Stone–Čech compactification Cardinality

Question

How I can prove $|\beta N| \geq | \beta Q|$. I need some hints. I find some help but I still face problem (consider any 1-1 map)

Answer 1

Let $f: \mathbb{N} \to \mathbb{Q}$ be any surjection (or bijection if you like; the rationals are countable). Any function on the discrete space $\mathbb{N}$ is continuous. Then we can extend the codomain and see $f$ as a map $\mathbb{N} \to \beta\mathbb{Q}$ with compact Hausdorff codomain. We can extend $f$ uniquely to a continuous $\beta f: \beta \mathbb{N} \to \beta\mathbb{Q}$. Now $\beta(f)$ is surjective as its image is a (compact hence) closed set that contains the dense set $\mathbb{Q}$, so equals the whole codomain. And (basic set theory) if $f: X \to Y$ is surjective then $|Y| \le |X|$ or equivalently $|X| \ge |Y|$.

Note that this works for any separable (Tychonoff) space $X$: $|\beta X| \le |\beta \mathbb{N}|$, using a bijection of the natural numbers with a countable dense subset of $X$ in the same way. So $|\beta \mathbb{R}| \le |\beta \mathbb{N}|$ as well, e.g.