nLab
filter space

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

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 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.

References

Revised on April 5, 2017 15:31:20 by Toby Bartels (64.89.54.53)