Countability conditions in locally solid convergence spaces

TL;DR

This paper investigates conditions under which certain convergence structures on Archimedean vector lattices are first countable, impacting the use of sequential methods in these mathematical contexts.

Contribution

It characterizes vector lattices where various types of convergence are strongly first countable, providing insights into sequential argument validity.

Findings

Characterization of vector lattices with first countable convergence structures

Implications for the validity of sequential arguments in these contexts

Conditions for strong first countability in different convergence types

Abstract

We study (strong) first countability of locally solid convergence structures on Archimedean vector lattices. Among other results, we characterise those vector lattices for which relatively unform-, order-, and $\sigma$-order convergence, respectively, is (strongly) first countable. The implications for the validity of sequential arguments in the contexts of these convergences are pointed out.