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Difference between revisions of "Neighbourhood"

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''of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662601.png" /> (of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662602.png" />) of a [[Topological space|topological space]]''
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''of a point $x$ (of a subset $A$) of a [[Topological space|topological space]]''
  
Any open subset of this space containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662603.png" /> (the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662604.png" />). Sometimes a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662605.png" /> (the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662606.png" />) is defined as any subset of this topological space containing the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662607.png" /> (the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066260/n0662608.png" />) in its interior (cf. also [[Interior of a set|Interior of a set]]).
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Any open subset of this space containing the point $x$ (the set $A$). Sometimes a neighbourhood of the point $x$ (the set $A$) is defined as any subset of this topological space containing the point $x$ (the set $A$) in its interior (cf. also [[Interior of a set|Interior of a set]]): in this case the first definition is that of an ''open neighorhood''.  A set $N$ is a neighbourhood of the set $A$ if and only if it is a neighbourhood of each point $x \in A$.
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In the first definition, the open neighbourhoods are precisely the open sets of the topology.
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In the second definition, the system of neighbourhoods $\mathfrak{N}(x)$ of a point $x$ satisfy the following four properties:
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# $x \in N$ for every $N \in \mathfrak{N}(x)$;
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# If $M \supset N$ for $N \in \mathfrak{N}(x)$, then $M \in \mathfrak{N}(x)$;
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# If $N_1, N_2 \in \mathfrak{N}(x)$ then $N_1 \cap N_2 \in \mathfrak{N}(x)$;
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# For each $N \in \mathfrak{N}(x)$ there exists $M \in \mathfrak{N}(x)$ such that $N \in \mathfrak{N}(y)$ for each $y \in M$.
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In the opposite direction, properties (1)--(4) may be taken as the definition of a topology; a set $N$ is open if it is in $\mathfrak{N}(x)$ for each of its elements $x$: these are the Hausdorff neighbourhood axioms.
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==References==
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* Franz, Wolfgang.  ''General topology'' (Harrap, 1967).

Latest revision as of 13:28, 12 December 2013

of a point $x$ (of a subset $A$) of a topological space

Any open subset of this space containing the point $x$ (the set $A$). Sometimes a neighbourhood of the point $x$ (the set $A$) is defined as any subset of this topological space containing the point $x$ (the set $A$) in its interior (cf. also Interior of a set): in this case the first definition is that of an open neighorhood. A set $N$ is a neighbourhood of the set $A$ if and only if it is a neighbourhood of each point $x \in A$.

In the first definition, the open neighbourhoods are precisely the open sets of the topology.

In the second definition, the system of neighbourhoods $\mathfrak{N}(x)$ of a point $x$ satisfy the following four properties:

  1. $x \in N$ for every $N \in \mathfrak{N}(x)$;
  2. If $M \supset N$ for $N \in \mathfrak{N}(x)$, then $M \in \mathfrak{N}(x)$;
  3. If $N_1, N_2 \in \mathfrak{N}(x)$ then $N_1 \cap N_2 \in \mathfrak{N}(x)$;
  4. For each $N \in \mathfrak{N}(x)$ there exists $M \in \mathfrak{N}(x)$ such that $N \in \mathfrak{N}(y)$ for each $y \in M$.

In the opposite direction, properties (1)--(4) may be taken as the definition of a topology; a set $N$ is open if it is in $\mathfrak{N}(x)$ for each of its elements $x$: these are the Hausdorff neighbourhood axioms.

References

  • Franz, Wolfgang. General topology (Harrap, 1967).
How to Cite This Entry:
Neighbourhood. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neighbourhood&oldid=15627
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article