The primary objective of this work is to construct spaces that are “pseudocompact but not countably compact”, abbreviated as P-NC, while endowing them with additional properties.
First, motivated by an old problem of van Douwen concerning first countable P-NC spaces with countable extent, we construct from CH a locally compact and locally countable first countable P-NC space with countable spread.
A space is deemed densely countably compact, denoted as DCC for brevity, if it possesses a dense, countably compact subspace. Moreover, a space qualifies as densely relatively countably compact, abbreviated as DRC, if it contains a dense subset D such that every infinite subset of D has an accumulation point in X.
A countably compact space is DCC, a DCC space is DRC, and a DRC space is evidently pseudocompact. The Tychonoff plank is a DCC space but is not countably compact. A Ψ-space belongs to the class of DRC spaces but is ¬DCC. Lastly, if is not a P-point, then , representing the type of p in , constitutes a pseudocompact subspace of that is ¬DRC.
When considering a topological property denoted as Q, we define a space X as “R-hereditarily Q” if every regular closed subspace of X also possesses property Q. The Tychonoff plank and the Ψ-space are not R-hereditary examples for separating the above-mentioned properties. However, the aforementioned space is an R-hereditary example, albeit not being first countable.
In this paper we want to find (first countable) examples which separate these properties R-hereditarily. We have obtained the following result.
- (1)There is a R-hereditarily “DCC, but not countably compact” space.
- (2)If CH holds, then there is a R-hereditarily “DRC, but ¬DCC” space.
- (3)If , then there is a first countable, R-hereditarily “pseudocompact, but ¬DRC” space.
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