Editing Approach space (section)

==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.