Pseudocompact versus countably compact in first countable spaces

Abstract

The primary objective of this work is to construct spaces that are "pseudocompact but not countably compact," abbreviated as PNC, while endowing them with additional properties. First, motivated by an old problem of van Douwen, we construct from CH a locally compact and locally countable first countable PNC space with countable spread. A space is deemed "densely countably compact", denoted as DCC for brevity, if it possesses a dense, countable 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 $\Psi$-space belongs to the class of DRC spaces but is not DCC. Lastly, if $p\in {\omega}^*$ is not a P-point, then $T(p)$, representing the type of $p$ in ${\omega}^*$, constitutes a pseudocompact subspace of ${\omega}^*$ that is not \DRC. When considering a topological property denoted as $Q$, we define a space $X$ as "hereditarily $Q$" if every regular closed subspace of $X$ also possesses property $Q$. The Tychonoff plank and the $\Psi$-spaces are not hereditary examples. However, the aforementioned space $T(p)$ is a hereditary example, albeit not being first countable. In this paper we want to find (first countable) examples which separates these properties hereditarily. We have obtained the following result. (1) There is a DCC space $X$ such that no $H\in RC(X)^+$ is countably compact. (2) If CH holds, then there is a DRC space $Y$ such that no $H\in RC(Y)^+$ is DCC. (3) If CH holds, then there is a first countable pseudocompact space $Z$ such that no $H\in RC(Z)^+$ is DRC.