The convergence-theoretic approach to g-first countable and symmetrizable spaces

TL;DR

This paper uses convergence space structures to characterize and analyze properties like symmetrizability and first-countability, providing algebraic proofs and new characterizations, especially for non-Hausdorff spaces.

Contribution

It offers convergence-theoretic characterizations of key topological notions and simplifies proofs of classical results, extending understanding to non-Hausdorff spaces.

Findings

Algebraic proof of when symmetrizable spaces are semi-metrizable

Characterization of spaces whose product with any metrizable space is weakly first-countable

Simplification of proofs regarding stability of symmetrizability under products

Abstract

This article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Fr\'echet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fr\'echet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.