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 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

A set $S$ is a preconvergence space if it comes with a binary relation $x \to c$ between the (large) set of all nets in $S$ and $S$ itself, for elements

$$x \in \bigcup_{I \in DirectedSet_\mathcal{U}} S^I$$

and $c \in S$

See also

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