Neighborhoods

Definition: Neighborhood of a Point

Let $(X,\tau )$ be a topological space and let $x\in X$.

A subset $N\subseteq X$ is a neighbourhood of $x$ iff there exists an open set $U$ such that $U\subseteq N$ and $x\in U$.

NOTATION

$$N(x)N_x$$

Definition: Neighbourhood of a Set

Let $(X,\tau )$ be a topological space and let $S$ be a subset of $X$.

A subset $N\subseteq X$ is a neighbourhood of $S$ iff there exists an open set $U$ such that

$$S\subseteq U\subseteq N$$

NOTATION

$$N_{S}N(S)$$