Spaces of set-valued functions
Abstract: If and are topological spaces, the set of all continuous functions from into , the space of nonempty, compact subsets of with the finite topology, contains a copy (with singleton sets substituted for points) of , the continuous point-valued functions from into . It is shown that is homeomorphic to this copy contained in (where all function spaces are assumed to have the compact-open topology) and that, if or is is homoemorphic to a subspace of . Further, if is , then these images of and are closed in and respectively.
Finally, it is shown that, under certain conditions, some elements of may be considered as elements of and that the induced - function between the subspaces is open.
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Keywords: Function spaces, finite topology, compact-open topology
Article copyright: © Copyright 1972 American Mathematical Society