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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)
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