What are the bounded lattices $\mathcal{L}=(Q,\lor,\land,\bot,\top)$ called that are isomorphic to topological spaces? I.e. those such that there exists a topological space $(X,\tau)$ and a bijection $f:Q\to\tau$ such that $f(\top)=X$ and $f(\bot)=\emptyset$ with $f(\bigvee_{i\in I}x_i)=\bigcup_{i\in I}f(x_i)$ and $f(a\land b)=f(a)\cap f(b)$?
It would seem the study of these lattices is essentially just the study of point-set topology, thus I imagine they must have a name? What are they called?
They are called spatial frames. They are, in particular, complete Heyting algebras. Their formal duals are called spatial locales.
Without the "spatial" condition you get a slightly more general class of objects called frames and locales which are more natural and better-behaved in some respects, and their study is sometimes called pointless topology.