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Whenever editing is allowed on the nLab again, this article should be ported over there.
The most general space where the notion of convergence and limit makes sense, but the term “convergence space” 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.
A set is a preconvergence space if it comes with a binary relation between the (large) set of all nets in
and itself.
A type is a preconvergence space if it comes with a binary relation between the type of all nets in
and itself.