{{Short description|Topological space where every open cover has a finite subcover}}
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.{{sfnp|de Groot|1969}}
==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{{sfnp|Bula|Nikiel|Tuncali|Tymchatyn|1992}} * Compact metrizable spaces (due originally to {{harvtxt|Strok|Szymański|1975}}, see also {{harvtxt|Mills|1979}}) <!-- * Compact topological groups, claimed by Charles F. Mills but I can't find the paper which apparently was not published --> * A product of supercompact spaces is supercompact (like a similar statement about compactness, Tychonoff's theorem, it is equivalent to the axiom of choice.){{sfnp|Banaschewski|1993}}
==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).{{sfnp|Bell|1978}}
A continuous image of a supercompact space need not be supercompact.{{sfnmp|1a1 = Verbeek|1y = 1972|2a1 = Mills|2a2 = van Mill|2y = 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.{{sfnp|Yang|1994}}
==Notes== {{reflist|colwidth=30em}}
==References== {{refbegin|colwidth=30em}} *{{citation |last = Banaschewski |first = B. |title = Supercompactness, products and the axiom of choice |journal = Kyungpook Math Journal |year = 1993 |volume = 33 |issue = 1 |pages = 111–114}} *{{citation |last = Bell |first = Murray G. |title = Not all compact Hausdorff spaces are supercompact |journal = General Topology and Its Applications |year = 1978 |volume = 8 |issue = 2 |pages = 151–155|doi = 10.1016/0016-660X(78)90046-6 }} *{{citation |last1 = Bula |first1 = W. |last2 = Nikiel |first2 = J. |last3 = Tuncali |first3 = H. M. |last4 = Tymchatyn |first4 = E. D. |title = Continuous images of ordered compacta are regular supercompact |journal = Topology and Its Applications |year = 1992 |volume = 45 |issue = 3 |pages = 203–221 |doi = 10.1016/0166-8641(92)90005-K}} *{{citation |last = de Groot |first= J. |author-link = Johannes de Groot |contribution = Supercompactness and superextensions |title = Contributions to extension theory of topological structures. Proceedings of the Symposium held in Berlin, August 14—19, 1967 |editor-last1 = Flachsmeyer |editor-first1 = J. |editor-last2 = Poppe |editor-first2 = H. |editor-last3 = Terpe |editor-first3 = F. |publisher = VEB Deutscher Verlag der Wissenschaften |location = Berlin |year = 1969 }} *{{citation |last = Engelking |first = R |author-link = Ryszard Engelking |title = General topology |publisher = Taylor & Francis |year = 1977 |isbn = 978-0-8002-0209-5}} *{{citation |last1 = Malykhin |first1 = VI |last2 = Ponomarev |first2 = VI |title = General topology (set-theoretic trend) |journal = Journal of Mathematical Sciences |publisher = Springer |publication-place = New York |volume = 7 |issue = 4 |year = 1977 |doi = 10.1007/BF01084982 |pages = 587–629 |s2cid = 120365836 }} *{{citation |last1 = Mills |first1 = Charles F. |title = A simpler proof that compact metric spaces are supercompact |mr = 518526 |year = 1979 |journal = Proceedings of the American Mathematical Society |volume = 73 |issue = 3 |pages = 388–390 |doi = 10.2307/2042369 |publisher = American Mathematical Society, Vol. 73, No. 3 |jstor = 2042369 |doi-access = free}} *{{citation |last1 = Mills |first1 = Charles F. |last2 = van Mill |first2 = Jan |title = A nonsupercompact continuous image of a supercompact space |journal = Houston Journal of Mathematics |year = 1979 |volume = 5 |issue = 2 |pages = 241–247}} *{{citation |last = Mysior |first = Adam |title = Universal compact T<sub>1</sub>-spaces |journal = Canadian Mathematical Bulletin |volume = 35 |issue = 2 |year = 1992 |pages = 261–266 |publisher = Canadian Mathematical Society |doi = 10.4153/CMB-1992-037-1 |doi-access =free }} *{{citation |last1 = Strok |first1 = M. |last2 = Szymański |first2 = A. |url = http://matwbn.icm.edu.pl/ksiazki/fm/fm89/fm8919.pdf |title = Compact metric spaces have binary bases |journal = Fundamenta Mathematicae |year = 1975 |volume = 89 |issue = 1 |pages = 81–91|doi = 10.4064/fm-89-1-81-91 }} *{{citation |last = van Mill |first = J. |title = Supercompactness and Wallman spaces (Mathematical Centre Tracts, No. 85.) |publisher = Mathematisch Centrum |location = Amsterdam |year = 1977 |isbn = 90-6196-151-3}} *{{citation |last = Verbeek |first = A. |title = Superextensions of topological spaces (Mathematical Centre tracts, No. 41) |publisher = Mathematisch Centrum |location = Amsterdam |year = 1972}} *{{citation |last = Yang |first = Zhong Qiang |year = 1994 |title = All cluster points of countable sets in supercompact spaces are the limits of nontrivial sequences |journal = Proceedings of the American Mathematical Society |volume = 122 |issue = 2 |pages = 591–595 |doi = 10.2307/2161053 |publisher = American Mathematical Society, Vol. 122, No. 2 |jstor = 2161053 |doi-access = free}} {{refend}}
Category:Compactness (mathematics) Category:Properties of topological spaces