# Approach space

A generalisation of the concept of metric space, formalising the notion of the distance from a point to a set. An approach space is a set $X$ together with a function $d$ on $X \times \mathcal{P}X$, where $\mathcal{P}X$ is the power set of $X$, talking values in the extended positive reals $[0,\infty]$, and satisfying $$d(x,\{x\}) = 0 \ ;$$ $$d(x,\emptyset) = \infty \ ;$$ $$d(x,A\cup B) = \min(d(x,A),d(x,B)) \ ;$$ $$d(x,A) \le d(x,A^u) + u \ ;$$ where for $u \in [0,\infty]$, we write $A^u = \{x \in X : d(x,A) \le u \}$.
A metric space $(X,\delta)$ has an approach structure via $$d(x,A) = \inf\{ \delta(x,a) : a \in A \} \ .$$ and a topological space $(X,{}^c)$, where ${}^c$ denotes the Kuratowski closure operator, via $$d(x,A) = \begin{cases} 0 & \ \text{if}\ x \in A^c \\ \infty & \ \text{otherwise} \end{cases} \ .$$
In the opposite direction, if $(X,d)$ is an approach space then the operation $$A^c = \{ x \in X : d(x,A) < \infty \}$$ is a Čech closure operator, giving $X$ the structure of a pre-topological space. However, the operation $$A^C = \{ x \in X : d(x,A) = 0 \}$$ is a closure operator giving a topological structure on $X$.