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Protective Planes Spaces

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4. a) Show that the projective space with a disk removed (more precisely, the image of a disk on S2 under the natural projection pi : S2 -> P2 removed) is a M¨obius band.
b) Let M be the connected sum of two copies of the projective space formed by removing a disk from each copy of P2 and gluing the resulting surfaces together along the boundaries of the holes. Which standard surface do you obtain?

This ugly paint file is what I believe is 1, if I am correct, then part b is just 2 mobius bands added together to make one big mobius band.

Klein Bottle + http://en.wikipedia.org/wiki/Klein_bottle

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The expert examines protective plane spaces.

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It seems you figured (a) yourself, and (b) is a Klein bottle, as you can see from carefully checking how you would identify the ...

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