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Note: It humble request to everyone please don't downvote my post .Im not genius or intelligent.Im new to the Algebraic Topology.Try to understand my level of understanding.I feel discourage whenever i get downvote

I have some confusion in this answer

Question : Is the Borsuk-Ulam theorem valid for a torus? In other words, for any map $f: S^1 \times S^1 \rightarrow \mathbb{R^2}$ there is a point $(x,y) \in S^1 \times S^1$ which $f(x,y)=f(-x,-y)?$

Here is the outlined of the given answer

No. With the usual torus embedded in $\mathbb{R}^3$, lying on the $OXY$ plane, one has a natural projection onto that plane, $p:S^1×S^1\to \mathbb{R}^2$, which is continuous.

Two points on the torus have the same image if they are one above the other, in the same vertical line. In particular, they are in the same meridian of the torus, i.e. they have the same first coordinate. So, if $p(a,b) = p(c,d)$, $a = c$. This implies that the Borsuk-Ulam theorem fails on the torus because if $x=-x$, and then $x=0\notin S^1$.

My confusion : Im not getting the statement "they are in the same meridian of the torus, i.e. they have the same first coordinate".

My attempt :If they are in the same meridian of the torus then $p(\theta_1, \theta_2)=p(\theta_1 +\frac{1}{2} , \theta_2 +\frac{1}{2})$

Now take $a=\theta_1$ and $c=\theta_1 +\frac{1}{2} $

$\theta_1=\theta_1 +\frac{1}{2} \implies \theta_1-\theta_1 =\frac{1}{2} \neq 0 \implies a \ne c$

and $\theta_1=\theta_1 +\frac{1}{2} \implies \frac{1}{2} \in S^1$

My question : why they have the same first coordinate in torus ?

jasmine
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    You are assuming that the downvotes on your previous questions have something to do with your intelligence. I don't think that is accurate. When you place your cursor over the down arrow it says "This question does not show any research effort; it is unclear or not useful." Having seen some of your questions, I think its reasonable to assume that some users agreed with that statement. I recommend you visit the Constructive Feedback chatroom to discuss how you can improve your future questions. – Michael Albanese Aug 28 '21 at 21:02

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Imagine projecting this torus on the horizontal plane:

enter image description here

Points that are mapped on the same point on the plane lie one above another, and so they have to both be on the same little circle (like the red one). E.g. here the upper point of the red circle will have the same image as its lower point. Points of other small circles, however, will have different projections on the plane.

The first coordinate of a point on a torus is specifying exactly the meridian (small circle) on which the point lies.

Арсений Кряжев
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