Is there a Hausdorff space $(X,\tau)$ such that $|X|>1$, and whenever $U, V\in \tau$ with $U\cong V$ (with the subspace topologies) we have $U=V$?

**Note.** There is an infinite $T_0$-space with this property, namely $(\omega, \omega+1)$. I don't know about $T_1$-spaces, though.