Compact C-closed spaces need not be sequential

Abstract

We obtain an independence result connected to the classic Moore–Mrowka problem. A property known to be intermediate between sequential and countable tightness in the class of compact spaces is the notion of a space being C-closed. A space is C-closed if every countably compact subset is closed. We prove it is consistent to have a compact C-closed space that is not sequential. Our example also answers a question of Arhangelskii by producing a compactification of the countable discrete space which is not itself sequential and yet it has a Fréchet–Urysohn remainder. Ismail and Nyikos showed that compact C-closed spaces are sequential if ${2^{\mathfrak{t}} > 2^\omega}$. We prove that compact C-closed spaces are sequential also holds in the standard Cohen model.