Example of a topological space $X$ and a contractible subspace $A$ such that $X$ and $X/A$ aren't homotopy equivalent.

Question

Can someone find an example of a topological space $X$ and a subspace $A\subset X$ such that:

• $A$ is contractible;
• $X$ and $X/A$ have different homotopy type.

I know it exists but I can't find one.
Thanks.

Answer 1

We can take $X=S^1$ and $A=S^1\setminus\{1\}$. Clearly, $A$ is contractible. Now, $X/A$ is the Sierpinski space and $X$ and $X/A$ are not homotopy-equivalent, because any continuous map $X/A\rightarrow X$ is constant (the two points of $X/A$ cannot be separated by neighborhoods and $X$ is hausdorff, so they must map to the same point in $X$) and $X$ is not contractible.