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I following the course Topology, and i want to ask some questions about the topics in the heading. We have the following definitions:

  • A topological space $X$ is totally separated iff for all $x\neq y$ there exists a clopen set $C\subset X$ such that $x\in C$ and $y\notin C$
  • X is called a Stone-space iff X is compact Hausdorff and zero-dimensional

We have also the following lemma: $X$ compact Hausdorff and totally separated implies $X$ zero-dimensional.

Now i want to answer the following statements:

  • Prove that for every $X\subset\Bbb{R}$ with empty interior is totally separeted
  • Prove that the one-point-compactification of every discrete space is a Stone-space
  • Let X be a discrete space: Prove that the linear span of the idempotents of $C_b(X,\Bbb{R})$ (the space of bounded continuous functions from $X$ to $\Bbb{R}$) is dense in $C_b(X,\Bbb{R})$.Now prove that the Stone-Chech-compactification $\beta X$ is a Stone space.

I have some ideas but i don't know if there are good. Here my ideas:

  • Choose $[a,b]$ and $(a,b)$ in $\Bbb{R}$ such that the intersection with $X$ is the same. Notice that $X$ does not contain an open interval because the interior is empty. Then $X\cap[a,b]=X\cap(a,b)$ is the clopen set we search for. Is this good or miss i something?
  • For the second one i want to prove that the one-point-compactification $X_{\infty}$ is totally separated. I know that this is true for $X$, because $X$ is discrete and thus all subsets are clopen. But what about the one-point-compactification (because then we can use the lemma to conclude that $X_{\infty}$ is a Stone-space).
  • We know that the functions in the space are bounded. Thus if we choose $\epsilon>0$ arbitrary we can find a minimal $n\in\Bbb{N}$ such that $|f|<n\epsilon$. Can we then make idempotent functions which are define in such a $\epsilon$-neighbourhood to conclude the result? I can't imagine another way to prove it. That $\beta X$ is a Stone-space i want to prove that $\beta X$ is totally separated because then we can use the Lemma, but how to do that?

I hope someone can help me?! Its a lot what i asked here, but if i can get solutions or hints i can go on with my work :) Thank you for the help :)

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1 Answer 1

The first answer is complete, the second almost so; for the third I’ve given a hint, but there’s still some real work to be done.

For the first question you must show that for all distinct $x,y\in X$ there is a clopen $C\subseteq X$ such that $x\in C$ and $y\notin C$. Let $x,y\in X$ with $x\ne y$. Without loss of generality we may assume that $x<y$. $X$ has empty interior, so $(x,y)\nsubseteq X$, and we may choose $z\in(x,y)\setminus X$. Now let $$C=(\leftarrow,z)\cap X=(\leftarrow,z]\cap X\;,$$ $C$ is clopen in $X$, $x\in C$, and $y\notin C$, just as we wanted.

Let $X$ be an infinite set with the discrete topology, and let $X^*=X\cup\{p\}$ be the one-point compactification of $X$, with $p$ the point at infinity. Let $x$ and $y$ be distinct points of $X^*$. If $x\in X$, show that $C=\{x\}$ is a clopen set containing $x$ and not $y$. The only other possibility is that $x=p$, in which case $C=X^*\setminus\{y\}$ works; why?

For the third problem note that $C_b(X,\Bbb R)$ is just the set of bounded functions from $X$ to $\Bbb R$, since all functions from $X$ to $\Bbb R$ are continuous. Now let $f\in C_b(X,\Bbb R)$, and let $A=f[X]$. $A$ is bounded, so for each $\epsilon>0$ there is a finite set $\{a_1,\dots,a_n\}\subseteq A$ such that for each $a\in A$ there is a $k\in\{1,\dots,n\}$ such that $|a-a_k|<\epsilon$. Now try to find idempotents $g_1,\dots,g_n$ such that $$\left\|f-\sum_{k=1}^na_kg_k\right\|_\infty<\epsilon\;.$$

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