Interior (topology)

In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty). If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists a real number such that is in whenever the distance This definition generalizes to topological spaces by replacing "open ball" with "open set". If is a subset of a topological space then is an of in if is contained in an open subset of that is completely contained in (Equivalently, is an interior point of if is a neighbourhood of ) The interior of a subset of a topological space denoted by or or can be defined in any of the following equivalent ways: is the largest open subset of contained in is the union of all open sets of contained in is the set of all interior points of If the space is understood from context then the shorter notation is usually preferred to In any space, the interior of the empty set is the empty set. In any space if then If is the real line (with the standard topology), then whereas the interior of the set of rational numbers is empty: If is the complex plane then In any Euclidean space, the interior of any finite set is the empty set.