Clopen sets in hyperspaces
HTML articles powered by AMS MathViewer
- by Paul Bankston PDF
- Proc. Amer. Math. Soc. 54 (1976), 298-302 Request permission
Abstract:
Let $X$ be a space and let $H(X)$ denote its hyperspace (= all nonempty closed subsets of $X$ topologized via the Vietoris topology). Then $X$ is Boolean (= totally disconnected compact Hausdorff) iff $H(X)$ is Boolean; and if $B$ denotes the characteristic algebra of clopen sets in $X$ then the corresponding algebra for $H(X)$ is the free algebra generated by $B$ modulo the ideal which âremembersâ the upper semilattice structure of $B$.References
- D. W. Curtis and R. M. Schori, $2^{x}$ and $C(X)$ are homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 80 (1974), 927â931. MR 353235, DOI 10.1090/S0002-9904-1974-13579-2
- J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22â36. MR 6505, DOI 10.1090/S0002-9947-1942-0006505-8 J. Nagata, Modern dimension theory, Bibliotheca Math., vol. 6, Interscience, New York, 1965. MR 34 #8380.
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; PaĆstwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 298-302
- MSC: Primary 54B20; Secondary 06A20
- DOI: https://doi.org/10.1090/S0002-9939-1976-0405332-5
- MathSciNet review: 0405332