In mathematics, a pseudometric space or semi-metric 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.
![X](/Images/godic/202501/27/02129bb861061d1a052c592e2dc6b3834833.png")
中任意元素![x,y](/Images/godic/202501/27/f10bc3c94b77e1d6b9f98106daf335c14833.png")
，若实值函数![d: (X,X) \to R](/Images/godic/202501/27/d8f578ca0545421950923be7d83d9b8a4834.png")
符合以下三个条件，称它为一个伪度量（pseudometric）。![d(x,x)=0](/Images/godic/202501/27/4c6db4a55aa298a1bdc2fbbf48c8147d4834.png")
![d(x,y)=d(y,x)](/Images/godic/202501/27/f2c22a820b8b5b12bc093e11247677724834.png")
![d(x,z) \le d(x,y) + d(y,z)](/Images/godic/202501/27/5570c108d5d442f6332e7127329235ef4834.png")
![d(x,y)=0](/Images/godic/202501/27/af57aa245c85ecb6c92e3a90759464864834.png")
。