Counting topologies. (English) Zbl 1228.54003
For an infinite set \(X\), the author computes the number of topologies on \(X\) which are \(P\), where \(P\) is some combination of compact (without \(T_2\)), connected, \(T_1\), \(T_2\), \(T_4\), \(T_5\).
E.g. he shows that this number is \(2^{2^{|X|}}\) if \(P\) is connected ore compact ore \(T_2\), but only \(2^{|X|}\) if \(P\) is compact and \(T_2\) (\(|X|\) is the cardinal number of \(X\)).
E.g. he shows that this number is \(2^{2^{|X|}}\) if \(P\) is connected ore compact ore \(T_2\), but only \(2^{|X|}\) if \(P\) is compact and \(T_2\) (\(|X|\) is the cardinal number of \(X\)).
Reviewer: Bernhard Behrens (Göteborg)
MSC:
| 54A10 | Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) |
| 54D80 | Special constructions of topological spaces (spaces of ultrafilters, etc.) |
Keywords:
(connected, compact, \(T_1\), \(T_2\), \(T_4\), \(T_5\)) toplogical space; cardinality; ultrafilterReferences:
| [1] | Comfort, W.W.; Negrepontis, S.: The Theory of Ultrafilters . Springer 1974. · Zbl 0298.02004 |
| [2] | Juhász, I.: Cardinal Functions in Topology . Mathematisch Centrum Amsterdam 1983. · Zbl 0479.54001 |
| [3] | Mazurkiewicz, S.; Sierpinski, W.: Contribution ‘a la topologie des ensembles dénombrables. Fund. Math. 1 (1920), 17-27. |
| [4] | Steen, L.A.; Seebach Jr., J.A.: Counterexamples in Topology . Dover 1995. · Zbl 1245.54001 |
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