Date: Apr 25, 2013 5:54 AM
Author: quasi
Subject: Re: Onto [0,1]
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