$\mathbf {The \ Problem \ is}:$ I am trying to identifying the quotient space $\frac {I×I}{I×\{0,1\}}$ (where $I =[0,1]$) with some known spaces.
$\mathbf {My \ approach}:$ I could only try that if we define $f : I×I \to I×I$ by $f(a,b) = (ab(1-b),e^{2πib})$; then I think the fibre becomes $I×\{0,1\}.$
But, I can't approach further . A small hint is warmly appreciated. Thanks in advance.
As far as i can tell, the description $\frac{I\times I}{I\times \{0,1\}}$ is the best description in terms of "known" spaces, but correct me if someone comes up with a better description. I have made a drawing of the quotient space if that helps you.
Another way to think of the space, is to describe it as a quotient of the cylinder $I\times S^1$. Namely, it can also be written as $\frac{I\times S^1}{(t,1)\sim(t',1)}$.