  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.
