Are there nonisomorphic $T_0$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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1$\begingroup$ For the specialization order? You mean like the usual topology on $\mathbb{R}$ and the discrete topology on $\mathbb{R}$? 😉 $\endgroup$– GroTsenCommented May 5 at 22:29

3$\begingroup$ I thought the OP is putting the natural $\subseteq$ order on $\tau_j$. $\endgroup$– Christian RemlingCommented May 5 at 22:31

1$\begingroup$ Won't any nonsober T0 space and it's soberification be an example? $\endgroup$– Benjamin SteinbergCommented May 6 at 20:18
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1 Answer
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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 $\mathbb Q$.