Existence of well-filterications of T0 topological spaces

Abstract

We prove that for every T0 space X, there is a well-filtered space W(X) and a continuous mapping ηx : X⭢ W(X), such that for any well-filtered space Y and any continuous mapping 𝒇 : X⭢Y there is a unique continuous mapping 𝒇^: W(X)⭢Y such that 𝒇=𝒇^∘ηX. Such a space W(X) will be called the well-filterification of X. This result gives a positive answer to one of the major open problems on well-filtered spaces. Another result on well-filtered spaces we will prove is that the product of two well-filtered spaces is well-filtered.