Editing Approach space (section)

==Definition==

Given a metric space (''X'', ''d''), or more generally, an [[Metric (mathematics)#Extending_the_range|extended]] [[Pseudometric_space|pseudo]][[quasimetric]] (which will be abbreviated ''∞pq-metric'' here), one can define an induced map '''d''': ''X'' × P(''X'') → [0,∞] by '''d'''(''x'', ''A'') = [[infimum|inf]]{''d''(''x'', ''a'') : ''a'' ∈ ''A''}.  With this example in mind, a '''distance''' on ''X'' is defined to be a map ''X'' × P(''X'') → [0,∞] satisfying for all ''x'' in ''X'' and ''A'', ''B'' ⊆ ''X'',
#'''d'''(''x'', {''x''}) = 0,
#'''d'''(''x'', Ø) = ∞,
#'''d'''(''x'', ''A''∪''B'') = min('''d'''(''x'', ''A''), '''d'''(''x'', ''B'')),
#For all 0 ≤ ε ≤ ∞, '''d'''(''x'', ''A'') ≤ '''d'''(''x'', ''A''<sup>(ε)</sup>) + ε,
where we define ''A''<sup>(ε)</sup> = {''x'' : '''d'''(''x'', ''A'') ≤ ε}.

(The "[[empty set|empty]] infimum is positive infinity" convention is like the [[Empty product#Nullary intersection|nullary intersection is everything]] convention.)

An approach space is defined to be a pair (''X'',&thinsp;'''d''') where '''d''' is a distance function on ''X''. Every approach space has a [[topological space|topology]], given by treating ''A''&nbsp;→&nbsp;''A''<sup>(0)</sup> as a [[Kuratowski closure axioms|Kuratowski closure operator]].

The appropriate maps between approach spaces are the ''contractions''. A map ''f'': (''X'',&thinsp;'''d''') → (''Y'',&thinsp;'''e''') is a contraction if '''e'''(''f''(''x''), ''f''[''A'']) ≤ '''d'''(''x'', ''A'') for all ''x'' ∈ ''X'' and ''A'' ⊆ ''X''.