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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: Onto [0,1]
Posted: Apr 25, 2013 5:54 AM

Butch Malahide wrote:
>quasi 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.

Yes, I realized as much myself.

In fact, the immediate counterexample that ocurred to me was
very similar to the one you gave above.

All in all, my claim was pretty silly.

quasi

Date Subject Author
4/21/13 William Elliot
4/21/13 Butch Malahide
4/21/13 William Elliot
4/22/13 David C. Ullrich
4/22/13 Butch Malahide
4/22/13 David C. Ullrich
4/21/13 Bacle H
4/21/13 William Elliot
4/21/13 Bacle H
4/21/13 Bacle H
4/21/13 Butch Malahide
4/21/13 Bacle H
4/25/13 William Elliot
4/25/13 Butch Malahide
4/27/13 William Elliot
4/27/13 Butch Malahide
4/27/13 Butch Malahide
4/29/13 Butch Malahide
4/29/13 William Elliot
4/25/13 William Elliot
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 David C. Ullrich
4/21/13
4/25/13 quasi
4/25/13 quasi
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 Tanu R.
4/25/13 quasi
4/25/13 Butch Malahide
4/25/13 quasi
4/26/13 Butch Malahide
4/26/13 quasi