## Proof That a Pseudometric Space is Not T0

A pseudometric on a setis a functionsuch that

1. D(x,y)=D(y,x)

2.

for all

The difference between a metric and a pseudometric space is that for a metric space with metric ifthenbut for a pseudometric space it is possible for the distance between distinct points to be equal to 0.

If for two pointsandin a pseudometric space,then every open neighbourhood ofcontainsand vice versa. This also means that a pseudometric space nis not T0 sinceandare distinct, but every open set containingcontainsand vice versa.