View source for Approach space

{{TEX|done}}{{MSC|54A05|54E05}}

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 $X$.

====References====
* Hofmann, Dirk (ed.); Seal, Gavin J. (ed.); Tholen, Walter (ed.) "Monoidal topology. A categorical approach to order, metric, and topology" Cambridge University Press (2014) {{ISBN|978-1-107-06394-5}} {{ZBL|1297.18001}}
* R. Lowen, "Approach spaces - a common supercategory of TOP and MET." ''Math. Nachr.'' '''141''' (1989) 183-226 {{ZBL|0676.54012}}
* R. Lowen, "Index Analysis: Approach Theory at Work", Springer (2015) {{ISBN|1-4471-6485-7}} {{ZBL|1311.54002}}