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Re: Onto [0,1]
Posted:
Apr 25, 2013 5:18 AM


On Apr 25, 3:56 am, quasi <qu...@null.set> wrote: > If X,Y are subsets of R, and f: X > Y is a monotonic function, > then f is continuous (with respect to the relative topologies on > X and Y inherited from R).
Hmm. Suppose X = [0,1] and Y = [0,1) union {2}. Let f: X > Y be an orderpreserving bijection, e.g., f(x) = x for x in [0,1), f(1) = 2. I don't believe that f is continuous with respect to the relative topologies on X and Y inherited from R. For one thing, X is compact and connected, while Y is neither. (I guess that was two things.) I guess you left out some assumptions.



