Closureinto

Closureinto is a term used to describe the closure of a subset within a specified ambient space. In topology, if X is a topological space with a topology τ and A is a subset of X, the closureinto of A in X is the smallest closed set in X that contains A. This is usually written as cl_X(A) and is sometimes referred to as the closure of A in X. The phrase “closure into X” emphasizes that the closure is taken with respect to the topology of X, which can differ from closures taken in other spaces or subspaces. Note that closureinto is not a formal standard term in most textbooks; it is an informal, descriptive way to discuss closures relative to a chosen ambient space.

Properties: Closureinto is increasing with respect to inclusion, meaning if A ⊆ B ⊆ X then closureinto(A) ⊆ closureinto(B). It is extensive (A ⊆ closureinto(A)) and idempotent (closureinto(closureinto(A)) = closureinto(A)). The closure includes A together with all limit points of A in X. In particular, a point x ∈ X lies in closureinto(A) iff every open neighborhood of x intersects A.

Examples: In the real line R with the usual topology, closureinto([0,1)) equals [0,1]. In a discrete topology on a set S, the closureinto of any subset A is A itself. If Y is a subspace of X and A ⊆ Y, then closureinto_Y(A) can differ from closureinto_X(A) when viewed inside X.

See also: closure (topology), limit point, boundary, interior, subspace topology.