Seminario de Geometría y Topología

"A closure operator for clopen topologies"

Gerald A. Beer

California State University-Los Ángeles.USA

13:00 h., Sala 224

A topology $\tau$ on a nonempty set $X$ is called a clopen topology provided each member of $\tau$ is both open and closed. Such topologies have been called both locally indiscrete and indiscretely generated in the literature. In joint work with Colin Bloomfield, to appear, Bull. Belgian Math. Soc., we show how such topologies arise from a natural closure operator familiar to any student of mathematics. 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 the 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, gaining a different perspective on an embedding theorem of Vedesinoff. Finally, we consider the operator $E \mapsto f(f^{-1}( (E))$ as a potential interior operator on the power set of $Y$.