Metric space

In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally: In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally: A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space. In fact, a 'metric' is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. In 1906 Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel. However the name is due to Felix Hausdorff. A metric space is an ordered pair ( M , d ) {displaystyle (M,d)} where M {displaystyle M} is a set and d {displaystyle d} is a metric on M {displaystyle M} , i.e., a function such that for any x , y , z ∈ M {displaystyle x,y,zin M} , the following holds: Given the above three axioms, we also have that d ( x , y ) ≥ 0 {displaystyle d(x,y)geq 0} for any x , y ∈ M {displaystyle x,yin M} . This is deduced as follows: The function d {displaystyle d} is also called distance function or simply distance. Often, d {displaystyle d} is omitted and one just writes M {displaystyle M} for a metric space if it is clear from the context what metric is used.