Examples for Stone–Čech compactification

Question

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!

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. — Daniel Fischer, May 16 '14 at 14:45
The identity is not a function into a compact Hausdorff space, I think you mean the embedding into $[0,1]$. — Stefan Hamcke, May 16 '14 at 14:46
For the second one, every continuous function $\omega_1\to\mathbb{R}$ is eventually constant. — Daniel Fischer, May 16 '14 at 14:48

score 8 · Accepted Answer · answered May 16 '14 at 14:47

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.

Examples for Stone–Čech compactification

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