Quotient topology on $\mathbb{R}^2$ such that the result is homeomorphic to a sphere, to a rectangle Define a quotient topology on $\mathbb{R}^2$ such that the result is homeomorphic to a sphere and second to a closed rectangle with all the interior points included.
For the first one: I know that $\mathbb{R}^2$ is homeomorphic to a sphere (without the nort pole). But I don't know how to get 'some extra point' with defining a quotient topology. I would rather say you get 'less points' by identifying some points. 
For the second: You have to identify all points outside some rectangle? Then you get a rectangle and one point that identifies all points outside the rectangle? Is that the result?
Thanks in advance 

Your second solution is actually very close to the answer for the first problem: identify all points outside some open disc to each other: the open disc is homeomorphic to a sphere minus the north pole, and the identification makes all the rest of the plane into said north pole.
As for the rectangle, draw the rectangle in the plane, and identify each point outside the rectangle with the point on the boundary of the rectangle that it is closest to.
