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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.
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Abstract

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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

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