In the first of these lectures (http://www.mpim-bonn.mpg.de/node/4436) given by M. Freedman he says that there exists (compact metric) spaces $X$ and $Y$ such that $X\times S^{1}$ is homeomorphic to $Y\times S^{1}$ but $X\times \mathbb{R}$ is not homeomorphic to $Y\times\mathbb{R}$.
Does anyone know an example of such spaces?