totally disconnected compact Hausdorff uniform space

Question

Let $(X, \mathcal{U})$ be a compact and Hausdorff uniform space. For $D\in\mathcal{U}$, the sequence $\{x_i\}_{i\in\mathbb{Z}}$ is called a $D$- chain if $(x_i, x_{i+1})\in D$ for all $i\in\mathbb{Z}$.

Let $X$ be totally disconnected and $E\in\mathcal{U}$ be given. Is it true that

There is $D\in\mathcal{U}$ such that for every $D$- chain $\{x_i\}_{i\in\mathbb{Z}}$, we have $(x_i, x_j)\in E$ for all $i, j\in\mathbb{Z}$?

Please help me to understand it.

Recently I must have to use another account, because I forgotted my password. Indeed the following link is mine. — Ali Barzanouni, Apr 15 '18 at 05:58
https://math.stackexchange.com/questions/2732762/question-about-totally-disconnected-compact-uniform-space — Ali Barzanouni, Apr 15 '18 at 06:01
Why do you even think it's true? It might very well be false. — Henno Brandsma, Apr 15 '18 at 06:09
In the case of metric space, it is true. But I dont know in the case of uniform space. Do you have an example to show that it is false? — Ali Barzanouni, Apr 15 '18 at 06:13
https://ac.els-cdn.com/0304397594902771/1-s2.0-0304397594902771-main.pdf?_tid=bc394bf9-e80c-4bbf-aba2-4496ba3d468e&acdnat=1523773290_48fc650a1576855499be7dd190493954 — Ali Barzanouni, Apr 15 '18 at 06:20
I don't see how that proof gives you the above result. AFAICS, it does not imply it. — Henno Brandsma, Apr 15 '18 at 06:23
Identity map on uniform space $X$ has shadowing, means that for every neighborhood $E$ of $\Delta_X$, there is a neighborhood $D$ of $\Delta_X$ such that for every $D$- chain ${x_i}$, we have $(x_i, x_j)\in E$. — Ali Barzanouni, Apr 15 '18 at 06:31

score 0 · Answer 1 · answered Jan 29 '19 at 07:14

Yes, this is true.

The slightly tricky part is showing that a compact Hausdorff totally disconnected space is totally separated. See here: Any two points in a Stone space can be disconnected by clopen sets. This means for any distinct $x,y$ there is a clopen $U$ containing $x$ but not $y.$ Since the space is compact, $U$ is "uniformly clopen" meaning there is $D\in\mathcal U$ such that there is no $D$-chain from $x$ to $y.$

For each $D\in\mathcal U$ let $C_D\subseteq X^2$ be the set of pairs connected by a $D$-chain. Each $C_D$ is clopen, and the family $\{C_D\}$ is closed under intersection: $C_{D\cap D'}=C_D\cap C_{D'}.$ We're given a neighborhood $E$ of the diagonal $\Delta=\{(x,x)\mid x\in X\}$ and we want to show that some $C_D\setminus E$ is empty. We can assume $E$ is open which makes these sets $C_D\setminus E$ closed. By the previous paragraph $\bigcap_{D\in\mathcal U} C_D=\Delta,$ so $\bigcap_{D\in\mathcal U} C_D\setminus E$ is empty. By compactness some $C_D\setminus E$ is empty.

totally disconnected compact Hausdorff uniform space

1 Answers1