Full Text
V. PROIZVOLOV
ON THE POINTWISE AND INTEGRAL CARDINALITY OF SYSTEMS OF SUBSETS OF A TOPOLOGICAL SPACE
(Presented by Academician P. S. Aleksandrov on 24 X 1964)
Corson and Michael proved [1] that a point-countable base of a countably compact space is countable. Here a stronger assertion is proved (Theorem 1). Theorem 2, in particular, answers the question of how the pointwise and integral cardinalities of a base of a finally compact space are related.
Definition. A system of open sets is called a \(t\)-pseudobase of a space \(X\) if, for every point \(x \in X\), it contains a subsystem whose elements have intersection exactly the point \(x\).
A topological space possessing a \(t\)-pseudobase is automatically a \(T_1\)-space. A base of a \(T_1\)-space is, evidently, a \(t\)-pseudobase.
Theorem 1. Let \(X\) be a countably compact space, and let \(B\) be its \(t\)-pseudobase, point-countable on an everywhere dense subset \(M\) of \(X\). Then the \(t\)-pseudobase \(B\) is countable.
Proof. We shall construct by induction countable sets \(C_i\) for all natural \(i\). Take an arbitrary point \(a \in M\) and its countable star from the \(t\)-pseudobase \(B\); denote the star by \(\gamma\). For every finite collection of elements \(U_1, U_2, \ldots, U_m \in \gamma\), choose one point
\[ x(U_1,\ldots,U_m)\in M \cap \left(X \setminus \bigcup_{j=1}^{m} U_j\right), \]
if such a point exists for the given collection. Denote by \(C_1\) the union of such points (over all finite collections of the star \(\gamma\)). Suppose \(C_i\) has been constructed. We shall construct \(C_{i+1}\). The system \(B_i\) is the union of all those elements of the \(t\)-pseudobase \(B\) which intersect at least one of the \(C_k\), \(k = 1,\ldots,i\). If \(U_1,\ldots,U_m \in B_i\) (where \(m\) is a natural number), then take a point
\[ x(U_1,\ldots,U_m)\in M \cap \left(X \setminus \bigcup_{j=1}^{m} U_j\right), \]
if such a point is found. The union of all such points, taken one for each finite collection of elements of \(B_i\), is the set \(C_{i+1}\).
Each of the sets \(C_i\) is countable. Put \(C=\bigcup_{i=1}^{\infty} C_i\); evidently, the set \(C\) is countable. We show that \([C]=X\). Suppose the contrary, \(X \setminus [C]\ne \Lambda\). Then there is a point \(a\in M\cap (X\setminus [C])\). For each point \(x\in [C]\) there is an element \(U_\alpha \in B\), containing the point \(x\) and such that \(U_\alpha \not\ni a\), since \(B\) is a \(t\)-pseudobase in \(X\). The union of such \(U_\alpha\) over all points \(x\in [C]\) forms a cover \(\omega=\{U_\alpha\}\) of the set \([C]\). The cover \(\omega\) is countable, since \(C\) is a countable everywhere dense subset of \([C]\) and since each point of the set \(C\subseteq M\), by the hypothesis of the theorem, belongs to no more than countably many elements of \(B\).
Since \([C]\), as a closed subspace of a countably compact space, is countably compact, from the countable cover \(\omega\) one can choose
finite cover \(\{U_1,\ldots,U_m\}\); \([C] \subseteq \bigcup_{j=1}^m U_j\). It is not hard to see that there is a natural number \(i\) such that \(U_1,U_2,\ldots,U_m \in B_i\). But then \(C_{i+1}\) must contain the point
\[ x(U_1,U_2,\ldots,U_m)\in M\cap\left(X\setminus\bigcup_{j=1}^m U_j\right), \]
if such a point exists. But such a point does exist: for example, the point \(a\). Thus, a point of \(C_{i+1}\) has been found which belongs to \(X\setminus\bigcup_{j=1}^m U_j\), which contradicts the fact that \(C_{i+1}\subseteq C\subseteq\bigcup_{j=1}^m U_j\). Hence, \([C]=X\), i.e. \(C\) is an everywhere dense countable set of the space \(X\).
Every point of the set \(C\subseteq M\) belongs to at most countably many elements of \(B\), and every element of \(B\) contains at least one point of the set \(C\); hence it follows that the \(t\)-pseudobase \(B\) is countable, as was required to prove.
Theorem 1 has the following generalization.
Theorem \(1'\). Let \(X\) be a space such that, from every open cover of cardinality \(\tau\leq \aleph_0\), one can extract a finite cover, and let \(B\) be its \(t\)-pseudobase of pointwise cardinality \(\leq\tau\) on an everywhere dense subset \(M\) of \(X\). Then the \(t\)-pseudobase \(B\) has cardinality \(\leq\tau\).
Corollary 1. Let \(X\) be a bicompact space, and let \(B\) be its \(t\)-pseudobase whose pointwise cardinality is \(\leq\tau\) on an everywhere dense subset \(M\) of \(X\). Then the cardinality of \(B\leq\tau\).
This corollary includes the theorem of A. Mishchenko \((^2)\) (together with the strengthening given to it in \((^3)\)). From Theorem 1 there follows the corresponding theorem of Corson and Michael from \((^1)\).
Corollary 2. Let \(X\) be a locally bicompact Hausdorff space, and let \(B\) be its base, pointwise countable on an everywhere dense subset \(M\) of \(X\). Then \(X\) is metrizable.
Corollary 3. Let \(X\) be a locally bicompact Hausdorff space, and let \(B\) be its base whose pointwise cardinality is \(\leq\tau\) on an everywhere dense subset \(M\) of \(X\). Then \(X\) decomposes into isolated subspaces of weight \(\leq\tau\). In particular, if \(X\) is connected, then its weight is \(\leq\tau\).
We derive Corollary 3. From \(B\) we select a subsystem \(B'\) of such elements whose closures are bicompact. The subsystem \(B'\), obviously, is a cover of \(X\), the elements of which have weight \(\leq\tau\), in view of Corollary 1. We shall call points \(x_0,x_1\in X\) equivalent if there exists a finite set of elements \(B_0,\ldots,B_n\in B'\) such that \(x_0\in B_0\), \(x_1\in B_n\), \(B_i\cap B_{i+1}\neq\Lambda\), \(i=1,2,\ldots,n-1\).
It is easy to see that \(X\) decomposes into open subspaces of equivalent points. Denote one of them by \(A\). Since each element of the cover \(B'\) intersects no more than \(\tau\) other elements, the locally bicompact subspace \(A\) is the union of no more than \(\tau\) subsets whose weights are \(\leq\tau\). By the addition theorem it follows that the weight of \(A\leq\tau\). The assertion is proved.
Examples are known \((^{1,2})\) of finally compact spaces with a pointwise countable base, but without a countable base. However, the following theorem is true, limiting the cardinality of pointwise countable bases.
Theorem 2. Let \(X\) be a finally compact space, and let \(B\) be its \(t\)-pseudobase, pointwise countable on an everywhere dense subset \(M\) of \(X\). Then the cardinality of the \(t\)-pseudobase \(B\leq 2^{\aleph_0}\).
Sketch of the proof. The proof of Theorem 2 has much in common with the proof of Theorem 1. We shall construct, by transfinite induction, sets \(C_\alpha\) for all ordinal numbers \(\alpha\) up to \(\omega_1\) exclusively (up to \(\omega_1\) because no countable sequence is cofinal with it—
ness). I take an arbitrary point \(a \in M\) and its countable star \(\gamma\) from the \(t\)-pseudobase \(B\). For every countable set of elements \(\{U_i\} \in \gamma\) choose one point
\[ x\{U_i\}\in M \cap \left(X \setminus \bigcup_{i=1}^{\infty} U_i\right), \]
if such a point exists. Denote by \(C_1\) the union of such points for all countable (or finite) sets of the star \(\gamma\). Suppose that \(C_\alpha\) have already been constructed for all ordinal numbers \(\alpha\) up to the ordinal number \(\mu\). We shall construct \(C_\mu\). The system \(B_\mu\) is the union of all those elements of the \(t\)-pseudobase \(B\) which intersect at least one of the \(C_\alpha\), where \(\alpha<\mu\). Let a countable subsystem \(\{U_i\}\in B^\mu\); then take a point
\[ x\{U_i\}\in M \cap \left(X \setminus \bigcup_{j=1}^{\infty} U_j\right), \]
if such a point exists. The union of such points, taken one for each countable (or finite) set of elements of \(B_\mu\), is the set \(C_\mu\). Each of the sets \(C_\alpha\) has cardinality \(\leq 2^{\aleph_0}\). Put \(C=\bigcup_\alpha C_\alpha\); obviously, the cardinality of \(C\leq 2^{\aleph_0}\).
It remains to prove that \([C]=X\), after which one may conclude that the cardinality of the \(t\)-pseudobase \(B\leq 2^{\aleph_0}\).
Theorem 2 has the following generalization.
Theorem \(2'\). Let \(X\) be a space such that from every open cover of cardinality \(2^\tau\) \((\tau\geq \aleph_0)\) one can select a cover of cardinality \(\leq \tau\), and let \(B\) be its \(t\)-pseudobase of point cardinality \(\leq \tau\) on an everywhere dense subset \(M\) of \(X\). Then the \(t\)-pseudobase \(B\) has cardinality \(\leq 2^\tau\).
For \(\tau=\aleph_0\), Theorem \(2'\) gives a stronger assertion than Theorem 2.
We now give a theorem on the cardinality of \(t\)-pseudobases in metric spaces.
Theorem 3. A metric space of weight \(\leq 2^\tau\) has a \(t\)-pseudobase of cardinality \(\leq \tau\). In particular, a metric space whose weight is \(2^{\aleph_0}\) has a countable \(t\)-pseudobase.
We shall preface the proof of the theorem with several lemmas.
Lemma 1. Suppose there is a condensation \(f:X\to Y\), where \(Y\) has a \(t\)-pseudobase of cardinality \(\tau\). Then \(X\) also has a \(t\)-pseudobase of cardinality \(\tau\).
Indeed, if \(\omega=\{\omega_\alpha\}\) is a \(t\)-pseudobase of cardinality \(\tau\) in \(Y\), then \(f^{-1}\omega=\{f^{-1}\omega_\alpha\}\) is a \(t\)-pseudobase in \(X\), in view of the fact that \(f\) is a condensation.
Lemma 2. The bicompactum \(I^\tau\) (the product of \(\tau\) copies of a line segment) has a \(t\)-pseudobase of cardinality \(\tau\).
On each segment factor \(I_\alpha\) there is a countable base \(\omega_\alpha=\{U_{\alpha i}\}\); a typical element of the \(t\)-pseudobase is the product of \(U_{\alpha i}\) with all the remaining segment factors (except \(I_\alpha\)).
Lemma 3. A space of isolated points of cardinality \(\leq 2^\tau\) has a \(t\)-pseudobase of cardinality \(\leq \tau\).
A space of isolated points of cardinality \(2^\tau\) condenses onto \(I^\tau\). Since \(I^\tau\), by Lemma 2, has a \(t\)-pseudobase of cardinality \(\tau\), the space of isolated points, by Lemma 1, also has a \(t\)-pseudobase of cardinality \(\tau\).
We prove Theorem 3. By Bing’s criterion, the metric space \(M\) has a \(\sigma\)-discrete base \(\gamma=\{\gamma_i\}\), where each \(\gamma_i\) is a discrete system of elements; \(\gamma=\{\gamma_{i\alpha}\}\). The system \(\gamma_i\) is isomorphic to a space of isolated points of cardinality \(\leq 2^\tau\). Consequently, by Lemma 3, there is such a collection of subsystems in \(\gamma_i\) of cardinality \(\leq \tau\) that every element of
\(\gamma_i\) can be obtained as the intersection of certain elements of this family; denote this family by \(\Gamma_i\). It is not hard to verify that \(\Gamma = \{\Gamma_i\}\), \(i = 1, 2, \ldots\), is a \(t\)-pseudobase in \(M\). The theorem is proved.
Remark. The cardinality of a space having a countable \(t\)-pseudobase is \(\leq 2^{\aleph_0}\). Indeed, every point, by the definition of a \(t\)-pseudobase, can be obtained as the intersection of some subsystem of elements of the \(t\)-pseudobase, and the set of subsystems of a countable system is \(\leq 2^{\aleph_0}\). Hence it follows that Theorem 3 is final.
In conclusion we pose a question. If a bicompactum has a countable \(t\)-pseudobase, does it also have a countable base? In general, does the minimal cardinality of a \(t\)-pseudobase of a bicompactum coincide with its weight?*
Moscow State University
named after M. V. Lomonosov
Received
23 X 1964
References
- H. H. Corson, E. Michael, Illinois J. Math., 8, No. 2, 351 (1964).
- A. Mishchenko, DAN, 144, No. 5, 985 (1962).
- V. Proizvolov, DAN, 154, No. 1, 55 (1964).
* Note added in proof. After the article had been submitted for typesetting, an affirmative answer to this question was obtained.