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The interior, boundary, and exterior of a set together partition the whole space into three blocks (or fewer when one or more of these is empty): = , where denotes the boundary of . [3] The interior and exterior are always open, while the boundary is closed.
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.
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 ...
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]
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 definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
The interior of , denoted , is the set of points in which have neighborhoods homeomorphic to an open subset of . The boundary of M {\displaystyle M} , denoted ∂ M {\displaystyle \partial M} , is the complement of Int M {\displaystyle \operatorname {Int} M} in M {\displaystyle M} .
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.