W. Sun, School of Mathematics and Statistics, Shandong University at Weihai, Weihai, Shandong, 264209, P. R. China, e-mail: whsun_maths@163.com; Y. Xu, School of Mathematics and System Sciences, Shandong University, Jinan, Shandong, 250100, P. R. China, e-mail: xuyuming@sdu.edu.cn; N. Li, School of Mathematics and Physics, Dalian Jiaotong University, Dalian, Liaoning,116028, P. R. China e-mail: lining@djtu.edu.cn.
Abstract: In this article we introduce the notion of strongly ${\KC}$-spaces, that is, those spaces in which countably compact subsets are closed. We find they have good properties. We prove that a space $(X, \tau)$ is maximal countably compact if and only if it is minimal strongly ${\KC}$, and apply this result to study some properties of minimal strongly ${\KC}$-spaces, some of which are not possessed by minimal ${\KC}$-spaces. We also give a positive answer to a question proposed by O. T. Alas and R. G. Wilson, who asked whether every countably compact ${\KC}$-space of cardinality less than $c$ has the ${\FDS}$-property. Using this we obtain a characterization of Katetov strongly ${\KC}$-spaces and finally, we generalize one result of Alas and Wilson on Katetov-${\KC}$ spaces.
Keywords: ${\KC}$-space, strongly ${\KC}$-space, ${\FDS}$-property, maximal (countably) compact
Classification (MSC 2000): 54A10, 54D25, 54D55
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