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Could any one tell me how to show that a totally disconnected compact Hausdorff space is homeomorphic to a closed subspace of a product of discrete two-point spaces?

I am not able to see a known example of such space, so I am not able to proceed through my intuition. Please help.

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The Cantor set is an example (or any convergent sequence including the limit point). –  Martin Jan 12 '13 at 17:18
    
@Martin Thank you very much –  El Angel Exterminador Jan 12 '13 at 17:19
    
Paul Halmos' book Lectures on Boolean Algbras has quite a bit on this. The category of totally disconnected compact Hausdorff spaces and continuous functions is the opposite of the category of Boolean algebras and homomorphisms. The set of homomorphisms from a Boolean algebra to the two-member Boolean algebra, with the topology of pointwise convergence of nets of homomorphisms, is a totally disconnected compact Hausdorff space. And the set of all of clopen subsets of a totally disconnected is a Boolean algebra. –  Michael Hardy Apr 27 at 18:11

1 Answer 1

up vote 7 down vote accepted

HINT: For each clopen set $H\subseteq X$ the function

$$f_H:X\to\{0,1\}:x\mapsto\begin{cases} 1,&\text{if }x\in H\\ 0,&\text{if }x\notin H \end{cases}$$

is a continuous function from $X$ to the discrete two-point space. Let $\mathscr{H}$ be the set of all clopen subsets of $X$, and for each $H\in\mathscr{H}$ let $D_H$ be a copy of the discrete two-point space. Define a map

$$\varphi:X\to\prod_{H\in\mathscr{H}}D_H:x\mapsto\langle f_H(x):H\in\mathscr{H}\rangle\;,$$

and show that $\varphi$ is a homeomorphism of $X$ onto $\varphi[X]$.

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