In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa[1][2] in 1934. 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.