Every $$k$$ k -separable Čech-complete space is subcompact

Abstract

We establish that a Cech-complete space $$X$$ must be subcompact if it has a dense subspace representable as the countable union of closed subcompact subspaces of $$X$$ . In particular, if a Cech-complete space contains a dense $$\sigma $$ -compact subspace then it is subcompact. This result is new even for separable Cech-complete spaces. Furthermore, if $$X$$ is a compact space of countable tightness then $$X\backslash A$$ is subcompact for any countable set $$A\subset X$$ . We also show that any $$G_\delta $$ -subset of a dyadic compact space is subcompact and give a comparatively simple proof of the fact that $$X\backslash A$$ is subcompact for any linearly ordered compact space $$X$$ and any countable set $$A\subset X$$ .