Order and uo-convergence in spaces of continuous functions

TL;DR

This paper characterizes uo-convergence and order convergence in various spaces of continuous functions, showing that uo-convergence is equivalent to pointwise convergence on co-meagre sets and providing new insights into order bounded sets.

Contribution

It extends previous results by providing new characterizations of uo-convergence and order convergence in spaces of continuous functions, including pointwise convergence on co-meagre sets.

Findings

uoconvergence iff pointwise convergence on co-meagre sets

characterization of order bounded sets in continuous function spaces

new criteria for order convergence

Abstract

We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions $C(\Omega)$, $C_b(\Omega)$, $C_0(\Omega)$, and $C^\infty(\Omega)$, extending results of [vdW18]. In particular, it is shown that a sequence uo-converges iff it converges pointwise on a co-meagre set. We also characterize order bounded sets in spaces of continuous functions. This leads to characterizations of order convergence.