Compact spaces and products of finite spaces
HTML articles powered by AMS MathViewer
- by Douglas Harris PDF
- Proc. Amer. Math. Soc. 35 (1972), 275-280 Request permission
Abstract:
It is shown that the compact spaces are precisely the extension closed subspaces of products of finite spaces, where a subspace is extension closed if every open cover of the subspace extends to an open cover of the entire space. Every closed subspace is extension closed, and for Hausdorff spaces the converse also holds. A single compact space U is constructed, such that every compact space is an extension closed subspace of a product of copies of U; this parallels precisely the property possessed by the unit interval with respect to compact Hausdorff spaces.References
- W. W. Comfort, A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1968), 97–110. MR 236880, DOI 10.4064/fm-63-1-97-110
- Douglas Harris, Extension closed and cluster closed subspaces, Canadian J. Math. 24 (1972), 1132–1136. MR 312466, DOI 10.4153/CJM-1972-119-8 —, Universal compact ${T_1}$ spaces, General Topology and Appl. (to appear).
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 35 (1972), 275-280
- MSC: Primary 54D30
- DOI: https://doi.org/10.1090/S0002-9939-1972-0298622-7
- MathSciNet review: 0298622