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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 27, 2013 7:24 AM
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On Apr 27, 5:14 am, Butch Malahide <> wrote:
> On Apr 27, 3:04 am, William Elliot <> wrote:

> > > Correctly: every *nonempty* compact metric space is a continuous image
> > > of the Cantor set. (Likewise, every nonempty separable complete metric
> > > space is a continuous image of the space of irrational numbers.)

> > [. . .]
> > Pointwise continuity of f, though seemingly possible, isn't apparent for
> > the mess of details.  Is there another way to show f is continuous?

> Oops. Now that you mention it, I don't see any reason for f to be
> continuous. I guess I needed to use *open* covers instead of closed
> covers. Does that work? I'm not going to think about it now. Need to
> get some sleep.

> > Is your proof an example of an inverse limit topology?
> No, it's an example of an invalid argument.

No, I was right the first time. There's nothing wrong with the proof I
gave (using *closed* covers), and it really *is* obvious. And now I
*really* need to get some sleep.

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