Proving that $X_{\infty}$ satisfy a property in connectedness!

Question

Let $X_n$ be an inverse limit that satisfy: for every n $X_n\neq{\emptyset}$ is totally disconnected, Prove that if $X_{\infty}\neq{\emptyset} $ , then $X_{\infty}$ is totally disconnected. How it can be true?

Answer 1

The inverse limit $X_\infty$ can be empty, unless all $X_n$ are compact Hausdorff, e.g.

(Example: $X_n = \Bbb N$ in the discrete topology, all bonding maps $f(n)=n+1$, if all bonding maps are onto, the inverse limit of a sequence is always non-empty, as you can just construct a point in it, so the non-surjectiveness of $f$ is no coincidence.)

Furthermore, $X_\infty$ is a subset of the product $\prod_n X_n$ and products of totally disconnected spaces keep that property, and likewise for subspaces.

(If $A \subseteq \prod_n X_n$ is connected, so are all $\pi_n[A]$, so..., and a connected subset of a subspace would also be connected in the superspace etc. These last facts are not hard at all.)