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