TITLE: Show that the Stone–Čech compactification $\beta \mathbb{Z}$ is not metrizable.
QUESTION [10 upvotes]: Show that the Stone–Čech compactification $\beta \mathbb{Z}$ is not metrizable (here $\mathbb{Z}$ denotes the set of integer numbers in discrete topology).
Definition. Let $X$ be a completely regular space.  We say $\beta (X)$ is a *Stone–Čech compactification * of $X$ if it is a compactification of $X$ such that any continuous map $f:X\rightarrow C$ of $X$ into a compact Hausdorff space $C$ extends uniquely to a continuous map $g:\beta (X)\rightarrow C$. 
We also have a theorem that states if $X$ is metrizable, then $X$ is first countable.  So, if we show that $\beta (\mathbb{Z}) $ is not first countable, then we have our conclusion.  
I've looked online for proofs of this fact and can't seem to find any.

REPLY [4 votes]: A compact metric space is separable.  
Take an uncountable family of subsets of $\mathbb Z$, any two of which have finite intersection.  Then their closures intersected with the "remainder" $\beta \mathbb Z \setminus \mathbb Z$ give you an uncountable family of pairwise disjoint open sets in the compact space $\beta \mathbb Z \setminus \mathbb Z$.  So $\beta \mathbb Z \setminus \mathbb Z$ is not metrizable, and therefore of course neither is $\beta \mathbb Z$.