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Are there non-isomorphic $T_0$-spaces $(X_i, \tau_i)$ for $i = 1,2$ such that $\tau_1 \cong \tau_2$ when considered as partially ordered sets?

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    $\begingroup$ For the specialization order? You mean like the usual topology on $\mathbb{R}$ and the discrete topology on $\mathbb{R}$? 😉 $\endgroup$
    – Gro-Tsen
    Commented May 5 at 22:29
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    $\begingroup$ I thought the OP is putting the natural $\subseteq$ order on $\tau_j$. $\endgroup$ Commented May 5 at 22:31
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    $\begingroup$ Won't any nonsober T0 space and it's soberification be an example? $\endgroup$ Commented May 6 at 20:18

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Yes. For the first space, topologize $\mathbb R$ by taking the intervals $(-\infty,r)$, $r\in \mathbb R$, for the nonempty proper open subsets. For the second space, take the subspace with underlying set $\mathbb Q$.

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