If $(X,\tau )$ is a topological space and $P$ is a poset, then $\tau$ may cease to be a topology after forcing with $P$, for example if new subsets of $X$ are added. Nevertheless, in the generic extension, $\tau$ is a basis for a topology, call it ${\tau }^P$, which is finer than $\tau$. One can then ask which properties of $\tau$ are inherited by ${\tau }^P$. In this talk, we will look at what happens to the Lindelöf property under different classes of forcing notions.