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(Redirected from Neighborhood (mathematics))
In topology and related areas of mathematics, a 'neighbourhood' (or 'neighborhood') is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can wiggle the point a bit without leaving the set.
This concept is closely related to the concepts of open set and interior.
If ''X'' is a topological space and ''p'' is a point in ''X'', a 'neighbourhood' of ''p'' is a set ''V'', which contains an open set ''U'' containing ''p'',
:
Note that the neighbourhood ''V'' need not be an open set itself. If ''V'' is open it is called an 'open neighbourhood'. Some authors require that neighbourhoods be open, so it is important to note conventions.
The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
If ''S'' is a subset of ''X'', a 'neighbourhood' of ''S'' is a set ''V'', which contains an open set ''U'' containing ''S''. It follows that a set ''V'' is a neighbourhood of ''S'', if and only if, it is a neighbourhood of all the points in ''S''.
In a metric space ''M'' = (''X'',''d''), a set ''V'' is a 'neighbourhood' of a point ''p'' if there exists an open ball with centre ''p'' and radius ''r'',
:
which is contained in ''V''.
''V'' is called 'uniform neighbourhood' of a set ''S'' if there exists a positive number ''r'' such that for all elements ''p'' of ''S'',
:
is contained in ''V''.
For ''r''>0 the '''r''-neighborhood' of a set ''S'' is the set of all points in ''X'' which are at distance less than ''r'' from ''S'' (or equivalently, is the union of all the open balls of radius ''r'' which are centered at a point in ''S'').
It directly follows that an ''r''-neighborhood is a uniform neighborhood, and that a set is a uniform neighborhood if and only if it contains an ''r''-neighborhood for some value of ''r''.
Given the set of real numbers 'R' with the usual Euclidean metric and a subset ''V'' defined as
:
then ''V'' is a neighbourhood for the set 'N' of natural numbers, but is ''not'' a uniform neighbourhood of this set.
The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.
A neighbourhood system on ''X'' is the assignment of a filter ''N(x)'' (on the set ''X'') to each ''x'' in ''X'', such that
# the point ''x'' is an element of each ''U'' in ''N(x)''
# each ''U'' in ''N(x)'' contains some ''V'' in ''N(x)'' such that for each ''y'' in ''V'', ''U'' is in ''N(y)''.
One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.
In a uniform space ''S'':=(''X'', δ) ''V'' is called a 'uniform neighbourhood' of ''P'' if ''P'' is not close to ''X'' ''V'', that is there exists no entourage containing ''P'' and ''X'' ''V''.
A ''punctured neighbourhood'' of a point ''p'' (sometimes called a ''deleted neighbourhood'') is a neighbourhood of ''p'', minus {''p''}. For instance, the interval (−1, 1) = {''y'' : −1 < ''y'' < 1} is a neighbourhood of ''p'' = 0 in the real line, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0. Note that a punctured neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of punctured neighbourhood occurs in the .
★ Tubular neighbourhood
★ Regular neighbourhood
★ Absolute neighbourhood retract
★ General topology, , John L., Kelley, New York: Springer-Verlag, ,
★ Topology and geometry, , Glen E., Bredon, New York: Springer-Verlag, ,
★ Set Theory and Metric Spaces, , Irving, Kaplansky, American Mathematical Society, ,
A set in the plane is a neighbourhood of a point if a small disk around is contained in
A rectangle is not a neighbourhood of any of its corners.
In topology and related areas of mathematics, a 'neighbourhood' (or 'neighborhood') is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can wiggle the point a bit without leaving the set.
This concept is closely related to the concepts of open set and interior.
| Contents |
| Definition |
| In a metric space |
| Examples |
| Topology from neighbourhoods |
| Uniform neighbourhoods |
| Punctured neighbourhood |
| See also |
| References |
Definition
If ''X'' is a topological space and ''p'' is a point in ''X'', a 'neighbourhood' of ''p'' is a set ''V'', which contains an open set ''U'' containing ''p'',
:
Note that the neighbourhood ''V'' need not be an open set itself. If ''V'' is open it is called an 'open neighbourhood'. Some authors require that neighbourhoods be open, so it is important to note conventions.
The collection of all neighbourhoods of a point is called the neighbourhood system at the point.
If ''S'' is a subset of ''X'', a 'neighbourhood' of ''S'' is a set ''V'', which contains an open set ''U'' containing ''S''. It follows that a set ''V'' is a neighbourhood of ''S'', if and only if, it is a neighbourhood of all the points in ''S''.
In a metric space
A set in the plane and a uniform neighbourhood of
In a metric space ''M'' = (''X'',''d''), a set ''V'' is a 'neighbourhood' of a point ''p'' if there exists an open ball with centre ''p'' and radius ''r'',
:
which is contained in ''V''.
''V'' is called 'uniform neighbourhood' of a set ''S'' if there exists a positive number ''r'' such that for all elements ''p'' of ''S'',
:
is contained in ''V''.
For ''r''>0 the '''r''-neighborhood' of a set ''S'' is the set of all points in ''X'' which are at distance less than ''r'' from ''S'' (or equivalently, is the union of all the open balls of radius ''r'' which are centered at a point in ''S'').
It directly follows that an ''r''-neighborhood is a uniform neighborhood, and that a set is a uniform neighborhood if and only if it contains an ''r''-neighborhood for some value of ''r''.
Examples
Given the set of real numbers 'R' with the usual Euclidean metric and a subset ''V'' defined as
:
then ''V'' is a neighbourhood for the set 'N' of natural numbers, but is ''not'' a uniform neighbourhood of this set.
Topology from neighbourhoods
The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points.
A neighbourhood system on ''X'' is the assignment of a filter ''N(x)'' (on the set ''X'') to each ''x'' in ''X'', such that
# the point ''x'' is an element of each ''U'' in ''N(x)''
# each ''U'' in ''N(x)'' contains some ''V'' in ''N(x)'' such that for each ''y'' in ''V'', ''U'' is in ''N(y)''.
One can show that both definitions are compatible, i.e. the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system.
Uniform neighbourhoods
In a uniform space ''S'':=(''X'', δ) ''V'' is called a 'uniform neighbourhood' of ''P'' if ''P'' is not close to ''X'' ''V'', that is there exists no entourage containing ''P'' and ''X'' ''V''.
Punctured neighbourhood
A ''punctured neighbourhood'' of a point ''p'' (sometimes called a ''deleted neighbourhood'') is a neighbourhood of ''p'', minus {''p''}. For instance, the interval (−1, 1) = {''y'' : −1 < ''y'' < 1} is a neighbourhood of ''p'' = 0 in the real line, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0. Note that a punctured neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of punctured neighbourhood occurs in the .
See also
★ Tubular neighbourhood
★ Regular neighbourhood
★ Absolute neighbourhood retract
References
★ General topology, , John L., Kelley, New York: Springer-Verlag, ,
★ Topology and geometry, , Glen E., Bredon, New York: Springer-Verlag, ,
★ Set Theory and Metric Spaces, , Irving, Kaplansky, American Mathematical Society, ,
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