Order convergence structure on <i>C(X)</i>

Abstract

This paper brings together three concepts which have not been related so far, namely, the concept of order convergence, the concept of convergence space and the concept of Hausdorff continuous functions. The order convergence on a poset P, which is generally not a topological convergence, can be studied through the concept of convergence space. Indeed, under certain mild assumptions there exists a convergence structure on P which induces the order convergence. In particular, the result is true for any vector lattice. The primary focus is on the set C(X) of all continuous real functions on a topological space X. The vector lattice C(X) gives a typical example when the order convergence cannot be induced by a topology, thus justifying our interest in the convergence vector structure inducing the order convergence. The completion of the respective convergence vector space is obtained through Hausdorff continuous functions.
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Quaestiones Mathematicae 28(2005), 425–457