Topological aspects of order in $C(X)$

TL;DR

This paper explores the interplay between order and topology in spaces of bounded continuous functions, providing characterizations of bands and projection bands, and addressing an open question on un-convergence in these spaces.

Contribution

It offers new topological characterizations of bands in $C_0(X)$ and $C_b(X)$, and solves an open problem related to un-convergence in these function spaces.

Findings

Characterization of order continuous restriction operators as closures of interior sets.

Descriptions of bands and projection bands via compactifications.

Resolution of an open question on lifting un-convergence.

Abstract

In this paper we consider the relationship between order and topology in the vector lattice $C_b(X)$ of all bounded continuous functions on a Hausdorff space $X$. We prove that the restriction of $f\in C_b(X)$ to a closed set $A$ induces an order continuous operator iff $A=\overline{\mathrm{Int} A}.$ This result enables us to easily characterize bands and projection bands in $C_0(X)$ and $C_b(X)$ through the one-point compactification and the Stone-\v{C}ech compactification of $X$, respectively. With these characterizations we describe order complete $C_0(X)$ and $C_b(X)$-spaces in terms of extremally disconnected spaces. Our results serve us to solve an open question on lifting un-convergence in the case of $C_0(X)$ and $C_b(X)$.