Topological equivalence of shapes,again 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 ?
Thanks ahead!
 A: If $X$ and $Y$ are topological spaces of the same cardinality, and one chooses $\mathcal{P}(X)$ and $\mathcal{P}(Y)$ as topologies for $X$ and $Y$ respectively, then indeed every bijection $f: X \rightarrow Y$ is a homeomorphism. So in that sense, choosing this topology for all surfaces makes all surfaces homeomorphic(since in general the cardinality of a surface is the cardinality of $\mathbb{R}$). However, when people talk about the torus for example, they mean the torus with the standard subspace topology inherited by $\mathbb{R}^3$. 
So when people say the torus is not homeomorphic to the sphere they mean that there is no bijective function $\varphi: S^2 \rightarrow  T $ with the property that $U \subset S^2$ is open with respect to the subspace toplogy of $S^2$ in $\mathbb{R}^3$ if and only if $\varphi(U) \subset T$ is open with respect to the subspace topology of $T$ in $\mathbb{R}^3$. There is not only a bijection between the two topological spaces, but the bijection is compatible with the topological structure. Just think of a topology as a way of saying in what way the points of the set you start with are glued together. Then there is a homeomorphism $\varphi: X \rightarrow Y$ between two spaces $X$ and $Y$ if you can assign to each point $ x \in X$ a point $\varphi(x) \in Y$ such that sets of points are glued together in $X$ if and only their images are glued together in $Y$.
I hope this helps.
