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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 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 ...
For another example, consider the relative interior of a closed disk in . It is locally closed since it is an intersection of the closed disk and an open ball. On the other hand, { ( x , y ) ∈ R 2 ∣ x ≠ 0 } ∪ { ( 0 , 0 ) } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\neq 0\}\cup \{(0,0)\}} is not a locally closed subset of R 2 ...
There are several equivalent definitions for the boundary of a subset of a topological space , which will be denoted by , , or simply if is understood: . It is the closure of minus the interior of in : := ¯ where ¯ = denotes the closure of in and denotes the topological interior of in .
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 .
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.
Let be a locally compact Hausdorff space, and let () be the smallest σ-algebra that contains the open sets of ; this is known as the σ-algebra of Borel sets.A Borel measure is any measure defined on the σ-algebra of Borel sets. [2]
The interior of every set except X is empty. The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, a property that characterizes trivial topological spaces. As a result of this, the closure of every open subset U of X is either ∅ (if U = ∅) or X (otherwise).