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,)=d(x, \emptyset) = \infty

  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

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}

References