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?

It might be helpful if you provided a link to your previous question, like so: https://math.stackexchange.com/questions/3620392/verify-that-the-coordinate-functions-determine-the-topology-on-mathcall — Markus Zetto, Apr 11 '20 at 14:13

Henno Brandsma · Accepted Answer · 2020-04-13T13:17:41.010

2

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

$\forall n: A_n \in \mathcal{A}_n$ (from being in the product of the $\mathcal{A}_n$.)
$\forall n: A_{n+1} \subsetneq A_n$ (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.

edited Apr 13 '20 at 13:17

answered Apr 11 '20 at 14:23

Henno Brandsma

242,131

what is the definition of the map explicitly? I do not understand. – Apr 13 '20 at 10:31
@Happy it maps the sequence of $A_n$ to its unique intersection point. It's the map $f$ from this question. – Henno Brandsma Apr 13 '20 at 10:44
In the first line, Is it a sequence in $\mathcal{A}$ or a sequence in $A$? – Apr 13 '20 at 11:26
@Happy a sequence in $\prod_n \mathcal{A}n$ of course. This is where $A\infty$ lives. – Henno Brandsma Apr 13 '20 at 11:47
No, I mean what you wrote not its definition .... your sequence is in $A_{n}$ not in $\mathcal{A}_{n}$ .... should it be corrected? – Apr 13 '20 at 11:55
@Happy no, I’m correct. – Henno Brandsma Apr 13 '20 at 12:07
ok, could you please, tell me the relation between $\mathcal{A}{n} $ and $A{n}$? – Apr 13 '20 at 12:11
Are we in a complete metric space to use Cantor intersection theorem? – Apr 13 '20 at 12:21
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$A_n$ is a clopen subset of $X$ and a member of the finite partition $\mathcal{A}_n$ of $X$. And $X$ is compact metric so certainly complete. – Henno Brandsma Apr 13 '20 at 12:26
Which theorem are you using? – Apr 13 '20 at 12:29
there is a theorem maybe called nested sequence theorem, does it need completeness? – Apr 13 '20 at 12:34
@Happy yes, but compactness gives non-empty intersection. The shrinking diameter gives unique point. – Henno Brandsma Apr 13 '20 at 12:38
I want to avoid the proof that my space is complete (I do not know how to prove this)? – Apr 13 '20 at 12:42
@Happy compactness is enough. But completeness is trivial to prove: a. Cauchy sequence with a convergent subsetequence is convergent. – Henno Brandsma Apr 13 '20 at 12:44
could you please show me the statement of the theorem that contains compactness only? – Apr 13 '20 at 12:45
In the first line, why you are writing $(A)n$ and not $(A{n})_n$? – Apr 13 '20 at 13:02
ohh I understood just now. Is $\mathcal{L} = A_{\infty}$? – Apr 13 '20 at 13:06
@Happy yes $\mathcal{L}$ is the inverse limit. – Henno Brandsma Apr 13 '20 at 13:08
What about my comment before last? – Apr 13 '20 at 13:09
Also to understand the relation between $\mathcal{A}{n}$ and $A{n}$ Does this means that $\mathcal{A}{1}.$ contains only $A{1}$? I am confused, I am seeing all $\mathcal{A}_{n}$ the same. – Apr 13 '20 at 13:14
Let us continue this discussion in chat. – Henno Brandsma Apr 13 '20 at 13:17

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

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