A connected pseudocompact space
In this article a space \(X\) is called pseudocompact if every discrete collection of open subsets of \(X\) is finite. Recall that a set \(A\) is said to be conditionally compact or relatively countably compact in a space \(X\) if every infinite subset of \(A\) has a limit point in \(X\). At the 1990 Summer Conference in General Topology at Long Island University the reviewer asked if there is a connected, pseudocompact, first countable, regular space which is not countably compact. The author obtains the very nice result: (CH) There is a first countable, separable, connected, pseudocompact, Tikhonov space without a dense relatively countably compact subset. The construction is a very complicated and clever blending of an Ostaszewski-type construction with a \(\Psi\)-space type of construction. The space produced actually contains \(\mathbb{R}\) as a dense open subspace. As was the case with the article by \textit{Ian J. Tree} [Topology Appl. 52, No. 3, 267-278 (1993; Zbl 0793.54018)], a most worthwhile accomplishment of the author is the generality achieved by devising a method of pseudocompactifying a given space \(X\) that involves adding to \(X\) something more than a disjoint closed discrete set. The author also obtains some results on pseudocompactifications, such as: Every locally pseudocompact first countable regular space with a dense locally conditionally compact subset can be embedded in a pseudocompact first countable regular space. Moreover, he introduces a property SLP which is implied by either locally countably compact or the possession of a dense conditionally compact subset, and yet implies locally pseudocompact such that every SLP, first countable, regular space can be embedded in a pseudocompact, first countable, regular space. (A space \(X\) is called strongly locally pseudocompact or SLP if there is a dense \(D\subset X\) such that, for all \(x\in X\), there is a closed neighborhood \(U\) of \(x\) such that \(D\cap U\) is relatively countably compact in \(U\).) These results partially answer the question raised by the reviewer in [ibid. 27, No. 1, 11-28 (1987; Zbl 0633.54012)]: Can every locally pseudocompact, first countable, regular space be embedded in a pseudocompact, first countable, regular space? For another partial answer to this question, see Theorem 2.2. from Tree's article. A number of open problems are considered, including the following one: Is there, in ZFC, a (locally) pseudocompact first countable regular (Tikhonov) space which is not SLP?
connected, pseudocompact, first countable, regular space
scientific article; zbMATH DE number 600035
scientific article