Klein bottle as the union of two Mobius bands
I am trying to calculate the homology groups of the Klein bottle. I want to use the Mayer-Vietoris sequence with the Klein bottle decomposed as the union of two Mobius bands (A and B which are homotopic equivalent to circles), now AUB is the Klein bottle, but I don't understand how according to
AnB is also homotopic equivalent to a circle, I would think that the intersection is the disjoint union of two Mobius bands, so it is homotopic equivalent to the disjoint union of 2 circles, hence should be . But according to what's written in wikipedia, is just . Why?
Am I thinking this all wrong?