Editing Approach space

In [[topology]], a branch of [[mathematics]], '''approach spaces''' are a generalization of [[metric space]]s, based on point-to-[[set (mathematics)|set]] distances, instead of point-to-point distances. They were introduced by Robert Lowen in 1989, in a series of papers on approach theory between 1988 and 1995.

==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''.

==Examples==

Every ∞pq-metric space (''X'', ''d'') can be ''distanced'' to (''X'',&thinsp;'''d'''), as described at the beginning of the definition.

Given a set ''X'', the ''discrete'' distance is given by '''d'''(''x'', ''A'') = 0 if ''x'' ∈ ''A'' and '''d'''(''x'', ''A'') = ∞ if ''x'' ∉ ''A''. The [[induced topology]] is the [[discrete topology]].

Given a set ''X'', the ''indiscrete'' distance is given by '''d'''(''x'', ''A'') = 0 if ''A'' is non-empty, and '''d'''(''x'', ''A'') = ∞ if ''A'' is empty. The induced topology is the indiscrete topology.

Given a [[topological space]] ''X'', a ''topological'' distance is given by '''d'''(''x'', ''A'') = 0 if ''x'' ∈ <span style="text-decoration: overline;">''A''</span>, and '''d'''(''x'', ''A'') = ∞ otherwise. The induced topology is the original topology. In fact, the only two-valued distances are the topological distances.

Let '''P''' = [0,&thinsp;∞] be the [[extended real numbers|extended]] non-negative [[real number|reals]]. Let '''d'''<sup>+</sup>(''x'', ''A'') = max(''x'' − [[supremum|sup]]&thinsp;''A'', 0) for ''x'' ∈ '''P''' and ''A'' ⊆ '''P'''. Given any approach space (''X'',&thinsp;'''d'''), the maps (for each ''A'' ⊆ ''X'') '''d'''(., ''A'')&nbsp;:&nbsp;(''X'', '''d''')&nbsp;→&nbsp;('''P''', '''d'''<sup>+</sup>) are contractions.

On '''P''', let '''e'''(''x'', ''A'') = inf{|''x'' − ''a''| : ''a'' ∈ ''A''} for ''x'' < ∞, let '''e'''(∞, ''A'') = 0 if ''A'' is unbounded, and let '''e'''(∞, ''A'') = ∞ if ''A'' is bounded. Then ('''P''',&thinsp;'''e''') is an approach space. Topologically, '''P''' is the one-point compactification of <nowiki>[0,&thinsp;∞)</nowiki>.  Note that '''e''' extends the ordinary Euclidean distance. This cannot be done with the ordinary Euclidean metric.

Let β'''N''' be the Stone–Čech compactification of the [[integer]]s. A point ''U'' ∈ β'''N''' is an ultrafilter on '''N'''. A subset ''A'' ⊆ β'''N''' induces a filter ''F''(''A'') = ∩&thinsp;{''U'' : ''U'' ∈ ''A''}. Let '''b'''(''U'', ''A'') = sup{ inf{ |''n'' − ''j''| : ''n'' ∈ ''X'', ''j'' ∈ ''E'' } : ''X'' ∈ ''U'', ''E'' ∈ ''F''(''A'') }. Then (β'''N''',&thinsp;'''b''') is an approach space that extends the ordinary Euclidean distance on '''N'''. In contrast, β'''N''' is not metrizable.

==Equivalent definitions==

Lowen has offered at least seven equivalent formulations. Two of them are below.

Let XPQ(''X'') denote the set of xpq-metrics on ''X''. A subfamily ''G'' of XPQ(''X'') is called a ''gauge'' if
#0 ∈ ''G'', where 0 is the zero metric, that is, 0(''x'', ''y'') = 0 for all ''x'', ''y'',
#''e'' ≤ ''d'' ∈ ''G'' implies ''e'' ∈ ''G'',
#''d'', ''e'' ∈ ''G'' implies max(''d'',''e'') ∈ ''G'' (the "max" here is the [[pointwise maximum]]),
#For all ''d'' ∈ XPQ(''X''), if for all ''x'' ∈ ''X'', ε > 0, ''N'' < ∞ there is ''e'' ∈ ''G'' such that min(''d''(''x'',''y''), ''N'') ≤ ''e''(''x'', ''y'') + ε for all ''y'', then ''d'' ∈ ''G''.

If ''G'' is a gauge on ''X'', then '''d'''(''x'',''A'') = sup {'''e'''(''x'', ''a'') } : ''e'' ∈&thinsp;''G''} is a distance function on ''X''. Conversely, given a distance function '''d''' on ''X'', the set of ''e'' ∈ XPQ(''X'') such that '''e''' ≤ '''d''' is a gauge on ''X''. The two operations are inverse to each other.

A contraction ''f'': (''X'',&thinsp;'''d''') → (''Y'',&thinsp;'''e''') is, in terms of associated gauges ''G'' and ''H'' respectively, a map such that for all ''d'' ∈ ''H'', ''d''(''f''(.), ''f''(.)) ∈ ''G''.

A ''tower'' on ''X'' is a set of maps ''A'' → ''A''<sup>[ε]</sup> for ''A'' ⊆ ''X'', ε ≥ 0, satisfying for all ''A'', ''B'' ⊆ ''X'' and δ, ε ≥ 0
#''A'' ⊆ ''A''<sup>[ε]</sup>,
#Ø<sup>[ε]</sup> = Ø,
#(''A''&thinsp;∪&thinsp;''B'')<sup>[ε]</sup> = ''A''<sup>[ε]</sup>&thinsp;∪&thinsp;''B''<sup>[ε]</sup>,
#''A''<sup>[ε][δ]</sup> ⊆ ''A''<sup>[ε+δ]</sup>,
#''A''<sup>[ε]</sup> = ∩<sub>δ>ε</sub>&thinsp;''A''<sup>[δ]</sup>.

Given a distance '''d''', the associated ''A'' → ''A''<sup>(ε)</sup> is a tower. Conversely, given a tower, the map '''d'''(''x'',''A'') = inf{ε : ''x'' ∈ ''A''<sup>[ε]</sup>} is a distance, and these two operations are inverses of each other.

A contraction ''f'':(''X'',&thinsp;'''d''')→(''Y'',&thinsp;'''e''') is, in terms of associated towers, a map such that for all ε ≥ 0, ''f''[''A''<sup>[ε]</sup>] ⊆ ''f''[''A'']<sup>[ε]</sup>.

==Categorical properties==

The main interest in approach spaces and their contractions is that they form a [[category (mathematics)|category]] with good properties, while still being quantitative like metric spaces. One can take arbitrary [[Product (category theory)|products]], [[Coproduct|coproducts]], and quotients, and the results appropriately generalize the corresponding results for topologies. One can even "distancize" such badly non-metrizable spaces like β'''N''', the [[Stone–Čech compactification]] of the integers.

Certain hyperspaces, [[measure space|measure spaces]], and [[Probabilistic metric space|probabilistic metric spaces]] turn out to be naturally endowed with a distance. Applications have also been made to [[approximation theory]].

==References==
{{reflist}}
* {{cite book | last=Lowen | first=Robert | title=Approach spaces: the missing link in the topology-uniformity-metric triad | series=Oxford Mathematical Monographs | location=Oxford | publisher=[[Clarendon Press]] | year=1997 | isbn=0-19-850030-0 | zbl=0891.54001 }}
* {{cite book | last=Lowen | first=Robert | title=Index Analysis: Approach Theory at Work  | publisher=Springer | year=2015}}

==External links==
* [http://www.math.ua.ac.be/TOP/ Robert Lowen]

{{Metric spaces}}

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