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 $X$ together with a distance $d\colon X \times \mathcal{P}(X) \to [0,\infty]$ (where $\mathcal{P}(X)$ denotes the power set) such that the following axioms hold for all $x \in X$

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

2. $d(x, \emptyset) = \infty$

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

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

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

Properties

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

Examples

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

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

$d^*(x, A) = \begin{cases} d(x, A\setminus\{\infty\}) & x \neq \infty \\ 0 & x = \infty\, and\, A\, is\, not\, precompact \\ \infty & x = \infty\, and\, A\, is\, precompact. \end{cases}$
• A gauge, or more generally a gauge base, $G$ on $X$ gives a distance on $X$ by $d_G(x, A) = \sup_{d \in G} \inf_{y\in A} d(x,y)$.
• Robert Lowen, Approach spaces: the missing link in the topology-uniformity-metric triad, Oxford Mathematical Monographs. 1997. (publisher link).