Mesocompactness and selection theory. (Q2898340)

Motivated by results of \textit{E. Michael} [Duke Math. J. 26, 647--651 (1959; Zbl 0151.30805)] and of \textit{M. M. Choban} [Tr. Mosk. Mat. Obshch. 23, 277--301 (1970; Zbl 0231.54013)], which characterize paracompactness and metacompactness in terms of the existence of some semi-continuous compact-valued selections, the first author characterized metalindelöf spaces by the existence of separable-valued l.s.c. selections in [\textit{P. Yan}, J. Math., Wuhan Univ. 17, No. 4, 547--551 (1997; Zbl 0930.54022)].NEWLINENEWLINEThe present paper is devoted to mesocompact (sequentially mesocompact) spaces, i.e. topological spaces such that their open covers have open refinements which are finite on compact sets (on convergent sequences). The authors show that regular mesocompact spaces can be characterized by the existence of l.s.c. compact-valued selections which preserve relative compactness for l.s.c. closed valued mappings to complete metric spaces.NEWLINENEWLINEA similar result is obtained for sequentially mesocompact spaces. The existence of pairs of an l.s.c. compact-valued selection and a larger compact-preserving selection is shown to be sufficient for mesocompactness. With the help of the existence of a suitable pair of selections, the sequential mesocompactness of normal spaces is characterized.NEWLINENEWLINEQuestions concerning possible improvements of the last result are posed.