prove that if we consider this shape in the picture below with the equivalency relation that : a & b are in one class if they are antipoles in inner or outer circles, then the induced quotient space is homeomorphic with the klein bottle.
please help me how to prove this,i've no idea.
and if anyone knows some book with shapes like this and with this kind of approach to topology please introduce it.
thank you
It's homeomorphic to the connected sum of two projective planes (on $\mathbb{R}$), hence it is the Klein bottle (up to iso, as usual).