Existence of Homeomorphism? 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 $T^2$.
 A: 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 $$\beta_n=\{\{k\}:k\leq n\}\cup\{\mathbb Z\}.$$
