Let $X$ be a compact, Hausdorff and totally disconnected space. By giving the field of two elements $\mathbb F_2$ the discrete topology, let $A$ denote the set of continuous maps from $X$ to $\mathbb F_2$. Clearly $A$ is a boolean ring. Then $\operatorname{Spec}A$ is again a compact Hausdorff and totally disconnected space.
For boolean ring, we know that for any $p\in \operatorname{Spec}A$, $A/p\simeq \mathbb F_2$. Thus we can write $\operatorname{Spec}A=\operatorname{Hom}(A,\mathbb F_2)$.
We can define a map:$\theta:X\rightarrow \operatorname{Spec}A$ by $\theta(x)(a):=a(x)$.
My goal is to show $\theta$ is a homeomorphism. I can show this map is continuous and injective. But I have difficulty on showing surjectivity. Could you provide some hints (or solutions, references) for me? Thanks a lot!
Best Answer
Let $\phi \in \textrm{Hom}(A, \Bbb F_2)$.
Note that if $C \subseteq X$ is clopen, then $\chi_C: X \to \Bbb F_2$ defined by $$\chi_C = \begin{cases} 1 & x \in C\\ 0 & x \notin C\end{cases}$$ is continuous, so that $\chi_C \in A$.
Consider $$\mathcal{F} = \{A \subseteq X\mid A \text{ clopen }, \phi(\chi_A)=1\}$$ and prove that $\mathcal{F}$ is a family of closed subsets of $X$ with the FIP. Compactness (plus Hausdorff, and total disconnectedness) gives us a unique $x \in \bigcap \mathcal F$, and show then that $\theta(x)=\phi$, showing onto-ness.
The link to ultrafilter connection is clear and this blog post also gives some helpful info.