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Premetrics

Relaxing the last three axioms leads to the notion of a premetric, i.e. a function satisfying the following conditions:

This is not a standard term. Sometimes it is used to refer to other generalizations of metrics such as pseudosemimetrics or pseudometrics; in translations of Russian books it sometimes appears as "prametric". A premetric that satisfies symmetry, i.e. a pseudosemimetric, is also called a distance.

Any premetric gives rise to a topology as follows. For a positive real r {\displaystyle r}, the r {\displaystyle r} -ball centered at a point p {\displaystyle p} is defined as

A set is called open if for any point p {\displaystyle p} in the set there is an r {\displaystyle r} -ball centered at p {\displaystyle p} which is contained in the set. Every premetric space is a topological space, and in fact a sequential space. In general, the r {\displaystyle r} -balls themselves need not be open sets with respect to this topology. As for metrics, the distance between two sets A {\displaystyle A} and B {\displaystyle B}, is defined as

This defines a premetric on the power set of a premetric space. If we start with a (pseudosemi-)metric space, we get a pseudosemimetric, i.e. a symmetric premetric.Any premetric gives rise to a preclosure operator c l {\displaystyle cl} as follows: