Strong disconnectedness properties and remainders in compactifications

We study when a Tychonoff space X is countably compact at infinity, that is, the remainder of X in some (in any) Hausdorff compactification of X is countably compact. Though no internal characterization of such spaces is given, we present some sufficient conditions for that. In particular, we prove that every dense-in-itself extremally disconnected space of countable tightness is countably compact at infinity. Therefore, every countable extremally disconnected space without isolated points is countably compact at infinity. We also show how to construct certain extremally disconnected spaces without isolated points.