nLab preconvergence space




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.


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

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

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

See also

Last revised on October 24, 2023 at 18:55:15. See the history of this page for a list of all contributions to it.