On some questions about \(KC\) and related spaces (Q1032912)

This paper gives an answer to some of the questions posed by \textit{O. T. Alas} and \textit{R. G. Wilson} in [Commentat. Math. Univ. Carol. 43, No.~4, 641--652 (2002; Zbl 1090.54015) and Houston J. Math. 32, No.~2, 493--504 (2006; Zbl 1100.54018)], and by \textit{O. T. Alas, M. G. Tkachenko, V. V. Tkachuk} and \textit{R. G. Wilson} in [Sci. Math. Jpn. 61, No.~3, 473-480 (2005; Zbl 1077.54010)] concerning \(KC\) spaces and minimal topological spaces: A topological space is \(KC\) if every compact subset is closed (consequently it is \(T_1\)). The authors construct in this paper a Hausdorff space that cannot be embedded in any compact \(KC\) space, and, using the notion of resolvable spaces, produce an example of a compact \(KC\) space in which every non-empty open set is dense. Two different spaces, one having countable pseudocharacter, are given to show that there are minimal Hausdorff spaces which are not \(k\)-spaces, i.e., a subspace \(A\) is closed in the space iff \(A \cap K\) for any compact subspace \(K\) is closed in \(K\). A \(Kat\breve{e}tov\) \(KC\) space is a topological space for which there exists a coarser, minimal \(KC\) topology. This paper gives an example of a countable \(KC\) space that does not contain any minimal \(KC\) topology, and shows that on any infinite set it is possible to define a \(KC\) topology which is not \(Kat\breve{e}tov\) \(KC\). A topological space is \(SC\) if a sequence \(S\) converging to \(x\) implies that \(S \cup \{x\}\) is closed. The authors show that any minimal \(SC\) space must be sequentially compact (and therefore compact by a previous result).