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 $x \in X$

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

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

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

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 $d_G(x, A) = \sup_{d \in G} \inf_{y\in A} d(x,y)$.