I begin by writing the definition below that tries to capture what a continuous family /path of manifolds is. The underlying motivation behind the definition is that the transition maps should be continuous in the time parameter.
Defintion: We say a family of $n$ dimensional topological manifolds $\{M_t\}_{t\in [0,1]}$ is continuous iff there is an index set $J$ and atlases $\{\phi_j^t:M_t\rightarrow \mathbb{R}^n\}$ on $M_t$ such that for any $j_1,j_2\in J$, we have:
- The set $S$ defined to be $\{(p,t):\phi_{j_2}^t([\phi_{j_2}^t]^{-1}(p))$ makes sense $\}$ is open in $\mathbb{R}^n\times [0,1]$.
- The map $(p,t)\mapsto \phi_{j_2}^t([\phi_{j_2}^t]^{-1}(p))$ from $S$ to $\mathbb{R}^n\times [0,1]$ is continuous.
Question: If $\{M_t\}_{t\in [0,1]}$ is a continuous family of $n$ dimensional topological manifolds such that $M_t$ is compact for every $t$, must $M_0$ and $M_1$ be homeomorphic ? What about the smooth case ? Will we have that $M_0,M_1$ are diffeomorphic?
