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. ^[1]

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 ^[2]
• 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.) ^[3]

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). ^[4]

A continuous image of a supercompact space need not be supercompact. ^[5]

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. ^[6]

Notes

1. de Groot (1969).
2. Bula et al. (1992).
3. Banaschewski (1993).
4. Bell (1978).
5. Verbeek (1972); Mills & van Mill (1979).
6. 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