Neighborhood

Definition

Definition. Let $(X,\tau )$ be a topological space, a set $N$ is said to be a neighborhood of a point $p\in X$, denoted as $N(x)$ if there exists an open set $U$ of $X$ such that $p\in U\subseteq N\subseteq X$.

Definition. Let $(X,d)$ be a metric space. A set $N\subset X$ is said to be a neighborhood of $a\in X$ if there exists $\delta >0$ such that the open ball $B_{<\delta }(a)$ is a subset of $N$.

References

@Tu11 pp. 318 and @Men90 Definition 4.1