For each $\displaystyle n\in\mathbb{N}$ give examples of subspaces of $\displaystyle \mathbb{R}^n$, which are homotopy equivalent but not homeomorphic to each other.
Give reasons for your answer.
Any help would be great. Thanks
Okay, so if I take an open ball $\displaystyle B_r({\bf{x}})$ in $\displaystyle \mathbb{R}^n$ this is clearly not homeomorphic to a point $\displaystyle {\bf{y}}$ in $\displaystyle \mathbb{R}^n$ as any continous map is not bijective.
So take a map $\displaystyle f:B_r({\bf{x}})\rightarrow {\bf{y}}$ given by $\displaystyle f({\bf{v}})={\bf{y}}$ for $\displaystyle {\bf{v}}\in B_r({\bf{x}})$ and then take another map $\displaystyle g:{\bf{y}}\rightarrow B_r({\bf{x}})$ given by $\displaystyle g({\bf{y}})={\bf{x}}$
How do I should that $\displaystyle g\circ f$ and $\displaystyle f\circ g$ are homotopic to the respective identities?