\}\) of zero-sets of the space \(X\) to a maximal, also countably centered, system \(\hat\eta\) of zero-sets. In view of the fact that \(X\) is a \(Q\)-space, necessarily
\[ \bigcap_{\Phi_\lambda\in\hat\eta}\Phi_\lambda\ne\Lambda. \]
Then \(\bigcap_\alpha f^{-1}Z_\alpha\ne\Lambda\), and consequently also
\[ \bigcap_\alpha Z_\alpha\ne\Lambda, \]
i.e. \(Y\) is a \(Q\)-space. Theorem 1 is proved.

Remark 1. In the proof of Theorem 1 we used only the countable compactness of \(f^{-1}y_0\).

Remark 2. This theorem is a partial answer to a question of V. I. Ponomarev from (4): is functional closedness of a normal space preserved under a perfect mapping? In Frolik’s paper (3) a theorem is formulated which gives a complete answer to the question, but the proof of this theorem contains a gap.

By the same method one also proves the following known result (see (4)).

Proposition (V. I. Ponomarev). An open perfect image of a normal \(Q\)-space is always a \(Q\)-space.

In proving this proposition the following assertion is used:

Lemma 1. The image \(f\Phi\) of a zero-set \(\Phi\) in a space \(X\) under an open perfect mapping \(f:X\to Y\) is a zero-set of the space \(Y\).

Theorem 2. Let \(X\) be a normal countably paracompact space. In order that \(X\) be a \(Q\)-space, it is necessary and sufficient that every maximal countably centered system of closed (not necessarily zero-) sets have nonempty intersection.

Proof. Necessity. Let \(\xi=\{F_\lambda\}\) be a maximal countably centered system of closed sets. Denote by \(\xi_0\) the collection of all zero-sets belonging to the system \(\xi\). Clearly, \(\xi_0\) is countably centered. We show that it is maximal.

Let \(Z\) be a zero-set which meets all elements of the system \(\xi_0=\{Z_\lambda\}\). If \(Z\notin\xi=\{F_\lambda\}\), then there is \(F_{\lambda_0}\in\xi\), \(F_{\lambda_0}\cap Z=\Lambda\). By the normality of \(X\) there exists a zero-set \(Z_0\) such that \(Z_0\supseteq F_{\lambda_0}\), \(Z_0\cap Z=\Lambda\). But since \(\xi\) is a maximal countably centered system, \(Z_0\in\xi\); consequently, \(Z_0\in\xi_0\), and then \(Z_0\cap Z\ne\Lambda\) (by the choice of \(Z\)). Thus \(Z\in\xi\), and consequently \(Z\in\xi_0\). Since \(X\) is a \(Q\)-space, we have
\[ \bigcap_{Z_\lambda\in\xi_0} Z_\lambda\ne\Lambda. \]

Furthermore, any closed subset \(F\subseteq X\), in particular \(F_\lambda\in\xi\), is representable as an intersection of zero-sets, i.e. \(F_\lambda=\bigcap_\alpha Z_\alpha^\lambda\); \(Z_\alpha^\lambda\in\xi_0\). Then
\[ \bigcap_\lambda F_\lambda = \bigcap_\lambda\left(\bigcap_\alpha Z_\alpha^\lambda\right) = \bigcap_{Z\in\xi_0}, \]
which proves the necessity.

Sufficiency. Let \(\xi_0=\{Z_\lambda\}\) be a maximal countably centered-

system of zero-sets of the space \(X\). Extend it to a maximal simply centered system \(\xi=\{F_\lambda\}\) of closed sets. We shall prove the countable centeredness of the system \(\xi\). Let \(\{F_{\lambda_i}\}\) be some countable subsystem of the system \(\xi\). We must prove that
\[ \bigcap_{i=1}^{\infty} F_{\lambda_i}\ne \Lambda . \]

Suppose the contrary, let
\[ \bigcap_{i=1}^{\infty} F_{\lambda_i}=\Lambda . \]
Then, by normality and countable paracompactness (see \((^2)\)) of the space \(X\), there exist neighborhoods \(OF_{\lambda_i}\) of the closed sets \(F_{\lambda_i}\), for which also
\[ \bigcap_{i=1}^{\infty} OF_{\lambda_i}=\Lambda . \]
Moreover, there exist zero-sets \(Z_i\) such that
\[ F_{\lambda_i}\subseteq Z_i\subseteq OF_{\lambda_i}. \]
Since \(F_{\lambda_i}\in\xi\), it follows that \(Z_i\in\xi\), and consequently \(Z_i\in\xi_0\). Therefore, on the one hand,
\[ \bigcap_{i=1}^{\infty} Z_i\subset \bigcap_{i=1}^{\infty} OF_{\lambda_i}\ne \Lambda, \]
and on the other hand, \(Z_i\in\xi_0\), i.e.
\[ \bigcap_{i=1}^{\infty} Z_i\ne \Lambda \]
by the countable centeredness of the system \(\xi_0\). We have obtained a contradiction. Hence the system \(\xi=\{F_\lambda\}\) is countably centered. Then
\[ \bigcap_\lambda F_\lambda\ne \Lambda, \]
and consequently, a fortiori,
\[ \bigcap_{Z\in\xi_0} Z\ne \Lambda, \]
i.e. the space \(X\) is a \(Q\)-space, which proves sufficiency, and with it the whole theorem.

Remark. In proving necessity we did not use the countable paracompactness of \(X\).

Theorem 3. Let \(f:X\to Y\) be a continuous mapping of a normal space \(X\) onto a hereditarily finally compact space \(Y\), such that the preimage \(f^{-1}y\) of every point \(y\in Y\) is a \(Q\)-space. Then \(X\) is a \(Q\)-space.

Proof. Denote by \(\xi=\{Z_\lambda\}\) some arbitrary maximal countably centered system of zero-sets of the space \(X\). We must prove that
\[ \bigcap_\lambda Z_\lambda\ne \Lambda . \]
Consider the system
\[ f\xi=\{fZ_\lambda\} \]
of sets in \(Y\). The system \(f\xi\) is also countably centered. We shall show that
\[ \bigcap_\lambda fZ_\lambda\ne \Lambda . \]
By the final compactness of the space \(Y\) we shall have
\[ \bigcap_\lambda [fZ_\lambda]_Y\ne \Lambda, \]
where \([fZ_\lambda]_Y\) is the closure of the set \(fZ_\lambda\) in \(Y\). Moreover, observe that
\[ \bigcap_\lambda [fZ_\lambda]_Y \]
consists of not more than one point.

Let
\[ \bigcap_\lambda [fZ_\lambda]=y_0 . \]
Fix some index \(\lambda_0\) and consider the system
\[ \eta_{\lambda_0}=\{fZ_{\lambda_0}\cap fZ_\lambda\}. \]
This system is countably centered and consists of sets belonging to the finally compact space
\[ fZ_{\lambda_0}\subseteq Y. \]
Therefore
\[ A=\bigcap_\lambda \bigl([fZ_{\lambda_0}\cap fZ_\lambda]_{fZ_{\lambda_0}}\bigr)\ne \Lambda . \]
But
\[ A\subseteq \bigcap_\lambda [fZ_{\lambda_0}\cap fZ_\lambda]_Y \subseteq \bigcap_\lambda \bigl([fZ_{\lambda_0}]_Y\cap [fZ_\lambda]_Y\bigr) \subseteq \bigcap_\lambda [fZ_\lambda]_Y=y_0 . \]
It follows (by the arbitrariness of \(\lambda_0\)) that
\[ y_0=\bigcap_\lambda fZ_\lambda . \]
Now consider the system
\[ \{f^{-1}y_0\cap Z_\lambda\}=\xi_{y_0}. \]
This system is countably centered and consists of zero-sets in \(f^{-1}y_0\). We shall prove its maximality.

Let \(Z_0\subseteq f^{-1}y_0\) be a zero-set of some function \(\varphi\) defined on \(f^{-1}y_0\). By normality of the space \(X\) (Urysohn’s theorem) there exists an extension \(\widetilde{\varphi}\) of the function \(\varphi\) to the whole space \(X\).

Consider the zero-set \(Z(\widetilde{\varphi})\) of the function \(\widetilde{\varphi}\). We have
\[ Z(\widetilde{\varphi})\cap f^{-1}y_0=Z_0. \]
Since \(y_0\) is a zero-set in \(Y\), \(f^{-1}y_0\) is a zero-set in \(X\). Therefore \(Z_0\) is a zero-set in all of \(X\) as the intersection of the zero-sets \(f^{-1}y_0\) and \(Z(\widetilde{\varphi})\). Let
\[ Z_0\cap (f^{-1}y_0\cap Z_\lambda)\ne \Lambda \]
for every \(\lambda\). Then
\[ Z_0\cap Z_\lambda\ne \Lambda \]
for all \(\lambda\), i.e. \(Z_0\in\xi=\{Z_\lambda\}\), and then \(Z_0\in\{f^{-1}y_0\cap Z_\lambda\}\), \(Z_0\subseteq\)

\(\subseteq f^{-1}y_0\), whereby the maximality of the system \(\xi_{y_0}\) is proved. Thus, \(\xi_{y_0}\) is a maximal countably centered system of zero-sets of the subspace \(f^{-1}y_0\). From the fact that \(f^{-1}y_0\) is a \(Q\)-space, it follows that
\[ \bigcap_\lambda (f^{-1}y_0 \cap Z_\lambda) \ne \Lambda, \]
and then all the more
\[ \bigcap_\lambda Z_\lambda \ne \Lambda, \]
which proves the theorem.

Remark. Theorem 3 strengthens B. A. Pasynkov’s theorem from \((^5)\). In B. A. Pasynkov’s theorem the space \(Y\) has a countable base.

Theorem 4. Let \(f:X\to Y\) be a closed continuous mapping of a normal space \(X\) onto a \(Q\)-space \(Y\). If every point \(y\in Y\) has type \(G_\delta\) in \(Y\) and its inverse image \(f^{-1}y\) is a \(Q\)-space, then \(X\) is also a \(Q\)-space.

The proof of Theorem 4 is simple, and we omit it.

Theorem 5. Let \(f:X\to Y\) be a closed continuous mapping of a countably paracompact normal space \(X\) onto a \(Q\)-space \(Y\), under which the inverse image \(f^{-1}y\) of every point \(y\in Y\) is functionally closed. Then the space \(X\) is also functionally closed.

Proof. Since the space \(X\) is normal and countably paracompact, by virtue of Theorem 2 it suffices for us to prove that every maximal countably centered system of closed (not necessarily zero-) sets has a nonempty intersection. Let \(\xi=\{F_\lambda\}\) be an arbitrary maximal countably centered system of closed sets of the space \(X\). Consider the system \(f\xi=\{fF_\lambda\}\) of closed subsets of \(Y\). The system \(f\xi\) is a maximal countably centered system of closed sets in \(Y\). Since \(Y\) is functionally closed, we have
\[ \bigcap_\lambda fF_\lambda \ne \Lambda. \]
Let
\[ y_0\in \bigcap_\lambda fF_\lambda. \]
Then \(f^{-1}y_0\cap F_\lambda\ne \lambda\) for all \(\lambda\). Hence it follows that \(f^{-1}y_0\in\xi\) by the maximality of the system \(\xi\). Therefore the system
\[ \{f^{-1}y_0\cap F_\lambda\}=\xi_{y_0} \]
is a countably centered system of closed sets in \(f^{-1}y_0\). It is easy to prove that \(\xi_{y_0}\) is maximal with respect to these properties. Since \(f^{-1}y_0\) is functionally closed, we have
\[ \bigcap_\lambda (f^{-1}y_0\cap F_\lambda)\ne \Lambda, \]
and then all the more
\[ \bigcap_\lambda F_\lambda\ne \Lambda. \]
Theorem 5 is proved.

I express my gratitude to V. I. Ponomarev for his help in carrying out this work.

Moscow State University
named after M. V. Lomonosov

Received
10 VI 1966

REFERENCES

1. E. Hewitt, Trans. Am. Math. Soc., 64, No. 1, 45 (1948).
2. C. H. Dowker, Canad. J. Math., 3, No. 2, 219 (1951).
3. Z. Frolik, Czechoslovak Math. J., vol. 13, No. 1, 127 (1963).
4. V. I. Ponomarev, DAN, 126, 716 (1959).
5. B. A. Pasynkov, Mat. Sb., 66, No. 1, 35 (1965).