Difference between revisions of "Defining system of neighbourhoods"
From Encyclopedia of Mathematics
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| − | ''of a set | + | {{TEX|done}} |
| + | ''of a set $A$ in a topological space $X$'' | ||
| − | Any family | + | 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 | + | 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
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