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The point x is an interior point of S. The point y is on the boundary of S. In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the ...
where is the dimension of the intersection (∩) of the interior (I), boundary (B), and exterior (E) of geometries a and b.. 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 ...
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.
If X is an n-dimensional compact connected submanifold of R n+1 (or S n+1) without boundary, its complement has 2 connected components. There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem , which states that the interior and the exterior planar regions determined by a Jordan curve in R 2 are ...
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 ...
Equivalently, for a bounded set the inner Jordan measure of is the Lebesgue measure of the topological interior of and the outer Jordan measure is the Lebesgue measure of the closure. [4] From this it follows that a bounded set is Jordan measurable if and only if its topological boundary has Lebesgue measure zero.
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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.