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Is there any homeomorphism between $(X,T^1)$ and $(X,T^2)$ where $T^1$ and $T^2$ are topologies on X such that $T^1$ is a proper subset of $T^2$.

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Suppose $X$ is a set and that you have an bilateral sequence $(\tau_n)_{n\in\mathbb Z}$ of topologies on $X$ such that $\tau_n\subsetneq\tau_{n+1}$ for all $n\in\mathbb Z$.

Let $\mathcal X=X\times\mathbb Z$. Consider

  • the topology $\tau$ which makes $\mathcal X$ the disjoint union of its subspaces $X\times\{n\}$, $n\in\mathbb Z$, each endowed with the topology $\tau_n$,

  • and the topology \tau' which makes $\mathcal X$ the disjoint union of its subspaces $X\times\{n\}$, $n\in\mathbb Z$, each endowed with the topology $\tau_{n+1}$.

The two topologies $\tau$ and \tau' are distinct (indeed, they induce on the subset $X\times\{0\}$ different topologies!) and the choice of the initial data implies that \tau\subsetneq\tau'. But there is an obvious homeomorphism (\mathcal X,\tau)\to(\mathcal X,\tau').

 

To construct the sequence of topologies that I started with you can do the following silly trick. Let $X=\mathbb Z$, and for each $n\in\mathbb Z$ let $\tau_n$ be the topology which has as a basis the set $\beta_n=\{\{k\}:k\leq n\}\cup\{\mathbb Z\}.$

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    Or, for an example where the topologies on $X$ are Hausdorff, let $X = {\mathbb R}$, $\tau_{-\infty}$ the usual topology, and $\tau_n$ the topology generated by $\tau_{-\infty}$ and the singletons $\{k\}$ for integers $k \le n$.2011-10-16