Nearopen

Nearopen is a term used in topology and analysis to describe subsets that are not open but are in a sense close to being open. The term is informal in many texts, and there is no single universally accepted definition. In published discussions, two common formulations appear.

Definition A (closure–interior criterion): A subset U of a topological space X is nearopen if U is contained in the interior of its closure, i.e., U ⊆ int(cl(U)). This ensures every point of U lies in an open neighborhood contained in cl(U).

Definition B (closure of interior criterion): A subset U is nearopen if the closure of its interior equals the closure of the set, i.e., cl(int(U)) = cl(U). This mirrors a compatibility between U and its interior with respect to closure.

Notes: These definitions are not equivalent in general; some authors favor one while others use another. Open sets satisfy both criteria. Regular open sets (A = int(cl(A))) are closely related to Definition B, because applying cl and int to A yields comparisons to A.

Examples: In the real line with the usual topology, the interval (0,1) is nearopen under both definitions; the singleton {0} fails both.

Applications: Nearopen concepts appear in studies of approximations of openness, convergence of sets, and in certain lattice-theoretic discussions of subset classes.

See also: Open set, Regular open set, Topological space.

Because of definitional variation, readers should consult the specific source for the precise meaning of "nearopen" in that context.