Geodesic
Mesocompactness and selection theory
Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 149-157.

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A topological space $X$ is called mesocompact (sequentially mesocompact) if for every open cover ${\mathcal U}$ of $X$, there exists an open refinement ${\mathcal V}$ of ${\mathcal U}$ such that $\{V\in {\mathcal V}: V\cap K\neq \emptyset\}$ is finite for every compact set (converging sequence including its limit point) $K$ in $X$. In this paper, we give some characterizations of mesocompact (sequentially mesocompact) spaces using selection theory.
Classification : 54C60, 54C65
Keywords: selections; l.s.c. set-valued maps; mesocompact; sequentially mesocompact; persevering compact set-valued maps 
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Yan, Peng-fei; Yang, Zhongqiang. Mesocompactness and selection theory. Commentationes Mathematicae Universitatis Carolinae, Tome 53 (2012) no. 1, pp. 149-157. http://geodesic.mathdoc.fr/item/CMUC_2012__53_1_a10/