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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 ).
For any set ,, where denotes the superset with equality holding if and only if the boundary of has no interior points, which will be the case for example if is either closed or open. Since the boundary of a set is closed, ∂ ∂ S = ∂ ∂ ∂ S {\displaystyle \partial \partial S=\partial \partial \partial S} for any set S . {\displaystyle S.}
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 regular open set. Similarly, the union (but not necessarily the intersection) of two regular closed sets is a regular ...
Formally, the relative interior of a set (denoted ()) is defined as its interior within the affine hull of . [1] In other words, ():= {: > ()}, where is the affine hull of , and () is a ball of radius centered on . Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.
The terms interior and boundary in this article are used in the sense used in algebraic topology and manifold theory, not in the sense used in general topology: for example, the interior of a line segment is the line segment without its endpoints, and its boundary is just the two endpoints (in general topology, the interior of a line segment in ...
An example of a convex polygon: a regular pentagon.. In geometry, a convex polygon is a polygon that is the boundary of a convex set.This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon.
The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense (, [5] Anmerkungen zu §10, /p. 590).
One of these components is bounded (the interior) and the other is unbounded (the exterior), and the curve is the boundary of each component. In contrast, the complement of a Jordan arc in the plane is connected.