On finally compact spaces (Q700342)

For an infinite cardinal \(\kappa\), \(\kappa^+\) denotes the smallest cardinal greater than \(\kappa\). A space \(X\) is called finally \(\kappa^+\)-compact if every open cover of \(X\) has a subcover with cardinality \(\leq\kappa\). The authors define weakly \(\kappa\overline{\theta}\)-refinable spaces and study conditions under which a countably compact space is finally \(\kappa^+\)-compact and under which a space in which discrete families are countable is finally \(\kappa^+\)-compact. A space \(X\) is weakly \(\kappa\overline{\theta}\)-refinable if every open cover of \(X\) has an open refinement \({\mathcal C}=\bigcup_{\alpha<\kappa}{\mathcal C}_\alpha\) such that for each \(x\in X\) there exists an \(\alpha(x)<\kappa\) with \(0<\text{ord}(x,{\mathcal C}_{\alpha(x)})<\kappa\) and the open cover \(\{\bigcup{\mathcal C}_\alpha:\alpha<\kappa\}\) is point-finite. In case \(\kappa=\omega\), this notion coincides with weak \(\overline{\theta}\)-refinability defined by J. C. Smith. Main results are: (1) A countably compact space \(X\) is finally \(\kappa^+\)-compact if and only if every open cover of \(X\) has an open refinement \({\mathcal C}=\bigcup_{n<\omega}{\mathcal C}_n\) such that for each \(x\in X\) there exists an \(n(x)<\omega\) such that \(0<\text{ord}(x,{\mathcal C}_{n(x)})\leq\kappa\). (2) A topological space in which discrete families are countable is finally \(\kappa^+\)-compact if it is weakly \(\kappa\overline{\theta}\)-refinable. The case \(\kappa=\omega\) of (1) is a theorem of \textit{H. H. Wicke} and \textit{J. M. Worrell jun.} [Proc. Am. Math. Soc. 55, 427-431 (1976; Zbl 0323.54013)] that a countably compact, weakly \(\delta\theta\)-refinable space is Lindelöf, and hence, compact. The case \(\kappa=\omega\) of (2) is a theorem of \textit{J. C. Smith} [ibid. 53, 511-517 (1975; Zbl 0338.54013)] that a space in which discrete families are countable is Lindelöf if it is weakly \(\overline{\theta}\)-refinable.