Give an example of topological spaces $X$,$Y$ with bijective continuous maps between them(so 2 maps in total!), such that $X$ is not homotopy equivalent to $Y$.Be aware of the theorem stating that bijective continuous map from compact space to Hausdorff space must be a homeomorphism. There are also examples such that $f$ and $g$ don't provide a homotopy equivalence, but still the spaces are.
P.S. The problem has already been asked 2 years ago, but there is no satisfactory answer: Example of topological spaces with continuous bijections that are not homotopy equivalent