Editing Metric space (section)

===Pseudoquasimetrics===
The prefixes ''pseudo-'', ''quasi-'' and ''semi-'' can also be combined, e.g., a '''pseudoquasimetric''' (sometimes called '''hemimetric''') relaxes both the indiscernibility axiom and the symmetry axiom and is simply a premetric satisfying the triangle inequality. For pseudoquasimetric spaces the open {{Nobr|<math>r</math>-balls}} form a basis of open sets. A very basic example of a pseudoquasimetric space is the set <math>\{0,1\}</math> with the premetric given by <math>d(0,1) = 1</math> and <math>d(1,0) = 0.</math> The associated topological space is the [[Sierpiński space]].

Sets equipped with an extended pseudoquasimetric were studied by [[William Lawvere]] as "generalized metric spaces".<ref>{{harvtxt|Lawvere|1973}}; {{harvtxt|Vickers|2005}}</ref> From a [[Category theory|categorical]] point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the [[category of metric spaces|metric space categories]]. One can take arbitrary products and coproducts and form quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients.

Lawvere also gave an alternate definition of such spaces as [[enriched category|enriched categories]].  The ordered set <math>(\mathbb{R},\geq)</math> can be seen as a [[Category (mathematics)|category]] with one [[morphism]] <math>a\to b</math> if <math>a\geq b</math> and none otherwise. Using {{math|+}} as the [[tensor product]] and 0 as the [[Identity element|identity]] makes this category into a [[monoidal category]] <math>R^*</math>.
Every (extended pseudoquasi-)metric space <math>(M,d)</math> can now be viewed as a category <math>M^*</math> enriched over <math>R^*</math>:
* The objects of the category are the points of {{mvar|M}}.
* For every pair of points {{mvar|x}} and {{mvar|y}} such that <math>d(x,y)<\infty</math>, there is a single morphism which is assigned the object <math>d(x,y)</math> of <math>R^*</math>.
* The triangle inequality and the fact that <math>d(x,x)=0</math> for all points {{mvar|x}} derive from the properties of composition and identity in an enriched category.
* Since <math>R^*</math> is a poset, all [[Diagram (category theory)|diagrams]] that are required for an enriched category commute automatically.