Supercompact Space
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. By the Alexander subbase theorem, every supercompact space is compact. Conversely, many (but not all) compact spaces are supercompact....
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. 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. Szymanski 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 compact Hausdorff spaces are not supercompact; such an example is given by the Stone ech compactification of the natural numbers (with the discrete topology) (Bell 1978).
