On Apr 27, 5:14 am, Butch Malahide <fred.gal...@gmail.com> wrote: > On Apr 27, 3:04 am, William Elliot <ma...@panix.com> 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.