nLab preconvergence space

Idea

By a preconvergence space we shall mean the most general notion of topological space for which the notion of convergence and of limit for filters and nets makes sense. Notice that the possibly more natural term “convergence space” for this notion is already taken to mean a set with filters that are isotone, centered, and directed, and that definition is not the most general definition for which the notions of convergence and limit makes sense for nets.

Definition

Given a set $S$, let $\mathcal{F}(S)$ denote the set of filters on $S$. A set $S$ is a preconvergence space if it comes with a binary relation $x \to c$ between the $\mathcal{F}(S)$ and $S$ itself, for elements $x \in \mathcal{F}(S)$ and $c \in S$.

The definition can also be phrased in terms of nets; a net $\nu$ converges to $x$ if and only if its eventuality filter converges to $x$.

The morphisms of preconvergence spaces are the pointwise continuous functions; a function $f$ between preconvergence spaces is pointwise continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{F(A) \;|\; A \in F\}$.

See also

• limit of a filter
• Hausdorff space
• pointwise continuous function
• sequential preconvergence space
• convergence space