Set-valued maps and some generalized metric spaces.

To give characterizations of monotonically countably paracompact spaces with set-valued maps, Yamazaki [22] introduced the notion of strictly increasing closed cover of a topological space with which the boundedness of a set-valued map was defined. In this paper, we show that most of generalized metric spaces such as stratifiable spaces, semi-metrizable spaces can be characterized with set-valued maps with values into the family of all closed nonempty subsets of a space which has a strictly increasing closed cover. Moreover, as an application, we use the results obtained to give characterizations of the corresponding spaces with generalized real-valued functions.