Problem. Prove that if $(X_{i}, \varphi_{ij})$ is a inverse system of nonempty sets and surjective maps indexed by a countable direct set, then the inverse limit is nonempty.
I proved some similar results with different hypothesis combining "finite", "compact", "Hausdorff". I know that if we exclude "countable direct set", the statment becomes false. I tried to adapt my others proof for this case, but they didn't work.
My approach in other case was use the discrete topology on $X_{i}$ (then $X_{i}$ becomes Hausdorff) and work with
If $X_{i}$ are nonempty compact Hausdorff space, then $X$ is nonempty.
But I think that this don't work with this problem, at least, I cannot see.
Bourbaki proof this result with additional hypothesis, but it uses some definitions that are not in Wilson's book. The idea for the proof is show that we can consider $I = \mathbb{N}$ and show that the maps $\varphi_{i}: X \to X_{i}$ are surjective where $(X,\varphi_{i})$ is the inverse limit of $(X_{i},\varphi_{ij})$.
Can someone help me?
Hint : $I$ is countable and directed, therefore it has a totally ordered countable cofinal subset, and we may assume it's $\mathbb{N}$.
By categorical abstract nonsense, the inverse limit over $I$ is the same as that of its cofinal subset, that is, over $\mathbb{N}$. Over $\mathbb{N}$ the proof is easy, you build an element of the limit by recursion.
(If you don't like abstract nonsense you can first find an element of the limit over the cofinal subset, and then show how it provides an element of the limit over $I$)