Investigating the relationship between $X$ and the inverse limit $\mathcal{L}$ of the tower.

Question

Let $X$ be totally disconnected, compact, metric and let $\mathcal{L}$ be the inverse limit of the sequence. (I mentioned in my previous question).

$(a)$ Write out what a typical element of $\mathcal{L}$ is in terms of subsets of $X$ and the inclusions of sets in other sets.

Could anyone help me in doing so, please?

Answer 1

A typical element of $\mathcal{L}$ is a sequence $(A_n)_n = (A_1, A_2 ,A_3, \ldots)$ obeying:

1. $\forall n: A_n \in \mathcal{A}_n$ (from being in the product of the $\mathcal{A}_n$.)
2. $\forall n: A_{n+1} \subsetneq A_n$ (so they get smaller and smaller), from the fact that $\psi(A_{n+1})=A_n$ and we have strict refinements going up.

Then of course, as by all the previous questions we know that all $A_n$ are clopen, non-empty and $\operatorname{diam}(A_n) < \frac1n$, there is a unique $x \in \bigcap_n A_n$ (Cantor intersection theorem, using that all $A_n$ are closed), so we have a map from $\mathcal{L}$ to $X$ (the image of $(A)_n$ being defined by this unique point), and then it's easy to verify that this map is 1-1 and onto and continuous, and so a homeomorphism.