Omlukning

Omlukning is a term used in Danish topology referring to what is commonly called the closure of a subset in a topological space. If X is a topological space and A is a subset of X, the omlukning of A is denoted cl(A).

One standard definition is that cl(A) is the intersection of all closed sets in X that contain A. Equivalently, cl(A) can be described as A together with all its limit points, or as the set of all x in X such that every open neighborhood of x intersects A. These perspectives lead to several key characterizations: cl(A) contains A, cl(A) is closed, and cl(cl(A)) = cl(A) (idempotence). If A is already closed, then cl(A) = A.

Omlukning is closely connected to the concept of limit points. The derived set A′ consists of all limit points of A, and cl(A) = A ∪ A′. The boundary of A is bd(A) = cl(A) \ int(A), where int(A) denotes the largest open set contained in A.

Common examples illustrate the idea. In the real numbers with the standard topology, omlukning of the open interval (0,1) is [0,1]. In a discrete topology, every subset is closed, so omlukning of A is simply A. In a subspace Y of X, cl_Y(A) = Y ∩ cl_X(A).

Omlukning is a fundamental operation in topology, used to define dense sets, continuity via preimages of closed sets, and various forms of convergence.