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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]]''
  
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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]]).

Revision as of 12:15, 29 September 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).

How to Cite This Entry:
Neighbourhood. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neighbourhood&oldid=30570
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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