What do we mean mathematically when we say that a coffee mug is topologically equivalent to a donut ? How do we decide which shapes can be transferred into which shapes ? I guess people would say "Homeomorphism " is the key to such questions. So , is it so that if we dan find a homeomorphism from one shape to another then we can say that shape $\ A$ can be transformed into shape$\ B$ . But , again homeomorphism depends on the topology that is chosen . If , I declare each and every subset of the power set of the set of points in both shapes $\ A$ and$\ B$ to be open , then define a mapping such that for every point in $\ A$there lies a unique point in $\ B$ and vice versa , then this mapping can be a homeomorphism .(Correct me , if I am wrong ). And by this logic all the shapes are topologically equivalent . But , I can smell something is fishy around here . Could someone help me in understanding this ? What is an example of a shape that is not topologically equivalent to a sphere? What is an example of a shape that can not be obtained from a square ?