Spaces of set-valued functions
HTML articles powered by AMS MathViewer
- by David N. O’Steen PDF
- Trans. Amer. Math. Soc. 169 (1972), 307-315 Request permission
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.References
-
C. Berge, Topological spaces, Macmillan, New York, 1963, pp. 109-115.
- Jane M. Day and Stanley P. Franklin, Spaces of continuous relations, Math. Ann. 169 (1967), 289–293. MR 210092, DOI 10.1007/BF01362352
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Phillip Zenor, On the completeness of the space of compact subsets, Proc. Amer. Math. Soc. 26 (1970), 190–192. MR 261538, DOI 10.1090/S0002-9939-1970-0261538-4
Additional Information
- © Copyright 1972 American Mathematical Society
- 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