Closure
A closure of a set is the union of the set with all of its limit points.
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Description
Openness of a set can be determined in either a topological space or a metric space. The core idea is that a set is closed if it contains all of its limit points, and open if it contains none of them.
The union of a set with all of its limit points is defined as its closure.
In circumstances where it isn't clear what topological space A is a subset of, the most common notation is cl(X, τ)(A).
A closure is always closed. The statements that 'a set is equal to its own closure' and that 'a set is closed' are equivalent.