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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 $T$.

Torus from a square

What am I doing wrong?

Thanks for your time.

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  • $\begingroup$ It might be easier to view it this way: The torus and square are just two different topologies on the set $[0,1]^2$, and $p$ is the identity map. $\endgroup$ – Henricus V. Jan 21 '18 at 12:52
  • $\begingroup$ @HenricusV. Not exactly. The torus is a topology on $[0,1)^2$ in this interpretation. $\endgroup$ – Arthur Jan 21 '18 at 13:13
  • $\begingroup$ @Arthur If you glue the edge so that the topology is not Hausdorff, this viewpoint is still valid. $\endgroup$ – Henricus V. Jan 21 '18 at 13:14
  • $\begingroup$ @HenricusV. If you mean analogously to the number line with two origins, only one dimension up, sure. It would make $p$ a bijection, but I wouldn't call the result a torus. $\endgroup$ – Arthur Jan 21 '18 at 13:17
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Hint: take a small open neighborhood around $(0,0)$.

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  • $\begingroup$ A small neighborhood like $[0,\epsilon) \times [0,\epsilon)$? How I prove that the projection image is not open? $\endgroup$ – Relure Jan 21 '18 at 12:04
  • $\begingroup$ I'm sorry Arthur but I think that my problem is understanding how $p$ send the points of the square to the torus $\endgroup$ – Relure Jan 21 '18 at 12:05
  • $\begingroup$ @Relure $p$ can be viewed as the identity map. $\endgroup$ – Henricus V. Jan 21 '18 at 12:51
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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.

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