Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Clopen sets in hyperspaces

Author: Paul Bankston
Journal: Proc. Amer. Math. Soc. 54 (1976), 298-302
MSC: Primary 54B20; Secondary 06A20
MathSciNet review: 0405332
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 ``remembers'' the upper semilattice structure of $ B$.

References [Enhancements On Off] (What's this?)

  • [1] D. Curtis and R. Schori, $ {2^x}$ and $ C(X)$ are homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 80 (1974), 927-931. MR 0353235 (50:5719)
  • [2] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 3, 315. MR 0006505 (3:315b)
  • [3] J. Nagata, Modern dimension theory, Bibliotheca Math., vol. 6, Interscience, New York, 1965. MR 34 #8380.
  • [4] K. Kuratowski, Topology. Vol. 1, 5th ed., PWN, Warsaw; Academic Press, New York, 1966. MR 36 #840. MR 0217751 (36:840)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B20, 06A20

Retrieve articles in all journals with MSC: 54B20, 06A20

Additional Information

Keywords: Hyperspace, Boolean space, Boolean algebra, clopen sets, characteristic algebra, Stone space
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society