V. V. PROIZVOLOV

CARDINALITIES OF A BASE OF AN \(H\)-CLOSED SPACE

(Presented by Academician P. S. Aleksandrov on 11 VII 1963)

In this note I shall prove

Theorem 1. Let \(X\) be an \(H\)-closed space and let \(B\) be its \(t\)-pseudobase\(^*\) such that, for every point \(x \in X\), the set \(B_x\) of all elements of the \(t\)-pseudobase \(B\) for which the point \(x\) is a point of contact has cardinality \(\leq a\), where \(a\) is some cardinal number. Then the cardinality of \(B\) is \(\leq a\).

In addition to this theorem, here a theorem on the cardinality of \(t\)-pseudobases in bicompact spaces is proved, proved by A. Mishchenko for bases \((^1)\). In conclusion an example is given of an \(H\)-closed space showing that Theorem 1 cannot be strengthened.

Proof of Theorem 1. I carry out the proof following \((^1)\). I shall call a pseudocovering of a space \(X\) any finite system \(\omega = \{U_\alpha\}\) of open sets such that the system \(\bar{\omega} = \{[\overline{U_\alpha}]\}\), composed of the closures of the elements of the system \(\omega\), covers the space \(X\). As is known \((^2)\), from any covering of an \(H\)-closed space one can choose a finite pseudocovering. I shall call a pseudocovering \(\omega\) minimal if it is finite and such that for every \(U_\alpha \in \omega\) there is a point \(x\) not belonging to any other elements of the pseudocovering \(\omega\). Let \(V\) be the set of possible minimal pseudocoverings of the \(H\)-closed space \(X\), composed of elements of the \(t\)-pseudobase \(B\).

Since each minimal pseudocovering is finite, the cardinality of the set \(V\) does not exceed the cardinality of \(B\). But the cardinality of the set \(V\) cannot be less than the cardinality of \(B\); this follows from the fact that for any \(\sigma \in B\) there exists a pseudocovering \(\omega \in V\) such that \(\sigma \in \omega\).

To prove the last assertion, take some point \(x \in \sigma\). For each point \(y \in X \setminus x\) choose an element \(U(y)\) of the \(t\)-pseudobase \(B\) containing it and lying in \(X \setminus x\). Denote by \(\Omega\) the covering consisting of all the just selected \(U(y)\) and of \(\sigma\). Let \(\omega\) be any minimal pseudocovering contained in \(\Omega\); if one adds \(\sigma\) to its elements (in the case when \(\sigma\) is not already in \(\omega\)), then again a minimal pseudocovering is obtained. Thus, the cardinality of the set \(V\) is equal to the cardinality of \(B\). Suppose that the cardinality of \(B > a\). Then, if \(V_n\) denotes the set of all minimal pseudocoverings consisting of \(n\) distinct elements of the \(t\)-pseudobase, there exists such an \(n_0\) that the cardinality of \(V_{n_0} > a\).

Let \(k \leq n_0\), and let \(\sigma_1,\ldots,\sigma_k\) be arbitrary distinct elements of the \(t\)-pseudobase \(B\). Denote by \(S_{\sigma_1\ldots\sigma_k}\) the set of all those pseudocoverings \(\omega \in V_{n_0}\) for which \(\sigma_i \in \omega\) \((i = 1,2,\ldots,k)\).

Let \(x\) be an arbitrary point of the space \(X\), and let \(B_x\) be the set of all elements of the \(t\)-pseudobase \(B\) for which the point \(x\) serves as a point of contact. Then the equality holds
\[ V_{n_0} = \bigcup_{\sigma \in B_x} S_\sigma . \tag{1} \]

\(^*\) A \(t\)-pseudobase of a topological space \(X\) is a system of open sets such that for every point \(x \in X\) there is in it a subsystem whose intersection is exactly the point \(x\). A base, obviously, is a \(t\)-pseudobase.

If \(\sigma_1,\ldots,\sigma_k\) are any (pairwise distinct) elements of the \(t\)-pseudobase \(B\) not containing the point \(x\), then

\[ S_{\sigma_1\ldots \sigma_k} = \bigcup_{\sigma\in B_x} S_{\sigma_1\ldots \sigma_k\sigma}, \quad \text{where } \sigma\ne \sigma_i \quad (i=1,\ldots,k). \tag{2} \]

Let \(x_1\in X\) be an arbitrary point. There exists \(\sigma_1\in B_{x_1}\) such that the cardinality of \(S_{\sigma_1}\) is \(>a\) (this follows from equality (1)). Suppose that for \(k<n_0\) such (pairwise distinct) elements \(\sigma_1,\ldots,\sigma_k\) of the \(t\)-pseudobase \(B\) have been found that the cardinality of \(S_{\sigma_1\ldots \sigma_k}\) is \(>a\). Since \(k<n_0\), there exists a point \(x_{k+1}\) that is contained in none of the sets \(\sigma_1,\ldots,\sigma_k\). But then, by virtue of equality (2), there exists also such a \(\sigma_{k+1}\in B_{x_{k+1}}\) that the cardinality of \(S_{\sigma_1\ldots \sigma_k\sigma_{k+1}}\) is \(>a\).

As a result, for all \(k\le n_0\), the sets \(S_{\sigma_1\ldots \sigma_k}\) have cardinality greater than \(a\). In particular, the cardinality of the set \(S_{\sigma_1\ldots \sigma_{n_0}}\) exceeds the cardinal number \(a\). But \(S_{\sigma_1\ldots \sigma_{n_0}}\subset V_{n_0}\), and therefore there exists only one cover \(\omega\in S_{\sigma_1\ldots \sigma_{n_0}}\), namely \(\omega=\{\sigma_1,\sigma_2,\ldots,\sigma_{n_0}\}\). The theorem is proved.

Theorem 2. Let \(X\) be a bicompact \(T_1\)-space and let \(B\) be its \(t\)-pseudobase such that, for every point \(x\in X\), the set \(B_x\) of all elements of the \(t\)-pseudobase \(B\) containing the point \(x\) has cardinality \(\le a\), where \(a\) is some cardinal number. Then the cardinality of \(B\) is \(\le a\).

The proof of this theorem differs hardly at all from the proof of the corresponding theorem in paper \((^1)\).

Now I shall construct an \(H\)-closed space \(M\) possessing a point-countable base but not possessing a countable base. This will show that Theorem 1 cannot be strengthened. The space is borrowed from paper \((^1)\).

Represent the interval \([0,1]\) as the sum of a disjoint system \(N_1\) of nonempty everywhere dense sets \(M_\alpha\):

\[ [0,1]=\bigcup_{\alpha<\omega_1} M_\alpha,\quad M_\alpha\cap M_\beta=\Lambda,\quad \alpha\ne\beta . \]

Let \(p_1,p_2\) be arbitrary rational numbers, \(p_1<p_2\). By \((p_1,p_2)\) we denote the interval of the number line with endpoints \(p_1\) and \(p_2\). Let \(x\) be an arbitrary point of the interval \([0,1]\), \(x\in M_\alpha\). Define the neighborhood \(U_{p_1p_2}(x)\) as the set

\[ U_{p_1p_2}(x)=(p_1,p_2)\cap\bigcup_{\beta\ge\alpha} M_\beta . \]

In \((^1)\) it is proved that \(M\) is a Hausdorff space without a countable, but with a point-countable, base. I shall prove that \(M\) is \(H\)-closed.

Take an arbitrary cover \(\omega\) of the space \(M\) by elements of the base. We shall prove that from the cover \(\omega\) one can extract a finite pseudocover \(\pi\). Let \(U_{p_1p_2}(x)\in\omega\); then the interval \((p_1,p_2)\in\overline{\omega}\); thus we obtain a cover \(\overline{\omega}\).

From the cover \(\overline{\omega}\) extract a finite cover \(\overline{\pi}\) of the interval \([0,1]\). Let \((p_1,p_2)\in\overline{\pi}\); then the corresponding \(U_{p_1p_2}(x)\in\pi\); thus we obtain \(\pi\). It is not hard to verify that the closure of each \(U_{p_1p_2}(x)\) is exactly the interval \([p_1,p_2]\). Hence it follows that \(\pi\) is a finite pseudocover.

Moscow State University
named after M. V. Lomonosov

Received
5 VII 1963

REFERENCES

\(^1\) A. Mishchenko, DAN, 144, No. 5, 985 (1962). \(^2\) P. S. Aleksandrov, P. S. Uryson, Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 31. Monograph, 1950.