Editing Pseudometric space (section)

==Topology==

The '''{{visible anchor|pseudometric topology}}''' is the [[Topology (structure)|topology]] generated by the [[open balls]]
<math display=block>B_r(p) = \{x \in X : d(p, x) < r\},</math>
which form a [[Basis (topology)|basis]] for the topology.<ref>{{planetmath reference|urlname=PseudometricTopology|title=Pseudometric topology}}</ref> A topological space is said to be a '''{{visible anchor|pseudometrizable space}}'''<ref>Willard, p. 23</ref> if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space.

The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is [[T0 space|T<sub>0</sub>]] (that is, distinct points are [[topologically distinguishable]]).

The definitions of [[Cauchy sequence]]s and [[Completion (metric space)|metric completion]] for metric spaces carry over to pseudometric spaces unchanged.<ref>{{Cite web|last=Cain|first=George|date=Summer 2000|title=Chapter 7: Complete pseudometric spaces|url=http://people.math.gatech.edu/~cain/summer00/ch7.pdf|url-status=live|archive-url=https://archive.today/20201007070509/http://people.math.gatech.edu/~cain/summer00/ch7.pdf|archive-date=7 October 2020|access-date=7 October 2020}}</ref>