Computably regular topological spaces

TL;DR

This paper advances the theory of computable topology by introducing new separation axioms, exploring their logical relations, and demonstrating conditions under which computably regular spaces embed into computable metric spaces.

Contribution

It introduces new computable separation axioms, investigates their logical relations, and extends known results on computable metrization under uniform assumptions.

Findings

Several computable T3- and Tychonoff separation axioms are introduced.

Many implications between these axioms are proved or disproved.

Computably regular spaces with non-empty base elements can embed into computable metric spaces.

Abstract

This article continues the study of computable elementary topology started by the author and T. Grubba in 2009 and extends the author's 2010 study of axioms of computable separation. Several computable T3- and Tychonoff separation axioms are introduced and their logical relation is investigated. A number of implications between these axioms are proved and several implications are excluded by counter examples, however, many questions have not yet been answered. Known results on computable metrization of T3-spaces from M. Schr/"oder (1998) and T. Grubba, M. Schr/"oder and the author (2007) are proved under uniform assumptions and with partly simpler proofs, in particular, the theorem that every computably regular computable topological space with non-empty base elements can be embedded into a computable metric space. Most of the computable separation axioms remain true for finite products of spaces.