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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 ...
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 ...
The interior of the boundary of a closed set is empty. [proof 1] Consequently, the interior of the boundary of the closure of a set is empty. The interior of the boundary of an open set is also empty. [proof 2] Consequently, the interior of the boundary of the interior of a set
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 collection of all bounded sets on a topological vector space is called the von Neumann bornology or the (canonical) bornology of .. A base or fundamental system of bounded sets of is a set of bounded subsets of such that every bounded subset of is a subset of some . [1] The set of all bounded subsets of trivially forms a fundamental system of bounded sets of .
Assume that is a subset of a vector space . The algebraic interior (or radial kernel) of with respect to is the set of all points at which is a radial set.A point is called an internal point of [1] [2] and is said to be radial at if for every there exists a real number > such that for every [,], +.
A subset S of a metric space (M, d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r. The metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For subsets of R n the two are equivalent.
Let (,) be a topological space; let () denote the Borel σ-algebra on , i.e. the smallest sigma algebra on that contains all open sets . Let be a measure on (, ()) Then the support (or spectrum) of is defined as the set of all points in for which every open neighbourhood of has positive measure: ():= {: (() >)}.