Uniformizable functional Alexandroff spaces

TL;DR

This paper characterizes when Alexandroff spaces are uniformizable, showing that such spaces are uniformizable if their smallest neighborhoods form a partition, and they are also functional if this partition consists of finite subsets.

Contribution

It provides a precise characterization of uniformizable and functional Alexandroff spaces based on the structure of their smallest neighborhoods.

Findings

Alexandroff space is uniformizable iff smallest neighborhoods form a partition.

Such spaces are functional Alexandroff if the partition consists of finite subsets.

The characterization links the topology of Alexandroff spaces to their uniformizability properties.

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.