Definition: $\beta X$ is the Stone-Čech compactification of $X$.
Theorem A: If $K$ is a compact Hausdorff space and $f\colon X \to K$ is
continuous, there is a continuous $F: \beta X \to K$ such that $F \circ e = f$, where $e\colon X\to\beta X$ is an embedding into a compact Hausdorff space.
Show that $\left|\beta\mathbb{N}\right|\geq\left|\beta\mathbb{Q}\right|$.
Let $f\colon\mathbb{N}\to\mathbb{Q}$ be a bijection. As any function from the discrete topology is continuous ($\mathbb{N}$ with the relative topology from $\mathbb{R}_\text{std.}$ is the discrete topology), we can enlarge the range. Therefore, we can enlarge the range to $\beta\mathbb{Q}$, which is a compact Hausdorff space, so that $f\colon\mathbb{N}\to\beta\mathbb{Q}$ is continous. By Theorem A we can extend $f$ uniquely to to a continuous function
$\beta f\colon\beta\mathbb{N}\to\beta\mathbb{Q}$.
I need to show that
$\beta f\colon\beta\mathbb{N}\to\beta\mathbb{Q}$ is surjective, i.e. $\beta f[\beta\mathbb{N}]=\beta\mathbb{Q}$. As for any mapping $\beta f[\beta\mathbb{N}]\subseteq\beta\mathbb{Q}$, it's enough to show that $\beta f[\beta\mathbb{N}]\supseteq\beta\mathbb{Q}$.
Best Answer
This proof looks correct, you just need a little push.
We will use the following result:
Lemma: If $X$ is a topological space, $D$ is a dense subset of $X$ and $A$ is a closed subset of $X$ such that $D\subseteq A$, then we have the equality $A=X$.
Since $\beta f$ extends the bijective function $f$, we get the relations $$\mathbb{Q}=f[\mathbb{N}] \subseteq \beta f[\beta\mathbb{N}].$$
Now, since $\beta f$ is continuous, this tells us that $\beta f[\beta\mathbb{N}]$ is a compact space (therefore, a closed subset of $\beta\mathbb{Q}$) that contains a dense subset of $\beta\mathbb{Q}$. So, if we let $X=\beta\mathbb{Q}$, $D=\mathbb{Q}$ and $A=\beta f[\beta\mathbb{N}]$, then we can apply the lemma to get that $\beta f[\beta\mathbb{N}]$ must be exactly $\beta\mathbb{Q}$; in other words, $\beta f$ is surjective.