Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-22T17:00:15.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Minimal first countable spaces

Published online by Cambridge University Press:  17 April 2009

Jack R. Porter
Jack R. Porter
Affiliation:
The University of Kansas, Lawrence, Kansas, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A topological space is E0 (resp. E1) provided every point is the countable intersection of neighborhoods (resp. closed neighborhoods). For i = 0 and i = 1, characterizations of minimal Ei. spaces (Ei. spaces with no strictly coarser Ei. topology) and Ei-closed spaces (Ei. spaces which are closed in every Ei. space containing them) are given; for example, the properties of minimal Ei. and minimal first countable Ti+1 are shown to be equivalent. Minimal E0 spaces are characterized as countable spaces with the cofinite topology, and minimal E1 spaces are characterized as E1-closed and semiregular spaces. E0-closed spaces are shown to be precisely the finite discrete spaces.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 1 , August 1970 , pp. 55 - 64
Copyright
Copyright © Australian Mathematical Society 1970

References

[1] Anderson, Frank W., “A lattice characterization of completely regular G δ -spaces”, Proc. Amer. Math. Soc. 6 (1955), 757–765.Google Scholar
[2] Aull, C.E., “A certain class of topological spaces”, Prace Mat. 11 (1967), 49–53.Google Scholar
[3] Bagley, R.W., Connell, E.H. and McKnight, J.D. Jr, “On properties characterizing pseudocompact spaces”, Proc. Amer. Math. Soc. 9 (1958), 500–506.CrossRefGoogle Scholar
[4] Berri, Manuel P., “Minimal topological spaces”, Trans. Amer. Math. Soc. 108 (1963), 97–105.CrossRefGoogle Scholar
[5] Berri, Manuel P., Porter, Jack R. and Stephenson, R.M. Jr, “A survey of minimal topological spaces”, (submitted).Google Scholar
[6] Berri, Manuel P. and Sorgenfrey, R.H., “Minimal regular spaces”, Proc. Amer. Math. Soc. 14 (1963), 454–458.CrossRefGoogle Scholar
[7] Bourbaki, Nicolas, General topology, Part 1 (Addison-Wesley, Reading, Massachussets; London, Ontario, 1966).Google Scholar
[8] Dugundji, James, Topology (Allyn and Bacon, Boston, 1966).Google Scholar
[9] Hewitt, Edwin, “A problem of set-theoretic topology”, Duke Math. J. 10 (1943), 309–333.CrossRefGoogle Scholar
[10] Katětov, Miroslav, “Über H-abgeschlossene und bikompakte Räume”, Časopis Pěst. Mat. Fys. 69 (1940), 36–49.CrossRefGoogle Scholar
[11] Liu, Chen-Tung, “Absolutely closed spaces”, Trans. Amer. Math. Soc. 130 (1968), 86–104.CrossRefGoogle Scholar
[12] Stephenson, R.M. Jr, “Minimal first countable topologies”, Trans. Amer. Math. Soc. 138 (1969), 115–127.CrossRefGoogle Scholar
[13] Stone, A.H., “Hereditarily compact spaces”, Amer. J. Math. 82 (1960), 901–916.CrossRefGoogle Scholar
[14] Viglino, Giovanni, “A co-topologlcal application to minimal spaces”, Pacific J. Math. 27 (1968), 197–200.CrossRefGoogle Scholar