A pseudometric space is a set $X$ together with a non-negative real-valued function $d: X \times X \longrightarrow \mathbb{R}$ (called a pseudometric) such that, for every $x,y,z \in X$

• $d(x,x) = 0$
• $d(x,y) = d(y,x)$
• $d(x,z) \leq d(x,y) + d(y,z)$

In other words, a pseudometric space is a generalization of a metric space in which we allow the possibility that $d(x,y)=0$ for distinct values of $x$ and $y$

Bibliography

1

L.A. Steen, J.A.Seebach, Jr., Counterexamples in topology, Holt, Rinehart and Winston, Inc., 1970.