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The real projective space and the space formed by pinching together the north and south poles of a 2 sphere, S2, are both the image of S2 under an exactly 2 to 1 map. (A map is said to be exactly 2 to 1 if it is onto, continuous, and the inverse of each point is exactly two points.) Clearly these spaces are not homeomorphic; however, from the standpoint of additive mod 2 homology, they are indistinguishable. In fact, from the mod 2 standpoint there is only one type of image.' Precisely,

The 𝑚𝑜𝑑2 homology of the image of an exactly 2 to 1 map from a sphere