MATHEMATICS

V. PROIZVOLOV

ON COMPACTIFICATIONS OF COMPLETELY REGULAR SPACES

(Presented by Academician P. S. Aleksandrov on 14 VI 1962)

In this note an example is constructed of a completely regular space that is not compactifiable in any normal space, and a theorem on compactifiability in bicompacta is proved, generalizing one result of I. L. Raukhvarger \((^{1})\).

Take the product \(P' = I \times W_\alpha\); \(I\) is an interval of the line, and \(W\) is the set of all ordinal numbers \(\leq \alpha\), where \(\alpha\) is any, but fixed once and for all, transfinite number such that no sequence of cardinality \(\leq 2^c\) is cofinal in it. From \(P'\) we remove the points whose first coordinate is irrational and whose second is equal to \(\alpha\); we obtain the space \(P\). The space \(P\) is completely regular and peripherally bicompact. We shall prove that \(P\) is not compactifiable in any normal space.

Consider the auxiliary space \(Q = (\Xi \times W_\alpha)\setminus(\sigma,\alpha)\), where \(\Xi\) is a countable sequence with limit point \(\sigma\), and \(W_\alpha\) is the set of all transfinite numbers up to \(\alpha\), inclusive.

Lemma. Let a compactification \(f: Q \to X\) be given, where \(X\) is completely regular. It is assumed here that the image of the set of all points of \(Q\) whose second coordinate is equal to \(\alpha\) has no limit point in \(X\). Then \(f\) is a homeomorphism.

Suppose that \(f\) is not a homeomorphism. Then \([fA]\ne fA\), where \(A\) is some closed subspace of \(Q\). Take a point \(\xi\in [fA]\setminus fA\) and an arbitrary neighborhood \(O\xi\) of it. We shall show that \([f^{-1}O\xi]\ni(\sigma,\alpha)\); here the closure is taken in the space \(Q\) augmented by the point \((\sigma,\alpha)\); the so augmented \(Q\) we denote by \(\overline Q\). We argue by contradiction. Let \((\sigma,\alpha)\notin [f^{-1}O\xi]\). Then \(A' = A\cap [f^{-1}O\xi]_{\overline Q}\) is bicompact. The bicompact \(fA'\) coincides with \(fA\cap [O\xi]\), since \(f[f^{-1}O\xi]=[O\xi]\). But \(\xi\in [fA]\setminus fA\), and therefore \(fA\cap [O\xi]\) cannot be bicompact. Thus \((\sigma,\alpha)\in [f^{-1}O\xi]_{\overline Q}\). Moreover, we shall show that \([f^{-1}O\xi]\) contains a countable subset of the set of points with second coordinate \(\alpha\). Indeed, since \((\sigma,\alpha)\in [f^{-1}O\xi]_{\overline Q}\), the open set \(f^{-1}O\xi\) contains a set \(V\) cofinal in the set of all points with first coordinate \(\sigma\). The set \([OV]\), where \(OV\) is an arbitrary neighborhood of \(V\), contains all points with second coordinate \(\alpha\), except, perhaps, finitely many of them. In particular, this property is possessed by \([f^{-1}O\xi]\). Since \(O\xi\) is an arbitrary neighborhood of \(\xi\), \(\xi\) is a limit point for the image of the set of points with second coordinate \(\alpha\). This contradicts the hypothesis of the lemma. Thus \(f\) is a homeomorphism.

We return to the space \(P\). Suppose that there exists a compactification \(f: P\to X\), where \(X\) is normal. Denote by \(G\) the set of points of \(P\) with second coordinate \(\alpha\). The cardinality \(|fG|\leq 2^c\), since \(G\) is countable. Every countable subset \(N\subseteq fG\) has at least one limit point. Indeed, suppose that some \(N\subseteq fG\) has no limit points in \(X\). Then \(f^{-1}N\subset (I,\alpha)\) has no limit points in \(P\). Take in \(P\) a closed subspace \(R\) such that \(R\) is homeomorphic to \(Q\) and

\(R \cap (I,\alpha) \subseteq f^{-1}N\). By the lemma, the mapping \(f\) on \(R\) is a homeomorphism. The set \(fR\) is closed in \(X\) and is not normal; hence \(X\) is a nonnormal space. Thus every \(N \subseteq fG\) has a limit point. Further, \([fG]\) is countable, since \(G\) is not embedded in an everywhere dense subset of a countable compactum, because a countable compactum has an isolated point, whereas \(G\) has none. It is not hard to show that \(D=[fG]\setminus G\) is everywhere dense in \([fG]\) and, in particular, \(D\) is not closed in \(X\). The set \(f^{-1}[fG]=f^{-1}D\cup G\) is a closed set, but then \(f^{-1}D\) is closed and bicompact, since the cardinality of \(f^{-1}D \leq 2^c\), and \(\alpha\) was chosen in a special way. But then \(D\) is also bicompact, which contradicts the fact that \(D\) is not closed in the space \(X\). Thus \(P\) is not embedded in a normal space.

I. Raukhvarger proved [1] the theorem: if a countable number of points is removed from a compactum, then the remaining space is embedded in a compactum.

Here the following will be proved.

Theorem. Let a bicompactum \(B\) be given which is a product of compacta,

\[ B=\prod_\alpha^\tau K_\alpha . \]

If a countable number of points is removed from \(B\), then the remaining space is embedded in a bicompactum.

It is necessary to prove that \(B'=B\setminus A\) is embedded in a bicompactum, where \(A\) is a countable subset, \(A=\{a_i\},\ i=1,2,\ldots\). We may suppose that no point \(a_i\) is isolated in \(B\): the union of all points of \(A\) isolated in \(B\) is an open set \(V_1\), so that \(B\setminus V_1\) is a bicompactum; if among \(A\cap V_1\) in the bicompactum \(B\setminus V_1\) there are isolated points, then in any case after a countable number of such steps we shall get rid of them, and everything will reduce to the case of a space without isolated points.

The point \(a_i\) has coordinate \(a_{i\alpha}\) with respect to the compactum \(K_\alpha\). Take the closed \(\frac1i\)-neighborhood \(Oa_{i\alpha}\) of the point \(a_{i\alpha}\) in \(K_\alpha\) (i.e. the set of all points at distance from \(a_{i\alpha}\) not greater than \(\frac1i\)). In the set \(\prod_\alpha^\tau Oa_{i\alpha}\) choose any point \(\xi_i\) distinct from \(a_i\). In choosing \(\xi_i\) for different \(i\), we shall ensure that they are pairwise distinct.

We now construct a decomposition of the bicompactum \(B\): the elements of the decomposition will be the pairs of points \((a_i,\xi_i)\) and the individual points of the bicompactum \(B\). We shall prove that this decomposition is continuous. Let \(U_\alpha\) be a one-index neighborhood of some element \(r\) of the decomposition. From Raukhvarger’s theorem for compacta it follows that in \(U_\alpha\) one can inscribe such a one-index neighborhood \(V_\alpha\) of the element \(r\) of the decomposition that, as soon as some element of the decomposition intersects \(V_\alpha\), it necessarily lies in \(U_\alpha\), which proves everything.

Moscow State University
named after M. V. Lomonosov

Received
13 VI 1962

REFERENCES

1. I. Raukhvarger, DAN, 66, No. 1, 13 (1949).