For a set A in a Topological space, its closure is the intersection of all Closed sets which contain A. It is thus the smallest closed set containing A.

Compare with Interior of a set

The closure of a subspace of a topological space X (a subset Y with the Subspace topology) is the intersection of the closure in the topology of X with the subset Y (see here for proof, using lemma that all closed sets in the subspace can be written as intersections with the subspace of closed sets in the larger space)

Proposition

The closure of a set is the union of the set with the set of all Limit points