It is a well-known fact, that given a topological manifold there can exists different smooth structure (or none at all). I'm interested in the converse question: Given a smooth structure (in the sense of a complete smooth atlas), are the multiple topologies with this smooth structure possible ?
More concretely formulated: Does there exist a non-empty set $M$
with two different topologies $\tau_{1}$ and $\tau_{2}$ on it, such
that in the intersection of the collection $\mathcal{C}_{1}$ of all
(topological) charts on $\tau_{1}$ with the collection $\mathcal{C}_{2}$
of all (topological) charts on $\tau_{2}$ there is contained smooth
atlas ?
(Such a smooth atlas gives then rise to a unique, complete
smooth atlas, which would be a smooth structure for two different
topologies on $M$.)
Topology is uniquely determined by the smooth structure: a set $A\subset M$ is open if and only if the images of all of its intersections with the charts in the smooth atlas in the respective local coordinates are open.