For the second problem. Let an homeomorphism. We denote by . is a homeomorphism. The first set has four connected components whereas the second has two.
Let R denote the real line and R^2 the real plane.
How do I show that R^2 and R^2 - {(0,0)} are not homeomorphic? I don't even know where to start since I cannot think of a topological property that is in one but not in the other. Can anyone give me a clue on what that property is?
A similar problem is this:
Consider S = the union of the x and y- axes together with the subspace topology inherited from R^2. Show that S and R are not homeomorphic.
Thanks in advance!