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Topic: Onto [0,1]
Replies: 40   Last Post: Apr 29, 2013 10:16 PM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Onto [0,1]
Posted: Apr 25, 2013 5:18 AM
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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
order-preserving 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.

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