A prime-ideal compactness tree for the finite–cofinite algebra

Enumerate the finite subsets as $F_0=\mathrm{\varnothing },F_1=\{0\},F_2=\{1\},F_3=\{0,1\},\dots$ by increasing binary code, and enumerate $B$ by $b(2n)=F_n$, $b(2n+1)=\omega \setminus F_n$. This is onto, including repetitions such as $b(0)=\mathrm{\varnothing }$ and $b(1)=\omega$.

At levels $0,1,2$ the generated algebra is $\{\mathrm{\varnothing },\omega \}$ and has its unique homomorphism to $\mathbf{2}$. At level $3$, after $\{0\}$ appears, the generated algebra has atoms $\{0\}$ and $\omega \setminus \{0\}$ and hence two homomorphisms, with respective values $1$ and $0$ on $\{0\}$. Level $4$ adds only its complement and has the same two nodes.

At level $5$, the generators include $\{0\}$ and $\{1\}$; the atoms are $\{0\}$, $\{1\}$, and $R_2=\omega \setminus \{0,1\}$. The three homomorphisms select these atoms and have value pairs $(1,0),(0,1),(0,0)$ on the two singletons. The last node restricts to the value-$0$ node at level $3$.

At any finite stage let $E$ be the finite union of all finite generators seen so far. The generated algebra has the finitely many atomic pieces inside $E$ and the single cofinite remainder $R=\omega \setminus E$. Evaluation at $R$ is the unique level node assigning $0$ to every finite member of that subalgebra and $1$ to every cofinite member. These nodes restrict coherently, so they form the branch illustrated by steps 2.1 and 3.1.

The union homomorphism is $h(A)=0$ when $A$ is finite and $h(A)=1$ when $A$ is cofinite. Its zero fibre is therefore $\mathrm{F}\mathrm{i}\mathrm{n}$. This is proper and prime: if $A,B$ are both cofinite then $A\cap B$ is cofinite, so $A\cap B$ can be finite only when at least one of $A,B$ is finite. This explicit prime ideal agrees with F2's existence conclusion.