spaces homotopic equivalent to $\mathbb R \backslash \mathbb Q$ I am looking for spaces who are homotopic equivalent to the irrational numbers!
Can you help me, if you have seen some examples or references?
 A: How about a homeomorphism? The space $\mathbb{Q}^2\cap S^1\subset\mathbb{R}^2$ is in one to one correspondence with the rational numbers $\mathbb{Q}$ via stereographic projection. Let $N$ be the north pole $(0,1)\in S^1$. As the stereogrphic projection map $\phi\colon S^1\setminus\{N\}\rightarrow\mathbb{R}$ is a homeomorphism, it follows that we may restrict $\phi$ to the domain $A=S^1\setminus\mathbb{Q}^2$ giving us a homeomorphism between $A$ and the image of $\phi|_A$ which is the usual set of irrational numbers as a subspace of $\mathbb{R}$.
Set, $\psi\colon A\rightarrow \mathbb{R}\setminus\mathbb{Q}$ to be equal to $\phi|_A$, then $\psi$ is a homeomorphism. (This actually also shows that $\phi^{-1}|_{\mathbb{R}\setminus\mathbb{Q}}$ is a compactification of the irrationals in to the circle as its image is dense in $S^1$.)
Spaces which are homotopy equivalent but not homeomorphic may be harder to find (except for obvious example such as taking products with contractible spaces or more generally fiber bundles with contractible fibers). Certainly such a space would be difficult to describe as it must still be uncountable and with uncountably many path components.
