Simultaneous Extensions of Topologies Through Traces of Neighbourhood Filters

Abstract

Let X be a set, X _i ⊂ X for i ∈ I and, for x ∈ X, $\mathfrak{s}_i (x)$ a filter in X _i. The paper gives necessary and sufficient conditions for the existence of a topology τ on X compatible with ${\text{(}}\mathfrak{s}_i )$, i.e. such that $\mathfrak{s}_i (x)$ is the trace on X _i of the τ-neighbourhood filter of x. It is shown that, among these compatible topologies,there are always a coarsest one and a finest one. Some applications are also given.