Order and uo-convergence in spaces of continuous functions

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.

arXiv comment13 pages