Consider a topological space $(X,\tau)$. I would like to know whether the set $\tau$ of open sets can be given its own topology in a canonical way. In that respect I'm asking for the same kind of thing as this previous question. However I'd also like to specify a property that this canonical topology should have, which the other question doesn't do, so I hope this won't be considered a duplicate.
Specifically, let $(X,\tau)$ and $(Y,\sigma)$ be topological spaces, and consider a continuous function $f\colon X\to Y$. We can define the pullback map as a function $f^*\colon \sigma\to\tau$ that maps open subsets of $Y$ to their preimages under $f$, which are open subsets of $X$.
I would like to know whether $f^*$ can itself be seen as a continuous function. That would mean defining the open subsets of $\sigma$ and $\tau$. I'm wondering whether these can be defined in a canonical way, such that every continuous map between any pair of topological spaces has a continuous pullback.
Of course this would be the case if we just defined the topology on $\tau$ to be the discrete topology for every topological space $(X,\tau)$, but I'm wondering if it's possible to do it in a way where they can be coarser than that.