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
Compact metrizable spaces (due originally to Strok & 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.)

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).

A continuous image of a supercompact space need not be supercompact.

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.

Notes

↑ de Groot (1969).
↑ Bula et al. (1992).
↑ Banaschewski (1993).
↑ Bell (1978).
↑ Verbeek (1972); Mills & van Mill (1979).
↑ Yang (1994).

References

Banaschewski, B. (1993), "Supercompactness, products and the axiom of choice", Kyungpook Math Journal, 33 (1): 111–114
Bell, Murray G. (1978), "Not all compact Hausdorff spaces are supercompact", General Topology and Its Applications, 8 (2): 151–155, doi:10.1016/0016-660X(78)90046-6
Bula, W.; Nikiel, J.; Tuncali, H. M.; Tymchatyn, E. D. (1992), "Continuous images of ordered compacta are regular supercompact", Topology and Its Applications, 45 (3): 203–221, doi:10.1016/0166-8641(92)90005-K
de Groot, J. (1969), "Supercompactness and superextensions", in Flachsmeyer, J.; Poppe, H.; Terpe, F. (eds.), Contributions to extension theory of topological structures. Proceedings of the Symposium held in Berlin, August 14—19, 1967, Berlin: VEB Deutscher Verlag der Wissenschaften
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, 7 (4), New York: Springer: 587–629, doi:10.1007/BF01084982, S2CID 120365836
Mills, Charles F. (1979), "A simpler proof that compact metric spaces are supercompact", Proceedings of the American Mathematical Society, 73 (3), American Mathematical Society, Vol. 73, No. 3: 388–390, doi:10.2307/2042369, JSTOR 2042369, MR 0518526
Mills, Charles F.; van Mill, Jan (1979), "A nonsupercompact continuous image of a supercompact space", Houston Journal of Mathematics, 5 (2): 241–247
Mysior, Adam (1992), "Universal compact T₁-spaces", Canadian Mathematical Bulletin, 35 (2), Canadian Mathematical Society: 261–266, doi:10.4153/CMB-1992-037-1
Strok, M.; Szymański, A. (1975), "Compact metric spaces have binary bases" (PDF), Fundamenta Mathematicae, 89 (1): 81–91, doi:10.4064/fm-89-1-81-91
van Mill, J. (1977), Supercompactness and Wallman spaces (Mathematical Centre Tracts, No. 85.), Amsterdam: Mathematisch Centrum, ISBN 90-6196-151-3
Verbeek, A. (1972), Superextensions of topological spaces (Mathematical Centre tracts, No. 41), Amsterdam: Mathematisch Centrum
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, 122 (2), American Mathematical Society, Vol. 122, No. 2: 591–595, doi:10.2307/2161053, JSTOR 2161053

[FOOTNOTEde_Groot1969-1] Groot (1969).

[FOOTNOTEBulaNikielTuncaliTymchatyn1992-2] Bula et al. (1992).

[FOOTNOTEBanaschewski1993-3] Banaschewski (1993).

[FOOTNOTEBell1978-4] Bell (1978).

[FOOTNOTEVerbeek1972Millsvan_Mill1979-5] Verbeek (1972); Mills & van Mill (1979).

[FOOTNOTEYang1994-6] Yang (1994).