I'm wondering if the following can be true:
Let Y be a second countable space and
$\pi_2:Y \times \mathbb{R}\rightarrow\mathbb{R}$ ($\mathbb{R}$ with its usual topology and
$\pi_2$ the projection onto the second factor)
be a closed map: do these assumptions imply
that Y is compact? (There is no assumption $T_0$, $T_1$ or $T_2$ on $Y$.)

thank you in advance.