MATHEMATICS

A. ARKHANGELSKII

ON EXTERNAL BASES OF SETS LYING IN BICOMPACTA

(Presented by Academician P. S. Aleksandrov, 22 I 1960)

In all that follows the spaces \(X\) and \(Y\) will be subspaces of some fixed space \(R\), and the terms “open,” “closed” will mean “open in \(R\),” respectively “closed in \(R\).”

Definition 1. A system of sets \(\gamma=\{E_\alpha:\alpha\in A\}\) of the space \(R\) is called a separating system of the pair of spaces \((X,Y)\) if, for every pair of points \(x\in X\), \(y\in Y\), there are \(E_\alpha\in\gamma\), \(E_\beta\in\gamma\) such that \(E_\alpha\ni x\), \(E_\beta\ni y\), and \(E_\alpha\cap E_\beta=\Lambda\).

If \(R\) is normal, then, as is easy to see, the following holds.

Lemma 1. Let \(\gamma=\{E_\alpha:\alpha\in A\}\) be a separating system of the pair \((X,Y)\) consisting of closed sets. Then there exists a system \(\tilde{\gamma}\) of open sets, separating the pair \((X,Y)\) and having the same cardinality as the system \(\gamma\).

For the proof of Lemma 1 it is enough to take, for each pair of disjoint sets \(E_\alpha,E_\beta\), \(\alpha,\beta\in A\), disjoint neighborhoods of them in the space \(R\): \(O_\alpha^\beta\) and \(O_\beta^\alpha\). By normality of the space \(R\), this can always be done. The totality of the open sets selected forms the required system \(\tilde{\gamma}\).

Definition 2. Let \((X,Y)\) be an arbitrary pair of subspaces of some containing space \(R\). We shall call a system \(B_Y^X\) of open sets of the space \(R\) a base of the space \(X\) relative to the space \(Y\) if, for every point \(x\in X\) and every one of its neighborhoods \(Ox\) (in the space \(R\)), there is a \(\Gamma_x\in B_Y^X\) such that \(x\in\Gamma_x\) and \(\Gamma_x\cap Y\subseteq Ox\). A special case: \(R=X\cup Y\). Still more special cases are:

a) \(Y=X\); then the base \(B_Y^X\) is a base of the space \(X\);

b) \(X\subset Y\); then \(B_Y^X\) is an external base of the space \(X\) in the space \(Y\);

c) \(X\cap [Y]=\Lambda\); then as \(B_Y^X\) one may take a system consisting of the single set \(X\).

Lemma 2. Let \(\gamma=\{E_\alpha:\alpha\in A\}\) be a system of open sets separating the pair \((X,Y)\), and suppose \(Y\) is bicompact. Then there exists a base of the space \(X\) relative to the space \(Y\) of the same cardinality as the system \(\gamma\).

Proof. Consider the system \(\tilde{\gamma}\) consisting of all possible finite intersections of sets of the system \(\gamma\). We shall show that \(\tilde{\gamma}\) forms a base of the space \(X\) relative to the space \(Y\). Let \(x_1\) be an arbitrary point of the space \(X\), and let \(Ox_1\ni x_1\) be an arbitrary neighborhood of it (in \(R\)). Then \(\Phi_1=Y\setminus Ox_1\) is a closed subset of \(Y\) and therefore is bicompact. For each point \(y\in\Phi_1\) there is a pair of sets \(A_y,C_y\) such that \(A_y\ni x_1\), \(C_y\ni y\), \(A_y\cap C_y=\Lambda\), and \(A_y\in\gamma\), \(C_y\in\gamma\).

The collection of sets \(C_y\), where \(y\) ranges over \(\Phi_1\), forms a cover of the set \(\Phi_1\), and, by bicompactness of \(\Phi_1\), from this cover one can choose...

a finite subcover \(\{C_{y_1},\ldots,C_{y_k}\}\). Then for the set

\[ A_{x_1}=\bigcap_{i=1}^{k} A_{y_i} \]

the following conditions are satisfied: \(A_{x_1}\in\widetilde{\gamma};\ A_{x_1}\ni x_1;\ A_{x_1}\cap Y\subset Ox_1\) (the last holds by virtue of the relation \((A_{x_1})\cap Y\setminus Ox_1=A_{x_1}\cap\Phi_1=\Lambda\)). Lemma 2 is proved.

Lemma 3. Let \(X\subseteq Y=R;\ X\) be a Borel set in \(Y\) of type \(G_{(\tau)}\). Then the cardinality of the system \(\Sigma=\{G\}\) of all open sets of the space \(Y\) participating in the given representation of the set \(X\) does not exceed \(\tau\), and for arbitrary points \(x\in X,\ y\in Y\setminus X\) there is a \(G\in\Sigma\) such that \(x\in G,\ y\notin G\).

Both assertions of the lemma are easily proved by induction (on the class of the representation of the set \(X\)). We confine ourselves to the proof of the second assertion. Let \(X\) be of class \(\lambda\). Two cases are possible: a) \(X=\bigcup_s X_\alpha\); b) \(X=\bigcap_\alpha X_\alpha\), where the \(X_\alpha\) are sets given by representations of classes \(<\lambda\). Then there exists an \(\alpha\) such that \(x\in X_\alpha,\ y\in R\setminus X_\alpha\), and, since \(X_\alpha\) is represented by operations over all or some \(G\in\Sigma\), but already by class \(<\lambda\), there exists, by the induction hypothesis, such a \(G\in\Sigma\) that \(x\in G,\ y\notin G\).

On the basis of the three lemmas proved above we easily obtain the proof of the main theorem of this note.

Main theorem. Let \(X\subseteq\Phi,\ \Phi\) be a bicompactum, and \(X\) a Borel set of type \(G_{(\tau)}\) in \(\Phi\). Then from the existence of a network \(*\) of the space \(X\) of cardinality \(\tau\) there follows the existence of an external base of the space \(X\) in the space \(\Phi\) having the same cardinality \(\tau\).

Proof. Consider \(\Sigma=\{G^\alpha\}\)—the system of open sets of the space \(\Phi\) participating in the given representation of the set \(X\), and put \(F^\alpha=\Phi\setminus G^\alpha\). The cardinality of the system \(\gamma_1=\{F^\alpha\}\) does not exceed \(\tau\) by Lemma 3. Moreover, by the same lemma, for any pair of points \(x\in X,\ y\in Y\setminus X\) there is an \(F^\alpha\ni y,\ F^\alpha\not\ni x\). Finally, all \(F^\alpha\) are closed sets in \(\Phi\) (and hence bicompacta). Denote by \(\gamma_2=\{P^\alpha\}\) the system consisting of the closures in \(\Phi\) of the elements of a network of cardinality \(\leq\tau\) of the space \(X\). The cardinality of the system \(\gamma_2\), obviously, does not exceed \(\tau\).

The system \(\gamma=\gamma_1\cup\gamma_2\) consists of closed sets of the space \(\Phi\); its cardinality does not exceed \(\tau\); by the properties of the systems \(\gamma_1\) and \(\gamma_2\), the system \(\gamma\) separates the pair \((X,\Phi)\). By Lemma 1 there exists a system of open sets of the space \(\Phi\) of the same cardinality which also separates the pair \((X,\Phi)\). By Lemma 2 there exists a base of the space \(X\) in the space \(\Phi\) of the same cardinality, and by the relation \(X\subseteq\Phi\) this base turns out to be an external base of the space \(X\) in the space \(\Phi\). The theorem is proved.

The theorem implies the following corollaries:

1. Under a continuous mapping onto a space of type \(G_{(\aleph_0)}\) in a bicompactum, in particular onto a complete space in the sense of Čech, the weight cannot increase.

2. In order that a space \(X\) of type \(F_{(\aleph_0)}\), where the original sets \(F\) are compacta (or, in general, arbitrary spaces with a countable base), be metrizable, it is sufficient that it be of type \(G_{(\aleph_0)}\) in some bicompactum \(\Phi\supseteq X\); moreover, then it itself has a countable base.

In conclusion the author expresses his deep gratitude to P. S. Aleksandrov, under whose supervision this work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
19 I 1960

References

1. A. Arhangel’skii, DAN, 126, No. 2, 239 (1959).

* By a network \((^1)\) of the space \(X\) is meant such a system \(\Sigma=\{A_\alpha\}\) of sets of this space that for any point \(x\in X\) and any neighborhood \(Ox\) of it there is an \(A_\alpha\in\Sigma\) such that \(x\in A_\alpha\subseteq Ox\).