I am independently working through the topology book called, "Introduction to Topology: Pure and Applied." I am currently in a chapter regarding manifolds. They attempt to show that a connected sum of a Torus and the Projective plane (T#P) is homeomorphic to the connected sum of a Klein Bottle and a Projective Plane (K#P). I can go through the detail if someone would like but the conclusion is, "Therefore, T#P is topologically equivalent to K#P. I am not having any trouble with the proof. I am having trouble with the conclusion. How can it be that through multiple quotient maps we can end up with a space that is topologically equivalent to what we started with?