Previous |  Up |  Next


selections; l.s.c. set-valued maps; mesocompact; sequentially mesocompact; persevering compact set-valued maps
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.
[1] Boone J.R.: Some characterization of paracompactness in $\chi $-space. Fund. Math. 72 (1971), 145–155. MR 0295291
[2] Choban M.: Many-valued mappings and Borel sets, II. Trans. Moscow Math. Soc. 23 (1970), 286–310.
[3] Engelking R.: General Topology. Revised and completed edition, Heldermann, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[4] Michael E.: A theorem on semicontinuous set-valued funtions. Duke Math. 26 (1956), 647–652. DOI 10.1215/S0012-7094-59-02662-6 | MR 0109343
[5] Michael E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152–182. DOI 10.1090/S0002-9947-1951-0042109-4 | MR 0042109 | Zbl 0043.37902
[6] Miyazaki K.: Characterizations of paracompact-like properties by means of set-valued semi-continuous selections. Proc. Amer. Math. Soc. 129 (2001), 2777–2782. DOI 10.1090/S0002-9939-01-06204-9 | MR 1838802 | Zbl 0973.54009
[7] Nedev S.: Selection and factorization theorems for set-valued mapings. Serdica 6 (1980), 291–317. MR 0644284
[8] Yan P.-F.: $\tau$ selections and its applictions on BCO. J. Math. (in Chinese) 17 (1997), 547–551. MR 1675535
Partner of
EuDML logo