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Difference between revisions of "Defining system of neighbourhoods"

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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306701.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306702.png" />''
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''of a set $A$ in a topological space $X$''
  
Any family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306703.png" /> of subsets of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306704.png" /> subject to the following two conditions: a) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306705.png" /> there is an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306706.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306707.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306708.png" />; b) for any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d0306709.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067010.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067011.png" /> there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067012.png" /> of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067013.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067014.png" />.
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Any family $\xi$ of subsets of the space $X$ subject to the following two conditions: a) for every $O\in\xi$ there is an open set $V$ in $X$ such that $O\supset V\supset A$; b) for any open set $W$ in $X$ containing $A$ there is an element $U$ of the family $\xi$ contained in $W$.
  
It is sometimes further supposed that all elements of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067015.png" /> are open sets. A defining system of neighbourhoods of a one-point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067016.png" /> in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067017.png" /> is called a defining system of neighbourhoods of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030670/d03067019.png" />.
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It is sometimes further supposed that all elements of the family $\xi$ are open sets. A defining system of neighbourhoods of a one-point set $\{x\}$ in a topological space $X$ is called a defining system of neighbourhoods of the point $x\in X$ in $X$.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
 
A defining system of neighbourhoods is also called a local base or a neighbourhood base.
 
A defining system of neighbourhoods is also called a local base or a neighbourhood base.
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[[Category:General topology]]

Latest revision as of 21:28, 8 November 2014

of a set $A$ in a topological space $X$

Any family $\xi$ of subsets of the space $X$ subject to the following two conditions: a) for every $O\in\xi$ there is an open set $V$ in $X$ such that $O\supset V\supset A$; b) for any open set $W$ in $X$ containing $A$ there is an element $U$ of the family $\xi$ contained in $W$.

It is sometimes further supposed that all elements of the family $\xi$ are open sets. A defining system of neighbourhoods of a one-point set $\{x\}$ in a topological space $X$ is called a defining system of neighbourhoods of the point $x\in X$ in $X$.

References

[1] A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)


Comments

A defining system of neighbourhoods is also called a local base or a neighbourhood base.

How to Cite This Entry:
Defining system of neighbourhoods. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Defining_system_of_neighbourhoods&oldid=12940
This article was adapted from an original article by A.V. Arkhangel'skii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article