Let $\displaystyle \pi: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, $\displaystyle (x, y) \mapsto \pi(x, y)=x$.

Let $\displaystyle A=(\mathbb{R}^{\geq 0} \times \mathbb{R}) \cup (\mathbb{R} \times \{0\})$.

Prove that $\displaystyle f=\pi|_A$ is a quotient map and that $\displaystyle f$ is neither an open map or a closed map.

[$\displaystyle \mathbb{R}^{\geq 0}=\{ x\in \mathbb{R}: x\geq 0\}$]

[This is Munkres §22.3]