# Defining system of neighbourhoods

From Encyclopedia of Mathematics

*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://www.encyclopediaofmath.org/index.php?title=Defining_system_of_neighbourhoods&oldid=34382

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