> [!quote] Idea > > A neighbourhood is a collection of points considered close to a point by the topology. > > The interior of the neighbourhood serves to separate their inside and outside. Points and sets may be separated if they can live in disjoint neighbourhoods. > [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]], $x \in X$ be a point, and $E \subseteq X$ be a set. A set $V$ is a **neighbourhood** of $x$ or $E$ if > $ > x \in V^o \quad E \subseteq V^o > $ > $x$ or $E$ are contained in its [[Interior|interior]]. > [!theorem] > > Let $\cn(x)^o$ be the collection of all [[Open Set|open]] neighbourhoods of $x$. Then $\cn(x)^o$ is a [[Directed Set|directed set]] under reverse inclusion. > > *Proof*. Firstly, $U \supset U \forall U \in \cn(x)^o$. Secondly, $U \supset V$ and $V \supset W$ implies that $U \supset W$. Lastly, for any $U, V \in \cn(x)^o$, we can find $W = U \cap V \in \cn(x)^o$ such that $U, V \supset W$.