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I ve been told that the real projective plane of dimension two can be expresses as the union of a disk and a mobius strip. The only way that this makes sense to me is that if an annulus with with antipodal points on the outer circle identifies gives a mobius strip. But I can't see if this is true can anyone please explain
Thanks in advance

TheGeometer
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  • Yes, attaching a disk to the moebius strip along the boundary circle of the strip gives something homotopy equivalent to $\Bbb RP^2$. This follows from deformation retracting the moebius strip to it's boundary circle, which results the space $D^2 \cup_f S^1$ with $f$ being a degree $2$ map. – Balarka Sen May 16 '15 at 13:33

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You can see how they are the same by following these drawings:

enter image description here

Dario
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  • amazing thank you very much – TheGeometer May 16 '15 at 11:49
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    The Projective plane is the quotient of the 2 sphere by the antipodal map. A strip around the equator maps to a Mobious band, The two polar caps are identified into a single disk and are attached to the Mobius band along its boundary circle. – Joe S May 16 '15 at 12:27
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    Shouldn't the green arrow in the top center diagram be pointing left? – jcai Sep 22 '17 at 22:20
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    @Arcinde: you're right, I edited the picture. Thanks. – Dario Sep 23 '17 at 16:52