Compact {C}-closed spaces need not be sequential
A space is C-closed, see \textit{M. Ismail} and \textit{P. Nyikos} [Topology Appl. 11, 281--292 (1980; Zbl 0434.54018)], if all of its countably compact subsets are closed. This property is implied by sequentiality and implies countable tightness in countably compact spaces. The author proves that in ``the'' Cohen model (any model obtained by adding Cohen reals to a model of the Continuum Hypothesis) compact C-closed spaces are sequentially compact and hence sequential. He then goes on to construct a model in which \(\mathbb{N}\) has a compactification \(\gamma\mathbb{N}\) with a remainder that is Fréchet-Urysohn but in which \(\mathbb{N}\) itself is completely divergent (every infinite subset has infinite subsets with disjoint closures). This space is C-closed but not sequential and answers three related questions, collected in [\textit{D. B. Shakhmatov}, in: Recent progress in general topology. Papers from the Prague Toposym 1991, held in Prague, Czechoslovakia, Aug. 19-23, 1991. Amsterdam: North-Holland. 571--640 (1992; Zbl 0801.54001)] in one fell swoop.
scientific article; zbMATH DE number 6837922