Topologies for $2^{x}$; set-valued functions and their graphs
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- by Louis J. Billera PDF
- Trans. Amer. Math. Soc. 155 (1971), 137-147 Request permission
Abstract:
We consider the problem of topologizing ${2^X}$, the set of all closed subsets of a topological space $X$, in such a way as to make continuous functions from a space $Y$ into ${2^X}$ precisely those functions with closed graphs. We show there is at most one topology with this property, and if $X$ is a regular space, the existence of such a topology implies that $X$ is locally compact. We then define the compact-open topology for ${2^X}$, which has the desired property for locally compact Hausdorff $X$. The space ${2^X}$ with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.References
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Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 155 (1971), 137-147
- MSC: Primary 54.65
- DOI: https://doi.org/10.1090/S0002-9947-1971-0273584-0
- MathSciNet review: 0273584