Editing Approach space (section)

==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>.