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!