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Note on Finite Topological Spaces

Published online by Cambridge University Press:  09 April 2009

J. Knopfmacher
J. Knopfmacher
Affiliation:
University of the WitwatersrandJohannesburg, South Africa
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In a recent paper [4], H. Sharp, Jr., has discussed the problem of finding formulae for the following naturally defined integers: the numbers t(n), tc(n), t0(n), tc0(n), and ts(n) of all homeomorphism classes of finite topological spaces with n elements, which are respectively (i) arbitrary, (ii) connected, (iii) T0, (iv) connected and T0, (v) symmetric. Here, a finite topological space X is called symmetric provided the following relation ≦ is symmetric: x ≦ y if and only if x ∈ Uv, the intersection of all open sets containing y.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 1-2 , February 1969 , pp. 252 - 256
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Harary, F., ‘The number of linear, directed, rooted and connected graphs’, Trans. Amer. Math. Soc. 78 (1955), 445–463.CrossRefGoogle Scholar
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[3]Schenkman, E., Group Theory (Van Nostrand Co., Princeton, 1965).Google Scholar
[4]Sharp, H. Jr, ‘Quasi-orderings and topologies on finite sets’, Proc. Amer. Math. Soc. 17 (1966) 1344–1349.CrossRefGoogle Scholar
[5]Stong, R. E., ‘Finite topological spaces’, Trans. Amer. Math. Soc. 123 (1966) 325–340.CrossRefGoogle Scholar