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  1. P

    R^2 and R^2 - {(0,0)} are not homeomorphic

    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...
  2. P

    Show that there exists no sequence of functions satisfying the following

    I found this interesting exercise on a topology book I'm reading, but I don't have a clue what to do. Show that there is no sequence {g_n} of continuous functions from R to R such that the sequence {(g_n)(x)} is bounded iff x is rational (where R = set of real numbers).