Approach theory and pointfree convergence

Abstract

The purpose of this note is to show that convergence approach spaces in the sense of [9] can be seen as instances of special convergence lattices (specifically, convergence frames) in the sense of [6], hence fitting them in a general theory of pointfree convergence. Interestingly, the standard construction of the space of points (or spectrum) of a convergence approach space seen as a convergence frame returns the structure of the original convergence approach space, but packaged as a convergence space. The notion of a centered convergence lattice appears even more clearly as the natural pointfree analog of the point-axiom: it was already observed in [6] that it encapsulates the point-axiom of convergence spaces and it turns out that it also captures the point-axiom for convergence approach spaces. This is half of a characterization of those convergence frame structures on a function space $[0, \infty]^X$ that represent a convergence approach space structure on $X$. Notions of closed and of open elements in the pointfree convergence setting turn out to entirely depend, in the case of a convergence approach space, on the reflection on convergence spaces.