I thimk this can be done by idemtifying points on the boundary but I am not sure how to show this
Any ideas? E.g. By drawing nets..
2025-01-13 18:23:33.1736792613
Making a Klein bottle from 2 Möbius bands
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The Klein bottle is defined to be the quotient space $[0, 1]^2/\sim$ where $\sim$ identifies the top and bottom by reversing orientation and the two sides by preserving orientation.
Consider the image of $[1/3, 2/3] \times [0, 1]$ and $([0, 1/3] \cup [2/3, 1]) \times [0, 1]$ under the quotient map $[0, 1]^2 \to [0, 1]^2/\sim$. These are Moebius strips, hence you obtain your decompositions.