The Klein bottle K is obtained from the unit square [0, 1] x [0, 1] by making the identifications (z,0) = (1-z, 1) and (0, y) = (1, y) for z, y = [0, 1]. Is it true that any loop in K that has no self intersections cuts K into two components? If not, then exhibit a loop that does not cut K into two components.

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