Partitioning bases of topological spaces

TL;DR

This paper explores the partitioning of bases in topological spaces, proving that certain classes can be split into two bases, while providing counterexamples in other cases, and discusses related open problems.

Contribution

It establishes conditions under which bases can be partitioned into two, and presents counterexamples showing limitations in other topological spaces.

Findings

Every base in a T_3 Lindelöf topology can be partitioned into two bases.

Existence of a space with a point countable base that cannot be partitioned into two bases.

Several related results and open problems are discussed.

Abstract

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a T_3 Lindel\"of topology can be partitioned into two bases while there exists a consistent example of a first countable, 0-dimensional, Hausdorff space of size continuum and weight \omega_1 which admits a point countable base without a partition to two bases. Several related results are proved and the paper finishes with a list of open problems.