## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 25, Number 1 (2018), 149-159.

### A closure operator for clopen topologies

Gerald Beer and Colin Bloomfield

#### 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 25, Number 1 (2018), 149-159.

**Dates**

First available in Project Euclid: 11 April 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1523412062

**Digital Object Identifier**

doi:10.36045/bbms/1523412062

**Mathematical Reviews number (MathSciNet)**

MR3784513

**Zentralblatt MATH identifier**

06882549

**Subjects**

Primary: 54A05: Topological spaces and generalizations (closure spaces, etc.) 54G99: None of the above, but in this section

Secondary: 54C05: Continuous maps 54C50: Special sets defined by functions [See also 26A21]

**Keywords**

closure operator clopen topology weak topology zero-dimensional space Kolmogorov quotient

#### Citation

Beer, Gerald; Bloomfield, Colin. A closure operator for clopen topologies. Bull. Belg. Math. Soc. Simon Stevin 25 (2018), no. 1, 149--159. doi:10.36045/bbms/1523412062. https://projecteuclid.org/euclid.bbms/1523412062