Pseudometric space

In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis.

When a topology is generated using a family of pseudometrics, the space is called a gauge space.

A pseudometric space ${\displaystyle (X,d)}$ is a set ${\displaystyle X}$ together with a non-negative real-valued function ${\displaystyle d\colon X\times X\longrightarrow \mathbb {R} _{\geq 0}}$ (called a pseudometric) such that, for every ${\displaystyle x,y,z\in X}$,

...
Wikipedia