Neighbourhood (topology): Difference between revisions

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imported>Peter Schmitt
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imported>Peter Schmitt
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Neighbourhoods are used to define
Neighbourhoods are used to define
[[limit (mathematics)|convergence]] and [[continuous functions]]:
[[limit (mathematics)|convergence]] and [[continuous functions]]:
''(Definition)''
: ''(Definition)'' A sequence converges to a point if and only if every neighbourhood of that point contains almost all <br> (i.e., all but finitely many) elements of the sequence.
A sequence converges to a point if and only if every neighbourhood
: ''(Definition)'' 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''.
of that point contains almost all (i.e., all but finitely many)
elements of the sequence.
''(Definition)''
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==
==Neighbourhood spaces==
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Accordingly, the system of neighbourhoods of a point
Accordingly, the system of neighbourhoods of a point
is also called the '''neighbourhood filter''' of the point.
is also called the '''neighbourhood filter''' of the point.
 
<br>
Axiom (4) defines how neighbourhood systems of distinct points interact.
Axiom (4) defines how neighbourhood systems of distinct points interact.


Line 37: Line 31:
to define a [[topological space]].
to define a [[topological space]].
The equivalence is obtained by the following definitions:
The equivalence is obtained by the following definitions:
In a neighbourhood space, a set is open if it a neighbourhood of all its points.
: ''(Definition)'' In a neighbourhood space, a set is '''open''' if it is a neighbourhood of all its points.
In a topological space, a set is a neighbourhood of a point
: ''(Definition)'' 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 '''base''' for the neighbourhood system of this point.)
if it contains an open set that contains the point.
(In other words, the open sets containing a point are
a '''base for the neighbourhood system''' of this point.)
 
 
 
 
In topology, a '''neighbourhood''' of a point ''x'' in a [[topological space]] ''X'' is a set ''N'' such that ''x'' is in the interior of ''N''; that is, there is an open set ''U'' such that <math>x \in U \subseteq N</math>.
A ''neighbourhood of a set'' ''A'' in ''X'' is a set ''N'' such that ''A'' is contained in the interior of ''N''; that is, there is an open set ''U'' such that <math>A \subseteq U \subseteq N</math>.
 
The family of neighourhoods of a point ''x'', denoted <math>\mathcal{N}_x</math> satisfies the properties
 
#<math>X \in \mathcal{N}_x ; \,</math>
#<math>\empty \not\in \mathcal{N}_x ; \,</math>
#<math>U,V \in \mathcal{N}_x \Rightarrow U \cap V \in \mathcal{N}_x ; \,</math>
#<math>U \in \mathcal{N}_x \mbox{ and } U \subseteq N \Rightarrow N \in \mathcal{N}_x . \,</math>
 
The properties are equivalent to stating that the neighbourhood system  <math>\mathcal{N}_x</math> is a [[filter (mathematics)|filter]], the ''neighbourhood filter'' of ''x''.
 
A topology may be defined in terms of its neighbourhood systems: a set is open if and only if it is a neighbourhood of each of its points.


==See also==
==See also==
* [[Topological space#Some topological notions]]
* [[Topological space#Some topological notions]]

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In topology, the notion of a neighbourhood is used to describe, in an abstract setting, the concept of points near a given point. 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 convergence and continuous functions:

(Definition) 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.
(Definition) 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) of sets, called neighbourhoods of x, which satisfies the following axioms:

  1. x is an element of every neighborhood of x.
  2. The intersection of any two (and therefore of any finite collection of) neighbourhoods of x is a neighbourhood of x.
  3. Any neighbourhood of x contains an open neighbourhood of x,
    i.e., a neighbourhood of x that belongs to N(y) for all of its elements y.

Axioms (1-3) imply, that N(x) is a filter. Accordingly, the system of neighbourhoods of a point is also called the neighbourhood filter of the point.
Axiom (4) defines how neighbourhood systems of distinct points interact.

Neighbourhood spaces are one of several equivalent means to define a topological space. The equivalence is obtained by the following definitions:

(Definition) In a neighbourhood space, a set is open if it is a neighbourhood of all its points.
(Definition) In a topological space, a set is a neighbourhood of a point if it contains an open set that contains the point.
(In other words, the open sets containing a point form a base for the neighbourhood system of this point.)

See also

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