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In this sense interior and closure are dual notions. The exterior of a set S is the complement of the closure of S; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).
Each clopen subset of (which includes and itself) is simultaneously a regular open subset and regular closed subset. The interior of a closed subset of is a regular open subset of and likewise, the closure of an open subset of is a regular closed subset of . [2] The intersection (but not necessarily the union) of two regular open sets is a ...
A set (in light blue) and its boundary (in dark blue). In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S.
A subset of a topological space is called a regular open set if (¯) = or equivalently, if (¯) = , where , , and ¯ denote, respectively, the topological boundary, interior, and closure of in .
A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior. [21] An example of a non-regular open set is the set U = (0,1) ∪ (1,2) in R with its normal topology, since 1 is in the interior of the closure of U , but not in U .
If is a closed submanifold-with-boundary of a manifold , then the relative interior (that is, interior as a manifold) of is locally closed in and the boundary of it as a manifold is the same as the boundary of it as a locally closed subset. [2]
A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the topological interior of C. A closed convex subset is strictly convex if and only if every one of its boundary points is an extreme point. [3] A set C is absolutely convex if it is convex and balanced.
The interior of a line segment in an at least two-dimensional ambient space is empty, but its relative interior is the line segment without its endpoints. The interior of a disc in an at least three-dimensional ambient space is empty, but its relative interior is the same disc without its circular edge.