Bounding the cardinality of the Stone-Cech compactification.

Question

Let $X$ be an infinite topological space with Stone-Cech compactification $\beta X$. I want to bound the cardinality as a function of the cardinality of $|X|$: $$|\beta X| \le f(|X|).$$

Attempt: We can realise $\beta X$ as the characters of $C_b(X)$, i.e. there is an embedding $$\beta_X \hookrightarrow \mathbb{C}^{C_b(X)}$$ and thus $$|\beta X| \le (2^{\aleph_0})^{|C_b(X)|} \le (2^{\aleph_0})^{|\mathbb{C}^X|} \le (2^{\aleph_0})^{\left((2^{\aleph_0})^{|X|}\right)}= 2^{2^{|X|}}$$

This is of course a bad estimate. How much better can we do?

Answer 1

$\beta X| \le 2^{2^{|X|}}$ is a sharp bound witnessed e.g. by the discrete space....Yes, it can be a huge space in general.