In topology and mathematical analysis, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set.
Definitions
Neighbourhood of a point If X {\displaystyle X} is a topological space and p {\displaystyle p} is a point in X , {\displaystyle X,} then a neighbourhood of p {\displaystyle p} is a subset V {\displaystyle V} of X {\displaystyle X} that includes an open set U {\displaystyle U} containing p {\displaystyle p} ,
p ∈ U ⊆ V ⊆ X . {\displaystyle p\in U\subseteq V\subseteq X.}
This is equivalent to the point p ∈ X {\displaystyle p\in X} belonging to the topological interior of V {\displaystyle V} in X . {\displaystyle X.}
The neighbourhood V {\displaystyle V} need not be an open subset of X . {\displaystyle X.} When V {\displaystyle V} is open (resp. closed, compact, etc.) in X , {\displaystyle X,} it is called an open neighbourhood (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open.
A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle. The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
Neighbourhood of a set If S {\displaystyle S} is a subset of a topological space X {\displaystyle X} , then a neighbourhood of S {\displaystyle S} is a set V {\displaystyle V} that includes an open set U {\displaystyle U} containing S {\displaystyle S} , S ⊆ U ⊆ V ⊆ X . {\displaystyle S\subseteq U\subseteq V\subseteq X.} It follows that a set V {\displaystyle V} is a neighbourhood of S {\displaystyle S} if and only if it is a neighbourhood of all the points in S . {\displaystyle S.} Furthermore, V {\displaystyle V} is a neighbourhood of S {\displaystyle S} if and only if S {\displaystyle S} is a subset of the interior of V . {\displaystyle V.}
A neighbourhood of S {\displaystyle S} that is also an open subset of X {\displaystyle X} is called an open neighbourhood of S . {\displaystyle S.}
The neighbourhood of a point is just a special case of this definition.
In a metric space
In a metric space M = ( X , d ) , {\displaystyle M=(X,d),} a set V {\displaystyle V} is a neighbourhood of a point p {\displaystyle p} if there exists a positive real number r {\displaystyle r} such that the open ball
B ( p ; r ) = { x ∈ X : d ( x , p ) < r } {\displaystyle B(p;r)=\{x\in X:d(x,p)<r\}}
with center p {\displaystyle p} and radius r {\displaystyle r} is contained in V . {\displaystyle V.}
V {\displaystyle V} is a neighbourhood of a set S {\displaystyle S} if, for each element p {\displaystyle p} of S , {\displaystyle S,} there exists a positive number r {\displaystyle r} such that B ( p ; r ) {\displaystyle B(p;r)} is contained in V . {\displaystyle V.} V {\displaystyle V} is called a uniform neighbourhood of a set S {\displaystyle S} if there exists a positive number r {\displaystyle r} such that for all elements p {\displaystyle p} of S , {\displaystyle S,} the open ball B ( p ; r ) {\displaystyle B(p;r)} is contained in V . {\displaystyle V.}
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