Neighbourhood (topology): Difference between revisions

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In [[topology]], a '''neighbourhood of a point''' is any set that belongs to the '''neighbourhood system''' at that point.
The notion of neighbourhood systems is used to describe,
in an abstract setting, the concept of points near a given point,
a concept that cannot be expressed by a single set.
It is modelled after the situation in [[real analysis]]
where the points in small balls are considered as near to the centre of the ball.
Neighbourhoods are used to define
[[limit (mathematics)|convergence]] and [[continuous functions]]:
: '''Convergence''' ''(Definition)'' <br> A sequence converges to a point if and only if every neighbourhood of that point contains almost all (i.e., all but finitely many) elements of the sequence.
: '''Continuity''' ''(Definition)'' <br> A function ''f'' is continuous at a point ''x'' if and only if for every neighbourhood ''U'' of ''f(x)'' there is a neighbourhood ''V'' of ''x'' for which the image ''f(V)'' under ''f'' is a subset of ''U''.
== Neighbourhood spaces ==
A set ''X'' is called a '''neighbourhood space'''
if for every ''x'' in ''X''
there is a nonempty family ''N(x)'' (the '''neighbourhood system''' at ''x'') of sets, called neighbourhoods of ''x'',
which satisfies the following axioms:
#  ''x'' is an element of every neighborhood of ''x''.
#  Any set that contains a neighbourhood of ''x'' is a neighbourhood of ''x''.
#  The intersection of any two (and therefore of any finite collection of) neighbourhoods of ''x'' is a neighbourhood of ''x''.
#  Any neighbourhood of ''x'' contains an '''open neighbourhood''' of ''x'',<br> i.e., a neighbourhood of ''x'' that belongs to ''N(y)'' for all of its elements ''y''.
Axioms (2-3) imply that ''N(x)'' is a [[filter (mathematics)|filter]].
Accordingly, the neighbourhood system at a point
is also called the '''neighbourhood filter''' of the point.
<br>
Axiom (4) defines how neighbourhood systems at distinct points interact.
=== Neighbourhood base ===
To define a neighbourhood space it is often more convenient to describe,
for all ''x'', only a base for the neighbourhood system.
<br>
A nonempty family ''B(x)'' of sets is a '''neighbourhood base''' at ''x'' if it satisfies the following axioms:
#  ''x'' is an element of every set in ''B(x)''.
#  The intersection of any two sets of ''B(x)'' contains a set of ''B(x)''.
#  Any set of ''B(x)'' contains an open neighbourhood ''V'' of ''x'',<br> i.e., a set ''V'' (not necessarily a member of ''B(x)'') that contains some set of ''B(y)'' for all of its elements ''y'' in ''V''.
Axiom (2) implies that ''B(x)'' is a [[filter (mathematics)|filter base]].
<br>
The family ''N(x)'' consisting of all sets containing a set of ''B(x)''
is the neighbourhood filter '''induced''' by ''B(x)''
<br>
Two neighbourhood bases ''B<sub>1</sub>(x)'' and ''B<sub>2</sub>(x)'' are called equivalent
if they induce the same neighbourhood system ''N(x)'' at ''x''.
(This is the case if and only if each set in ''B<sub>1</sub>(x)'' contains a set in ''B<sub>2</sub>(x)'',
and if, vice versa, each set in ''B<sub>2</sub>(x)'' contains a set in ''B<sub>1</sub>(x)''.)
<br>
If there exist [[countable]] neighbourhood bases at all ''x'' in ''X'',
then the corresponding topological (or, equivalently, neighbourhood) space is said to be '''[[first-countable]]'''.
=== Example: Metric spaces ===
In a [[metric space]] the (open or closed) balls with centre ''x'' form a neighbourhood base at ''x''
and define the topology induced by the metric.
<br>
Moreover, it is sufficient to take the balls with radius ''1/n'' for all natural numbers ''n''
(that is, a [[countable]] set for each point ''x''), therefore metric spaces are first-countable.
The classical example (in calculus or real analysis) is <math> \mathbb R^d </math>:
:  A neighbourhood base at <math> x \in \mathbb R^d </math> is given by
:: <math> \mathcal B(x) = \left\{ B(x,n) \mid n \in \mathbb N \right\} </math>
:  where
:: <math> B(x,n) = \left\{ y \left\vert \left| y-x \right| < {1\over n} \right. , y \in \mathbb R^d \right\} \subset \mathbb R^d</math>
:  are the open balls with centre <math>x</math> and radius <math> 1/n </math>.
=== Neighbourhood of a set ===
=== Neighbourhood of a set ===


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if ''U'' is a neighbourhood for all points of ''S'' or, equivalently,
if ''U'' is a neighbourhood for all points of ''S'' or, equivalently,
if ''U'' contains an open set that contains ''S''.
if ''U'' contains an open set that contains ''S''.
=== Relation to topological spaces ===
Neighbourhood spaces are one of several equivalent means
to define a [[topological space]].
The equivalence is obtained by the following definitions:
: ''(Definition)'' <br> In a neighbourhood space, a set is '''open''' if it is a neighbourhood of all its points.
: ''(Definition)'' <br> In a topological space, a set is a '''neighbourhood''' of a point if it contains an open set that contains the point. <br> (In other words, the open sets containing a point form a '''neighbourhood base''' at this point.)

Revision as of 19:38, 28 May 2009

Neighbourhood of a set

A set U is called neighbourhood of the set S if U is a neighbourhood for all points of S or, equivalently, if U contains an open set that contains S.

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