nLab filter space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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.


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

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

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

The morphisms of filter spaces are the continuous functions; a function ff between filter spaces is continuous if FxF \to x implies that f(F)f(x)f(F) \to f(x), where f(F)f(F) is the filter generated by the filterbase {F(A)|AF}\{F(A) \;|\; A \in F\}. In this way, filter spaces form a concrete category FiltFilt, 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.


Last revised on April 5, 2017 at 19:31:20. See the history of this page for a list of all contributions to it.