Extending the discrete finite chain condition
Recall that a space \(X\) is feebly compact if every locally finite family of open subsets of \(X\) is finite; has the discrete finite chain condition (is DFCC) if every discrete collection of nonempty open sets of \(X\) is finite; and is pseudocompact if every continuous real valued function defined on \(X\) is bounded. It is known that in a regular space the first two of these properties are equivalent, and in a completely regular space all of them are equivalent. A space is called locally feebly compact (DFCC, pseudocompact) if each of its points has a feebly compact (DFCC, pseudocompact) neighborhood. In [Pac. J. Math. 36, 819-825 (1971; Zbl 0216.192)] the reviewer proved that every locally feebly compact, first countable zero-dimensional \(T_ 1\)-space has a feebly compact first countable zero-dimensional \(T_ 1\)- extension space. In [Fundam. Math. 91, 189-202 (1976; Zbl 0341.54016)] \textit{G. M. Reed} proved that every locally compact, separable Moore space can be embedded in a locally compact pseudocompact Moore space, and he asked whether every locally compact separable Moore space can be densely embedded. \textit{T. Terada} and \textit{J. Terasawa} proved in [Proc. Am. Math. Soc. 85, 95-99 (1982; Zbl 0477.54012)] that for Tykhonov spaces, every first countable locally pseudocompact (locally compact) space \(Y\) has a first countable (locally compact) pseudocompactification which is locally compact if \(Y\) is. Then in [Topology Appl. 27, No. 1, 11-28 (1987; Zbl 0633.54012)] the reviewer proved the following: Let \(Y\) be a first countable \(T_ 3\)-space in which the set \(B = \{y \in Y : Y\) is not locally feebly compact at \(y\}\) is countable, and there is a dense subset \(D\) of \(Y\) such that each point of \(D\) has a neighborhood base consisting of feebly compact open subsets of \(Y\). Then \(Y\) has a first countable feebly compact \(T_ 3\)-extension space \(X\). In the latter article the reviewer asked (1) if every locally feebly compact, first countable \(T_ 3\)-space has a first countable feebly compact \(T_ 3\)- extension space, and (2) if every locally feebly compact, separable Moore space has a feebly compact Moore extension space. More recently, in [Some Moore pseudocompactifications, presented at the University of South Carolina 1991 Fall Topology Conference] \textit{P. J. Nyikos} established the theorem: A Tykhonov locally pseudocompact Moore space \(Y\) can be densely embedded in a pseudocompact (feebly compact) Moore space if and only if \(Y\) is separable. Nyikos also obtained as a corollary to his proof the result: Let \(Y\) be a locally compact Moore space. Then \(Y\) can be densely embedded in a locally compact, pseudocompact Moore space if and only if \(Y\) is separable. Nyikos' results answered Reed's question the the Tykhonov version of question (2). In this article the author develops some new, complicated techniques and uses them to obtain the following very nice Theorem 2.2 (CH). Every (first countable) locally DFCC, \(T_ 3\)-space \(X\) with \(\pi\)-weight at most \(2^{\omega}\) can be embedded in a (first countable) DFCC, \(T_ 3\)-space \(Y\) as a dense open subspace. From 2.2 he obtains Corollary 2.3 (CH). If \(X\) is a first countable locally DFCC, \(T_ 3\)-space and \(| X| \leq 2^ \omega\), then \(X\) can be embedded in a first countable DFCC, \(T_ 3\)-space \(Y\) as a dense open subspace. Corollary 2.3 partially answers (1). In contrast with earlier constructions, the space \(Y\setminus X\) in 2.2 is not discrete (and does not even have a \(\sigma\)- closed discrete dense subset). Moreover, the author constructs a first countable locally compact Hausdorff space \(X\) such that \(Y \setminus X\) is not (closed) discrete whenever \(Y\) is a first countable DFCC, \(T_ 3\)-space containing \(X\) as a dense subspace. The author's last theorem, 3.3, states: Let \(Y\) be a first countable DFCC, \(T_ 3\)-space, \(X\) connected, locally connected, locally DFCC, and such that \(\overline{X} = Y\). If \(Y \setminus X\) is discrete then \(Y\) is locally connected. Theorem 3.3 is used to show that if \(X = (0,1]\) and \(Y\) is any first countable DFCC, \(T_ 3\)-space such that \(\overline{X} = Y\) and \(Y \setminus X\) is discrete, then \(Y\) is (homeomorphic) with \([0,1]\).
scientific article; zbMATH DE number 474696
scientific article