Defining System Of Neighbourhoods

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 $X$.

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