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A distance function on a set satisfying positivity, symmetry, and the triangle inequality.
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 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 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.
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.
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.