Editing Metric space (section)

=== Graphs and finite metric spaces ===
A {{visible anchor|metric space is ''discrete''|Discrete metric space}} if its induced topology is the [[discrete topology]].  Although many concepts, such as completeness and compactness, are not interesting for such spaces, they are nevertheless an object of study in several branches of mathematics.  In particular, {{visible anchor|finite metric spaces|Finite metric space}} (those having a [[finite set|finite]] number of points) are studied in [[combinatorics]] and [[theoretical computer science]].<ref>{{cite book |chapter=Finite metric-spaces—combinatorics, geometry and algorithms |last1=Linial |first1=Nathan |author-link1=Nati Linial |title=Proceedings of the ICM, Beijing 2002 |year=2003 |volume=3 |pages=573–586 |arxiv=math/0304466}}</ref>  Embeddings in other metric spaces are particularly well-studied.  For example, not every finite metric space can be [[isometry|isometrically embedded]] in a Euclidean space or in [[Hilbert space]].  On the other hand, in the worst case the required distortion (bilipschitz constant) is only logarithmic in the number of points.<ref>{{cite journal|doi=10.1007/BF02776078|doi-access=|title=On lipschitz embedding of finite metric spaces in Hilbert space|year=1985|last1=Bourgain|first1=J. |author-link1=Jean Bourgain |journal=[[Israel Journal of Mathematics]]|volume=52|issue=1–2|pages=46–52|s2cid=121649019}}</ref><ref>[[Jiří Matoušek (mathematician)|Jiří Matoušek]] and [[Assaf Naor]], ed. [http://kam.mff.cuni.cz/~matousek/metrop.ps "Open problems on embeddings of finite metric spaces"]. {{webarchive|url=https://web.archive.org/web/20101226232112/http://kam.mff.cuni.cz/~matousek/metrop.ps |date=2010-12-26 }}.</ref>

For any [[graph (discrete mathematics)|undirected connected graph]] {{mvar|G}}, the set {{mvar|V}} of vertices of {{mvar|G}} can be turned into a metric space by defining the [[distance (graph theory)|distance]] between vertices {{mvar|x}} and {{mvar|y}} to be the length of the shortest edge path connecting them. This is also called ''shortest-path distance'' or ''geodesic distance''. In [[geometric group theory]] this construction is applied to the [[Cayley graph]] of a (typically infinite) [[finitely-generated group]], yielding the [[word metric]].  Up to a bilipschitz homeomorphism, the word metric depends only on the group and not on the chosen finite generating set.{{sfn|Margalit|Thomas|2017}}