Prove that the projection from a square to a torus is not open

Question

I'm stuck with this exercises of my notes:

Let's consider a square $Q=[0,1]\times[0,1]\subset \mathbb{R^2}$, with the usual topology. Let's $p: Q \to T$ be the canonical projection on the torus $T$. Prove that $p$ is not open.

I think that maybe I'm not correct about my way of thinking on the "canonical projection", because if I think that the torus is constructed identifying the sides of that square and doing continuously deformations to it (see the image below), every open set on $[0,1]\times[0,1]$ leads me to an open set on $T$.

What am I doing wrong?

Thanks for your time.

Answer 1

Hint: take a small open neighborhood around $(0,0)$.

Answer 2

You're doing nothing wrong, the map does identify the sides like in your picture, except a small neighbourhood of a boundary point in the square gets mapped to only one "half" of an open set.