If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose models in $\mathbf{Set}$ are precisely the suplattices. Write $\mathbf{TopSupLat}$ for the category of models of $\mathsf{SupLat}$ in $\mathbf{Top}.$ There is a forgetful functor $$U : \mathbf{TopSupLat} \rightarrow \mathbf{Top}.$$
Does it have a left-adjoint $F$? If so, I'd like to see an explicit description of $F(X)$ for an arbitrary topological space $X$.