This is problem 11 (b) from the first chapter of "Basic Topology" by M.A. Armstrong. The author hasn't had time to develop many theorems or mathematical machinery, so this problem should be able to be solved by just picturing a series of intermediate steps. It goes
Imagine all the spaces shown in Fig. 1.23 to be made of rubber. For each pair of spaces X, Y, convince yourself that X can be continuously deformed into Y.
I'm having trouble with one of the pairs of spaces (the other examples in the problem are unrelated, so I neglected to draw them). The two spaces which I can't seem to think of a continuous deformation for are
The caption for the first picture reads "X = punctured torus", while the caption for the second picture is "Y = Two cylinders glued together over a square patch". I'm trying to think of some intermediate steps in the problem. Working backwards, I can see how each of the cylinders in the second picture could be deformed to spheres with two punctures each, but I'm having trouble seeing how the "handle" on the torus is created.