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 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.


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

x IDirectedSet 𝒰S Ix \in \bigcup_{I \in DirectedSet_\mathcal{U}} S^I

and cSc \in S

See also

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