Editing Metric space (section)

===Metametrics or partial metrics===

In a ''metametric'', all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric are:

# <math>d(x,y)\geq 0</math>
# <math>d(x,y)=0 \implies x=y</math>
# <math>d(x,y)=d(y,x)</math>
# <math>d(x,z)\leq d(x,y)+d(y,z).</math>

Metametrics appear in the study of [[Δ-hyperbolic space|Gromov hyperbolic metric spaces]] and their boundaries. The ''visual metametric'' on such a space satisfies <math>d(x,x)=0</math> for points <math>x</math> on the boundary, but otherwise <math>d(x,x)</math> is approximately the distance from ''<math>x</math>'' to the boundary. Metametrics were first defined by Jussi Väisälä.{{sfn|Väisälä|2005}}  In other work, a function satisfying these axioms is called a ''partial metric''<ref>{{cite web|url=http://www.dcs.warwick.ac.uk/pmetric/|title=Partial metrics: welcome|website=www.dcs.warwick.ac.uk|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20170727003912/http://www.dcs.warwick.ac.uk/pmetric/|archive-date=27 July 2017}}</ref><ref>{{cite journal |last1=Bukatin |first1=Michael |last2=Kopperman |first2=Ralph |last3=Matthews |first3=Steve |last4=Pajoohesh |first4=Homeira |title=Partial Metric Spaces |journal=American Mathematical Monthly |date=1 October 2009 |volume=116 |issue=8 |pages=708–718 |doi=10.4169/193009709X460831 |s2cid=13969183 |url=https://www.dcs.warwick.ac.uk/pmetric/monthly708-718.pdf}}</ref> or a ''dislocated metric''.{{sfn|Hitzler|Seda|2016|loc=Definition 4.2.1}}