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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Spaces of set-valued functions


Author: David N. O’Steen
Journal: Trans. Amer. Math. Soc. 169 (1972), 307-315
MSC: Primary 54C60
MathSciNet review: 0336699
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Abstract: If $ X$ and $ Y$ are topological spaces, the set of all continuous functions from $ X$ into $ CY$, the space of nonempty, compact subsets of $ Y$ with the finite topology, contains a copy (with singleton sets substituted for points) of $ {Y^X}$, the continuous point-valued functions from $ X$ into $ Y$. It is shown that $ {Y^X}$ is homeomorphic to this copy contained in $ {(CY)^X}$ (where all function spaces are assumed to have the compact-open topology) and that, if $ X$ or $ Y$ is $ {T_2},{(CY)^X}$ is homoemorphic to a subspace of $ {(CY)^{CX}}$. Further, if $ Y$ is $ {T_2}$, then these images of $ {Y^X}$ and $ {(CY)^X}$ are closed in $ {(CY)^X}$ and $ {(CY)^{CX}}$ respectively.

Finally, it is shown that, under certain conditions, some elements of $ {X^Y}$ may be considered as elements of $ {(CY)^X}$ and that the induced $ 1$-$ 1$ function between the subspaces is open.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0336699-5
Keywords: Function spaces, finite topology, compact-open topology
Article copyright: © Copyright 1972 American Mathematical Society