nearclosed

Nearclosed is a term used in some mathematical contexts to denote a relaxed form of closedness for subsets of a topological or measure space. A common definition appears in measure-theoretic contexts: in a measure space (X, Σ, μ), a subset A ⊆ X is called nearclosed if the closure of A differs from A by a μ-null set; that is, μ(closure(A) \ A) = 0. Equivalently, A is nearclosed if A = closure(A) up to a μ-null set. This captures the idea that A is "almost" closed from the measure-theoretic perspective.

In spaces with a finite measure, closed sets are nearclosed, and any nearclosed set has a boundary of measure zero. The class is closed under finite unions and intersections: if A and B are nearclosed, then μ(closure(A∪B) \ (A∪B)) ≤ μ(closure(A)\A) + μ(closure(B)\B) = 0, so A∪B is nearclosed; similarly for intersections.

Examples: In R with the Lebesgue measure, a finite union of disjoint open intervals is nearclosed since its boundary is a finite set and hence μ(∂A) = 0. Open sets with boundaries of positive measure need not be nearclosed.

Relation to other notions: nearclosed is closely related to the concept of sets equal to their closure up to a null set, i.e., almost closed sets, and to the idea of boundary having measure zero.

Limitations: The term is not universally standardized; definitions may vary among authors, and some use “almost closed” instead.

References: See Lebesgue measure, measure-zero sets, boundary.