Judging a restricted projection is a closed map Suppose that $p_1:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is the projection onto the first index. Let $Y=\mathbb{R}\times\{0\}\cup[0,+\infty)\times\mathbb{R}\subset\mathbb{R}^2$, $f=p_1\mid_Y$($p_i$ restricted on $Y$). Now I need a counterexample to show that $f$ is not closed, i.e., there exists a closed set $F\subset Y$ such that $f(F)=p_1(F)$ is not closed in $\mathbb{R}$.
I've figured out that $f$ is a quotient map and not open, but really stuck at this point to find a closed set with not-closed image. Any help would be aprreciated! Thanks!

$F = \{(x,\frac1x\mid x >0\}$ is closed in $Y$, but $f[F] = (0,+\infty)$ is not closed in $\Bbb R$.
