A topological space is called mesocompact (sequentially mesocompact) if for every open cover of , there exists an open refinement of such that is finite for every compact set (converging sequence including its limit point) in . In this paper, we give some characterizations of mesocompact (sequentially mesocompact) spaces using selection theory.
A version of Michael's theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a common constant) is proven. Uniform equi-LCⁿ property for such families of cells is checked. An example is given showing that the assumption about the common Lipschitz constant cannot be omitted.
A brief account of the connections between Carathéodory multifunctions, Scorza-Dragoni multifunctions, product-measurable multifunctions, and superpositionally measurable multifunctions of two variables is given.