nLab
approach space

Idea

The concept of approach space generalized the concept of metric space. The idea is that the distance describes not only the distance between two points but the distance of a point to a subset. This relatively tangible generalization gives the missing link in the triad of the concepts of uniformity, topology, and metric spaces.

Definition

An approach space is a set XX together with a distance d:X×𝒫(X)[0,]d\colon X \times \mathcal{P}(X) \to [0,\infty] (where 𝒫(X)\mathcal{P}(X) denotes the power set) such that the following axioms hold for all xXx \in X

  1. d(x,{x})=0d(x, \{x\}) = 0

  2. d(x,)=0d(x, \emptyset) = 0

  3. for all A,B𝒫A, B \in \mathcal{P}: d(x,AB)=min{d(x,A),d(x,B)}d(x, A\cup B) = \min\{ d(x, A), d(x, B) \}

  4. for all A𝒫A \in \mathcal{P} and ε[0,]\varepsilon \in [0,\infty]: d(x,A)d(x,A ε])+εd(x, A ) \leq d(x, A^{\varepsilon]} ) + \varepsilon

where A ε]{xXd(x,A)ε}A^{\varepsilon]} \coloneqq \{x \in X \mid d(x, A) \leq \varepsilon \}.

Properties

Every approach space dd induces a topology on XX via the closure operator Cl d(A)={xXd(x,A)=0}Cl_d(A) = \{x \in X \mid d(x, A) = 0 \}.

Examples

  • Every topological space is induced by a canonical approach structure given by d(x,A)=0d(x, A) = 0 if xCl(X)x \in Cl(X) and d(x,A)=d(x, A) = \infty otherwise.

  • The one-point compactification of a metric space dd can be metrised in a canonical way as an approach space by

d *(x,A)={d(x,A{}) x 0 x=andAisnotprecompact x=andAisprecompact. d^*(x, A) = \begin{cases} d(x, A\setminus\{\infty\}) & x \neq \infty \\ 0 & x = \infty\, and\, A\, is\, not\, precompact \\ \infty & x = \infty\, and\, A\, is\, precompact. \end{cases}
  • A gauge, or more generally a gauge base, GG on XX gives a distance on XX by d G(x,A)=sup dGinf yAd(x,y)d_G(x, A) = \sup_{d \in G} \inf_{y\in A} d(x,y).

References

  • Robert Lowen, Approach spaces: the missing link in the topology-uniformity-metric triad, Oxford Mathematical Monographs. 1997. (publisher link).

Last revised on March 16, 2019 at 02:35:43. See the history of this page for a list of all contributions to it.