Semi-order units in vector lattices

Abstract

The space C(X) of real-valued continuous functions on a topological space X is a vector lattice and a locally convex topological vector space, but what is the interaction between these structures? In the case of a compact space X, the norm and the order are closely related to one another. Indeed, one may define the norm through the order structure. We aim to generalize these results to a non-compact space. Let X be a Tychonoff space and consider C(X) equipped with the compact-open topology. We will establish a relationship between this topology and the order structure on C(X)