**analysis** (differential/integral calculus, functional analysis, topology)

metric space, normed vector space

open ball, open subset, neighbourhood

convergence, limit of a sequence

compactness, sequential compactness

continuous metric space valued function on compact metric space is uniformly continuous

…

…

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 $S$ 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 $c \in S$

Last revised on December 5, 2022 at 04:57:10. See the history of this page for a list of all contributions to it.