So, though I am aware that it is conventional to use Alexander's duality theorem to prove that no closed, non-orientable surface $S$ can be embedded into $\mathbb{R}^{3}$; I was hoping perhaps to find a more elementary version of this proof by the consideration of the Euler characteristic of $S$ (as is done for the answer to this question along with Alexander's duality). I wish to prove this theorem for the sake of proving the inscribed rectangle conjecture.

The argument I am considering goes like this (with reference in particular to the closed surface formed by pasting the boundary of a disk $D$ to that of a Mobius loop $M$, whose boundary is mapped to a jordan curve. We assume that this mapping is a homeomorphism, in order to cause a contradiction): To begin with, as $\chi(M) = 0$ and $\chi(D) = 1$. Then, we know that $\chi(M \cup D) = \chi(M) + \chi(D) -\chi(M \cap D) $ As the boundary of a Disk (i.e. the space $M \cap D$) is a loop (with Euler characteristic $0$),
$$\chi(M \cup D) = \chi(M) + \chi(D) -\chi(M \cap D) = 0 +1-0 =1$$ Which is odd.
It is here that I get stuck, as I do not know how to use this to show somehow that this leads to a contradiction to our assumption that the function mapping the Mobius loop into $\mathbb{R}^{3}$ (where the boundary is a jordan curve) leads to a contradiction.
Is there any theorem that says, for example, that any closed surface in $\mathbb{R}^{3}$ must have an Euler characteristic that is an even number?

If this proof is insufficient, then how else may I prove it in a manner that is at-least semi rigourous whilst still being accessible to an undergraduate who has just started to learn topology?


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