Basis for a Topology

Let $(X,\mathcal{T})$ be a topological space. A basis for the topology $\mathcal{T}$ is a family

with the following property: for every point $x\in X$ and every open set $U\in\mathcal{T}$ satisfying $x\in U$, there exists a set $B\in\mathcal{B}$ such that