5
$\begingroup$

I'm finding it a bit though to "feel" this topic of Stone–Čech compactification. For example, I want to show that $[0,1]$ is not a Stone–Čech compactification of $(0,1]$ and on the other hand $\omega_1^*$ is a Stone–Čech compactification of $\omega_1$.

For the first one I believe it's easier — I need to find a continuous and bounded function on $(0,1]$ that can't be extended to $[0,1]$. Isn't the identity enough? It just seems like I'm missing some point around this subject.

As for the second one — I'm not really sure how to start.

Any help would be appreciated!

$\endgroup$
  • $\begingroup$ The identity can be extended without a problem. What you need is a continuous and bounded function $f$ such that $\lim\limits_{x\to 0^+} f(x)$ does not exist. $\endgroup$ – Daniel Fischer May 16 '14 at 14:45
  • $\begingroup$ The identity is not a function into a compact Hausdorff space, I think you mean the embedding into $[0,1]$. $\endgroup$ – Stefan Hamcke May 16 '14 at 14:46
  • 1
    $\begingroup$ For the second one, every continuous function $\omega_1\to\mathbb{R}$ is eventually constant. $\endgroup$ – Daniel Fischer May 16 '14 at 14:48
6
$\begingroup$

The identity map $(0,1] \to (0,1]$ can certainly be extended to $[0,1]$---by the identity function! You need something a bit more convoluted than that.

What about the function $f(x) = \sin(1/x)$? This is certainly bounded, and it is also continuous on $(0, 1]$, but there is no extension of this to $[0,1]$, since $\lim_{t \to 0} \sin(1/x)$ does not exist.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.