This thread has it compactness theorem can be derived from Tychonoff theorem. I'm interested in how this can be done, but got stuck.
Here's how far I understand:
Following the version of campactness theorem in A Mathematical Introduction to Logic, Herbert B. Enderton(2ed):
A set of wffs (well-formed formula) is satisfiable iff every finite subset is satisfiable.
Let $\Sigma$ be a set of wffs, each of which is generated by a set of sentence $A$ whose elements can be indexed by $I$. Then the truth value of each finite subset $\Sigma_{\alpha}$ is determined by the truth assignment of $A$, which can be expressed as a function in the space $\{T, F\}^I$. For each finite subset $\Sigma_{\alpha}$, there is a non-empty subset $J_{\alpha}$ of $\{T, F\}^I$ which makes $\Sigma_{\alpha}$ true. Suppose all finite subsets of $\Sigma$ can be indexed by $B$, then the compactness theorem says $\bigcap_{\alpha \in B}J_{\alpha} \neq \varnothing$
I got stuck on how to define the topology of $\{T, F\}^I$. It seems to me, since $\Sigma_{\alpha}$ is a finite set of wffs, its truth value should only depend on the truth values of a finite number of sentences in $A$.
Supposedly, Tychonoff Theorem could serve as a hint, but I don't know how to proceed.