Sluiting

Sluiting is a term used in topology to denote the closure of a subset within a topological space. If X is a topological space and A is a subset of X, the sluiting cl(A) is the smallest closed set that contains A. Equivalently, it can be described as the intersection of all closed sets containing A, or as the set of all points x in X for which every open neighborhood of x intersects A.

Key properties of sluiting include: A is always contained in cl(A); cl(A) is a closed set; applying sluiting twice yields the same result (cl(cl(A)) = cl(A)); sluiting distributes over unions (cl(A ∪ B) = cl(A) ∪ cl(B)); and cl(A ∩ B) ⊆ cl(A) ∩ cl(B). The operation is monotone: if A ⊆ B, then cl(A) ⊆ cl(B). In a metric space, x belongs to cl(A) if and only if every neighborhood of x contains a point of A, and equivalently x is either in A or a limit point of A. A common characterization is that cl(A) = A ∪ A', where A' is the set of limit points of A.

Cl(A) is used to define other concepts, such as the interior int(A) and the boundary ∂A = cl(A) \ int(A). Notation varies by language; in Dutch texts, sluiting is the standard term for this concept, often written as overline{A} in English-language literature.

Typical examples include the closure of the interval (0,1) in the real line with the standard topology, which is [0,1], and the closure of a finite set is the set itself. The sluitening operation is fundamental to convergence, continuity, and the study of topological properties.