nLab filter space

Idea

A filter space is a generalisation of a topological space based on the concept of convergence of filters (or nets) as fundamental. The category of filter spaces is a quasitopos and may be thought of as a nice category of spaces that includes Top as a full subcategory. Filter spaces include convergence spaces, Choquet spaces, and Kuratowski limit spaces as full sub-quasitoposes.

Definitions

A filter space is a set $S$ together with a relation $\to$ from $\mathcal{F}S$ to $S$, where $\mathcal{F}S$ is the set of filters on $S$; if $F \to x$, we say that $F$ converges to $x$ or that $x$ is a limit of $F$. This must satisfy some axioms:

1. Centred: The principal ultrafilter $F_x = \{ A \;|\; x \in A \}$ at $x$ converges to $x$;
2. Isotone: If $F \subseteq G$ and $F \to x$, then $G \to x$;

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 filter spaces are the continuous functions; a function $f$ between filter spaces is continuous if $F \to x$ implies that $f(F) \to f(x)$, where $f(F)$ is the filter generated by the filterbase $\{f(A) \;|\; A \in F\}$. In this way, filter spaces form a concrete category $Filt$, which is a quasitopos.

A filter space that satisfies an additional directedness criterion is precisely a convergence space; see there for a variety of intermediate notions leading up to ordinary topological spaces.

Related concepts

• preconvergence space

• convergence space

References

• Martin Hyland, Filter spaces and continuous functionals, 1979 PDF