Supercompact space

In mathematics, in the field of topology, a topological space is called supercompact if there is a subbasis such that every open cover of the topological space from elements of the subbasis has a subcover with at most two subbasis elements. Supercompactness and the related notion of superextension was introduced by J. de Groot in 1967.

Examples

By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not all) compact spaces are supercompact. The following are examples of supercompact spaces:

• Compact linearly ordered spaces with the order topology and all continuous images of such spaces (Bula et al. 1992)
• Compact metrizable spaces (due originally to M. Strok and A. Szymański 1975, see also Mills 1979)
• A product of supercompact spaces is supercompact (like a similar statement about compactness, Tychonoff's theorem, it is equivalent to the axiom of choice, Banaschewski 1993)

Some Properties

Some compact Hausdorff spaces are not supercompact; such an example is given by the Stone–Čech compactification of the natural numbers (with the discrete topology) (Bell 1978).

A continuous image of a supercompact space need not be supercompact (Verbeek 1972, Mills—van Mill 1979).

In a supercompact space (or any continuous image of one), the cluster point of any countable subset is the limit of a nontrivial convergent sequence. (Yang 1994)

References

• B. Banaschewski, "Supercompactness, products and the axiom of choice." Kyungpook Math. J. 33 (1993), no. 1, 111—114.
• Bula, W.; Nikiel, J.; Tuncali, H. M.; Tymchatyn, E. D. "Continuous images of ordered compacta are regular supercompact." Proceedings of the Tsukuba Topology Symposium (Tsukuba, 1990). Topology Appl. 45 (1992), no. 3, 203—221.
• Murray G. Bell. "Not all compact Hausdorff spaces are supercompact." General Topology and Appl. 8 (1978), no. 2, 151—155.
• J. de Groot, "Supercompactness and superextensions." Contributions to extension theory of topological structures. Proceedings of the Symposium held in Berlin, August 14—19, 1967. Edited by J. Flachsmeyer, H. Poppe and F. Terpe. VEB Deutscher Verlag der Wissenschaften, Berlin 1969 279 pp.
• Engelking, R (1977), General topology, Taylor & Francis, ISBN 978-0-8002-0209-5.
• Malykhin, VI; Ponomarev, VI (1977), "General topology (set-theoretic trend)", Journal of Mathematical Sciences (New York: Springer) 7 (4): 587–629, doi:10.1007/BF01084982
• Mills, Charles F. (1979), "A simpler proof that compact metric spaces are supercompact", Proceedings of the American Mathematical Society (Proceedings of the American Mathematical Society, Vol. 73, No. 3) 73 (3): 388–390, doi:10.2307/2042369, JSTOR 2042369, MR 518526
• Mills, Charles F.; van Mill, Jan, "A nonsupercompact continuous image of a supercompact space." Houston J. Math. 5 (1979), no. 2, 241—247.
• Mysior, Adam (1992), "Universal compact T₁-spaces", Canadian Mathematical Bulletin (Canadian Mathematical Society) 35 (2): 261–266, doi:10.4153/CMB-1992-037-1.
• J. van Mill, Supercompactness and Wallman spaces. Mathematical Centre Tracts, No. 85. Mathematisch Centrum, Amsterdam, 1977. iv+238 pp. ISBN 90-6196-151-3
• M. Strok and A. Szymanski, "Compact metric spaces have binary bases. " Fund. Math. 89 (1975), no. 1, 81—91.
• A. Verbeek, Superextensions of topological spaces. Mathematical Centre Tracts, No. 41. Mathematisch Centrum, Amsterdam, 1972. iv+155 pp.
• Yang, Zhong Qiang (1994), "All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences", Proceedings of the American Mathematical Society (Proceedings of the American Mathematical Society, Vol. 122, No. 2) 122 (2): 591–595, doi:10.2307/2161053, JSTOR 2161053