**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

…

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

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

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

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