Exercise 15, § 1

Let $X$ be the subspace of $[0,1]\times [0,1]$ formed by the pairs $(x, y)$ such that $x\in \{0,1\}$, or $y\in \{0,1\}$, or $x= 0\not$ and $1/x\in \mathbf{N}$. Let $A =\{0\} \times [0,1]$. Prove that the canonical mapping of X onto $X/A$ is not a homeotopy, though A is contractible.