Given any two topological spaces $X$ and $Y$, and given any continuous function $f:X\to Y$, There exists a unique extension $\beta f:\beta X\to \beta Y$.
If it is also given that $f$ is an embedding - does it follow that $\beta f$ must also an embedding?
I think I know how to prove this claim only in the specific case where $f(X)$ is a set from which every continuous function $g:f(X)\to\mathbb{R}$ is extendable to a continuous $\tilde{g}:Y\to\mathbb{R}$.
Plus if you can recommend on a good introduction to the subject of Stone–Čech compactifications, I would be happy for that as well :)
The answer is no. Let $\omega$ be the first countable ordinal, and $\omega_1$ be the first uncountable ordinal. It is (perhaps widely) known that $\beta\omega_1=\omega_1+1$. If $\beta\omega$ is homeomorphic to a subspace $K$ of $\omega_1+1$, then $K$ is compact, and $\omega$ is dense in $K$. Hence $K=\omega+1$. But $\beta\omega$ is way more complicated than $\omega+1$.