Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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 – La Belle Noiseuse 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 '14 at 18:11
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 $\varphi[X]$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.