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The set of points within a given radius of a center point in a metric space, using non-strict inequality.
A closed ball in a metric space is a set of the form
where and .
Closed balls are closely related to open balls and are closed sets in the metric-induced topology.
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.
A closed set in a topological space is a subset such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
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.