Uniformizable functional Alexandroff spaces

Abstract

In the following text we show that the Alexandroff space $X$ is uniformizable if and only if the collection of all smallest neighbourhoods is a partition of $X$. Moreover the Alexandroff space $X$ is uniformizable and functional Alexandroff ($k-$primal) if and only if the collection of all smallest neighbourhoods is a partition of $X$ into its finite subsets.Comment: 8 page