Example of topological spaces with continuous bijections that are not homotopy equivalent
Solution 1:
The interval $[0,1)$ admits a continuous bijection to the circle (use $\exp(i 2\pi t)$ as your map). But the two spaces are not homotopy equivalent since the circle is not simply connected.
Edit: If you want to have more than one continuous bijection, just compose my map with a rotation of the circle.