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Everyone knows that the Earth's surface is a 2-sphere, or a geoid.

Flat earthers propose that Earth's surface is a disk or some variation of that, and there is lots of discussion on why its not true that the Earth's surface is not a disk and so on. I don't believe in flat earth theory, but this gives me the following idea.

What if we were to try to think of the Earth's surface as the real projective $2$-dimensional space $\mathbb{R}P^2$, that is $\{x\in \mathbb{R}^2: \|x\|\leq 1\}/\sim$ where $\sim$ is the equivalence relation generated by $x\sim -x$ for $x\in S^1 = \{x\in \mathbb{R}^2 : \|x\| = 1\}$? In other words, the opposite points of the circle are identified with each other on the $2$-dimensional disk.

What would be arguments to refute the possibility that earth can be modeled as $\mathbb{R}P^2$ in some way?

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  • $\begingroup$ I find this question hard to understand. Clearly, the earth is neither a 2-sphere or geoid, strictly speaking. It is a rather rough surface. And noone thinks that earth is mathematically 2D. So you need to tell me what exactly it is you want to prove. I.e. "up to what" do you want to prove earth to be a 2-sphere. Homoemorphism? Homotopies? Isotopies? Those are all too weak. $\endgroup$
    – Confuse-ray30
    Commented Feb 7 at 16:54
  • $\begingroup$ Even though I've phrased the objects here mathematically, my question is not mathematical. I don't expect anyone to formally prove or disprove anything. $\endgroup$
    – Rumpelstiltskin
    Commented Feb 7 at 17:07

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Often, we treat the Earth's surface as being embedded in 3-dimensional space. Since the real projective plane cannot be embedded in 3-space, the Earth's surface is clearly not equivalent to it in any relevant way.

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  • $\begingroup$ Related: math.stackexchange.com/questions/2869349/… $\endgroup$
    – Confuse-ray30
    Commented Feb 7 at 16:57
  • $\begingroup$ Also, note that in embedding, the person answering already talks about a map, which in general is not the "identity" inclusion map (whatever that means). So again, refer to my comment under OPs question. $\endgroup$
    – Confuse-ray30
    Commented Feb 7 at 16:58

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