Let $(X,\tau)$ be a Hausdorff space and let ${\cal D}$ denote the collection of dense subsets of $(X,\tau)$. Is it possible that there is another Hausdorff topology $\tau\_1 \neq \tau$ on $X$ such that the collection of dense subsets of the space $(X,\tau\_1)$ also equals ${\cal D}$?



Collection of dense subsets as a "fingerprint" for Hausdorff topologies?
The standard topology and [the lower limit topology](https://en.wikipedia.org/wiki/Lower_limit_topology) on $\mathbb{R}$ have the same dense subsets. They are two different topologies(even up to homeomorphism) on the real line.


So the next question could be "Is the collection of **open** dense subsets a “fingerprint” for Hausdorff topologies?


