View source for Defining system of neighbourhoods

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''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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Arkhangel'skii,   V.I. Ponomarev,   "Fundamentals of general topology: problems and exercises" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>



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