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march 2018 A closure operator for clopen topologies
Gerald Beer, Colin Bloomfield
Bull. Belg. Math. Soc. Simon Stevin 25(1): 149-159 (march 2018). DOI: 10.36045/bbms/1523412062

Abstract

A topology $\tau$ on a nonempty set $X$ is called a clopen topology provided each member of $\tau$ is both open and closed. Given a function $f$ from $X$ to $Y$, the operator $E \mapsto f^{-1}(f(E))$ is a closure operator on the power set of $X$ whose fixed points are closed subsets corresponding to a clopen topology on $X$. Conversely, for each clopen topology $\tau$ on $X$, we produce a function $f$ with domain $X$ such that $\tau = \{E \subseteq X : E = f^{-1}(f(E))\}$. We characterize the clopen topologies on $X$ as those that are weak topologies determined by a surjective function with values in some discrete topological space. Paralleling this result, we show that a topology admits a clopen base if and only if it is a weak topology determined by a family of functions with values in discrete spaces.

Citation

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Gerald Beer. Colin Bloomfield. "A closure operator for clopen topologies." Bull. Belg. Math. Soc. Simon Stevin 25 (1) 149 - 159, march 2018. https://doi.org/10.36045/bbms/1523412062

Information

Published: march 2018
First available in Project Euclid: 11 April 2018

zbMATH: 06882549
MathSciNet: MR3784513
Digital Object Identifier: 10.36045/bbms/1523412062

Subjects:
Primary: 54A05 , 54G99
Secondary: 54C05 , 54C50

Keywords: clopen topology , closure operator , Kolmogorov quotient , weak topology , zero-dimensional space

Rights: Copyright © 2018 The Belgian Mathematical Society

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Vol.25 • No. 1 • march 2018
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