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In topology, approach spaces are a generalization of metric spaces, based on point-to-set distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989. ==Definition== Given a metric space (''X'',''d''), or more generally, an extended pseudoquasimetric (which will be abbreviated ''∞pq-metric'' here), one can define an induced map d:''X''×P(''X'')→() by d(x,''A'') = inf . With this example in mind, a distance on ''X'' is defined to be a map ''X''×P(''X'')→() satisfying for all ''x'' in ''X'' and ''A'', ''B'' ⊆ ''X'', #d(''x'',) = 0 ; #d(''x'',Ø) = ∞ ; #d(''x'',''A''∪''B'') = min d(''x'',''A''),d(''x'',''B'') ; #For all ε, 0≤ε≤∞, d(''x'',''A'') ≤ d(''x'',''A''(ε)) + ε ; where ''A''(ε) = by definition. (The "empty infimum is positive infinity" convention is like the nullary intersection is everything convention.) An approach space is defined to be a pair (''X'',d) where d is a distance function on ''X''. Every approach space has a topology, given by treating ''A'' → ''A''(0) as a Kuratowski closure operator. The appropriate maps between approach spaces are the ''contractions''. A map ''f'':(''X'',d)→(''Y'',e) is a contraction if e(''f''(''x''),''f''()) ≤ d(''x'',''A'') for all ''x'' ∈ ''X'', ''A'' ⊆ ''X''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「approach space」の詳細全文を読む スポンサード リンク
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