In algebraic topology, we have the definition of homotopy equivalence of topological spaces:
Topological spaces $X$ and $Y$ are homotopy equivalent if there exist continuous maps $\phi:X \rightarrow Y$ and $\psi:Y\rightarrow X$ such that $\psi \circ \phi \simeq Id_X$ and $\phi \circ \psi \simeq Id_Y$.
If you compare this with the definition of homeomorphism, there are various differences.
However, in terms of finding examples where two spaces are homotopy equivalent and not homeomorphic, I struggle. The only examples I can find are the cases where we involve 1-point sets. For example, a disk in $\mathbb{R}^2$ centred at the origin is not generally homeomorphic to a 1-point set, but it is homotopy equivalent to, say, the origin.
The motivation for this question is that if I were asked whether two spaces were homotopy equivalent – and neither are a 1-point set, could I just check whether they were homeomorphic?
Best Answer
One non-trivial example is that the Möbius strip is homotopy equivalent to a cylinder. These spaces are not homeomorphic, since the boundary of a cylinder (as a manifold) consists of two circles, whereas the boundary of a Möbius strip consists of just one.
To see that they are homotopy equivalent, it suffices to note that they are both homotopic to a circle. One can see this by noting that we can deformation retract either figure "linearly" to a circle down the middle, by pulling the boundaries together.
We can also see that some general constructions hold to find pairs of homotopy equivalent spaces. In particular, if $X$ and $X'$ are homotopy equivalent, as are $Y$ and $Y'$, then so are $X\times X'$ and $Y\times Y'$. In particular, this implies that if a space $Y$ is contractible, then $X\times Y$ is homotopic to $Y$.
One can also find that if the pairs $(X,x)$ and $(X',x')$ are homotopy equivalent and $(Y,y)$ and $(Y,y')$ are homotopy equivalent, then $X\vee Y$ is homotopy equivalent to $X'\vee Y'$, where $\vee$ is the wedge sum. In a sense, once we have a few examples of things being homotopy equivalent, any space we can build out of them becomes homotopy equivalent too - so contractible spaces are a whole class that homotopy equivalence cannot see.