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A set equipped with a metric that measures distances between its points.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
A set is an object for which statements of the form (read “ is an element of ”) are meaningful, and whose identity is determined by extensionality: for sets ,
A metric on a set is a function such that for all :
A metric is the basic structure underlying a metric space; it determines open balls and hence the metric-induced topology.
A topological space is an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
The metric-induced topology on a metric space is the collection of subsets such that for every there exists with , where is the open ball of radius centered at .
This makes a topological space; equivalently, the family of open balls forms a basis for , and the open sets are precisely unions of open balls.
An open ball in a metric space is a set of the form
where and .
Open balls are open sets in the metric-induced topology and they form a basis for that topology; in particular, they are the basic neighborhoods in metric spaces.