Every k-separable Cech-complete space is subcompact

We establish that a Čech-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 Čech-complete space contains a dense σ-compact subspace then it is subcompact. This result is new even for separable Čech-complete spaces. Furthermore, if X is a compact space of countable tightness then X/A is subcompact for any countable set A ⊂ X. We also show that any Gδ-subset of a dyadic compact space is subcompact and give a comparatively simple proof of the fact that X/A is subcompact for any linearly ordered compact space X and any countable set A ⊂ X.