Continuity in the Stone–Čech compactification Let be $(X,  \tau)$ a compact and Hausdorff topological space, and let be $(x_n)_n \subseteq X$, a sequence of elements of X. I need to prove that the function $g:  \beta\mathbb{N}\rightarrow{}X$ defined by $g(F)=\displaystyle\lim_{n \to{}F}{} x_n$ is continuous.
I have $ \beta\mathbb{N}=\left\{{F\subseteq \mathbb{N}: F \text{ is an ultrafilter on } \mathbb{N}}\right\}$ and for $A\subseteq\mathbb{N}$, we defined $[A]:=\left\{{P \in{}  \beta\mathbb{N}: A \in{} P}\right\}\subseteq\beta\mathbb{N}$, and we know that $ \digamma:=\left\{{[A]: A \subseteq \mathbb{N}}\right\}\subseteq \mathbb{P}(\beta\ \mathbb{N})$ is a basis for a topology on $\beta\mathbb{N}$.
Also, I know that for any $F \in{}  \beta\mathbb{N}$, $\lim_{n \to{}F}{} x_n$ exist and it's unique because X is a compact and Hausdorff topological space. Then $g$ is well defined.
 A: Let $U$ be an open set in $X$. We want to show $g^{-1}[U]$ is open. For each $F\in g^{-1}[U]$, it suffices to find a neighborhood $\widehat{U}$ of $F$ in $\beta\mathbb{N}$ such that $\widehat{U}\subseteq g^{-1}[U]$.
In $X$, use compactness to find a pair of disjoint open sets $U'$ and $V$ such that  $g(F) \in U'\subseteq U$ and $(X\setminus U)\subseteq V$ (compact Hausdorff spaces are $T_4$: any two disjoint closed sets can be separated by disjoint open sets). Let $A = \{n\in \mathbb{N}\mid x_n\in U'\}$, and let $\widehat{U} = [A]$. Since $U'$ is an open neighborhood of $g(F)$, $A\in F$, so $F\in \widehat{U}$. It remains to show that $g[\widehat{U}]\subseteq U$.
Indeed, suppose for contradiction that $F'\in \widehat{U}$ (so $A\in F'$), but $g(F')\notin U$. Then $g(F') \in V$. Let $B = \{n\in \mathbb{N}\mid x_n\in V\}$. Then also $B\in F'$, but since $U'$ and $V$ are disjoint, $A$ and $B$ are disjoint, so $\emptyset\in F'$, contradiction.
By the way, this exercise is just the universal property of $\beta\mathbb{N}$.
