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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.

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 $U=V$?

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.

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Hausdorff space such that the open sets are pairwise non-isomorphic

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 $U=V$?