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 complement of S. In this sense interior and closure are dual notions. 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).
Definitions
Interior point If S {\displaystyle S} is a subset of a Euclidean space, then x {\displaystyle x} is an interior point of S {\displaystyle S} if there exists an open ball centered at x {\displaystyle x} which is completely contained in S . {\displaystyle S.}
(This is illustrated in the introductory section to this article.) This definition generalizes to any subset S {\displaystyle S} of a metric space X {\displaystyle X} with metric d {\displaystyle d} : x {\displaystyle x} is an interior point of S {\displaystyle S} if there exists a real number r > 0 , {\displaystyle r>0,} such that y {\displaystyle y} is in S {\displaystyle S} whenever the distance d ( x , y ) < r . {\displaystyle d(x,y)<r.}
This definition generalizes to topological spaces by replacing "open ball" with "open set". If S {\displaystyle S} is a subset of a topological space X {\displaystyle X} then x {\displaystyle x} is an interior point of S {\displaystyle S} in X {\displaystyle X} if x {\displaystyle x} is contained in an open subset of X {\displaystyle X} that is completely contained in S . {\displaystyle S.}
(Equivalently, x {\displaystyle x} is an interior point of S {\displaystyle S} if S {\displaystyle S} is a neighbourhood of x . {\displaystyle x.} )
Interior of a set The interior of a subset S {\displaystyle S} of a topological space X , {\displaystyle X,} denoted by int X S {\displaystyle \operatorname {int} _{X}S} or int S {\displaystyle \operatorname {int} S} or S ∘ , {\displaystyle S^{\circ },} can be defined in any of the following equivalent ways:
int S {\displaystyle \operatorname {int} S} is the largest open subset of X {\displaystyle X} contained in S . {\displaystyle S.}
int S {\displaystyle \operatorname {int} S} is the union of all open sets of X {\displaystyle X} contained in S . {\displaystyle S.}
int S {\displaystyle \operatorname {int} S} is the set of all interior points of S . {\displaystyle S.}
If the space X {\displaystyle X} is understood from context then the shorter notation int S {\displaystyle \operatorname {int} S} is usually preferred to int X S . {\displaystyle \operatorname {int} _{X}S.}
Examples
In any space, the interior of the empty set is the empty set. In any space X , {\displaystyle X,} if S ⊆ X , {\displaystyle S\subseteq X,} then int S ⊆ S . {\displaystyle \operatorname {int} S\subseteq S.}
If X {\displaystyle X} is the real line R {\displaystyle \mathbb {R} } (with the standard topology), then int ( [ 0 , 1 ] ) = ( 0 , 1 ) {\displaystyle \operatorname {int} ([0,1])=(0,1)} whereas the interior of the set Q {\displaystyle \mathbb {Q} } of rational numbers is empty: int Q = ∅ . {\displaystyle \operatorname {int} \mathbb {Q} =\varnothing .}
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