> [!quote] Idea > > Simpler subcollection of all the neighbourhoods of a point. Allows inferring properties about neighbourhoods by considering smaller neighbourhoods contained in them. > [!definition] > > Let $(X, \topo)$ be a [[Topological Space|topological space]], $x \in X$ be a point, and $\cn \subset \ct$ be a family of [[Open Set|open sets]]. The family $\cn$ is a **[[Neighbourhood|neighbourhood]] base** for $\topo$ at $x$ if > - $x \in V$ for all $V \in \cn$ > - For any $U \in \topo, U \ni x$, there exists $V \in \cn$ such that $V \subset U$. > [!theorem] > > Let $(X, \topo)$ be a topological space, $x \in X$, and $\cm(x)$ be a neighbourhood base at $x$. Then $\cm(x)$ is a [[Cofinal Subset|cofinal]] subset of the collection $\cn(x)^o$ of [[Neighbourhood|neighbourhoods]] of $x$. > > *Proof*. Second property in the definition.