A Set with this property contains all points with neighborhoods partly in the set, and so also contains points with neighborhoods partly inside and partly outside; such points are on the boundaries of the set.
A subset of a Polish space with this property has countably many isolated points by the Cantor-Bendixson theorem. A set with this property each of whose points is an accumulation point is called perfect; in general, sets with this property contain all of their accumulation points. Subsets of Euclidian space that are bounded and compact necessarily have this property by the Bolzano-Weierstrass theorem. A Set with this property contains all points with neighborhoods partly in the set, and so also contains points with neighborhoods partly inside and partly outside; such points are on the boundaries of the set. For 10 points, name this property exemplified by a range of real numbers including end values and possessed by sets whose complements are open.
A subset of a Polish space with this property has countably many isolated points by the Cantor-Bendixson theorem.
{closed} (accept word forms like {closure})