Given any subset each point of lies in exactly one of the three sets and Said differently, and these three sets are pairwise disjoint. Consequently, if these set are not empty then they form a partition of A point is a boundary point of a set if and only if every neighborhood of contains at least one point in the set and at least one point not in the set. The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set. Conceptual Venn diagram showing the relationships among different points of a subset of = set of limit points of set of boundary points of area shaded green = set of interior points of area shaded yellow = set of isolated points of areas shaded black = empty sets. Every point of is either an interior point or a boundary point. Also, every point of is either an accumulation point or an isolated point. Likewise, every boundary point of is either an accumulation point or an isolated point. Isolated points are always boundary points. Examples Characterizations and general examples The boundary of a set is equal to the boundary of the set's complement: A set is a dense open subset of if and only if The interior of the boundary of a closed set is the empty set. Consequently, the interior of the boundary of the closure of a set is the empty set. The interior of the boundary of an open set is also the empty set. Consequently, the interior of the boundary of the interior of a set is the empty set. In particular, if is a closed or open subset of then there does not exist any non-empty subset such that is also an open subset of This fact is important for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces. A set is the boundary of some open set if and only if it is closed and nowhere dense. The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set). Concrete examples Consider the real line with the usual topology (that is, the topology whose basis sets are open intervals) and the subset of rational numbers (whose topological interior in is empty). Then