Topological property of the Cantor set

[hedipaldi]

Let X be a metric separable metric and zero dimensional space.Then X is homeomorphic to a subset of Cantor set.
How can it be proved?
Thank's a lot,
Hedi

 

[micromass]

Hint: prove the following theorem:

• Let $X$ be a $T_1$ space (= singletons are closed). Let $\{f_i~\vert~i\in I\}$ be a collection of functions $f_i:X\rightarrow X_i$ which separates points from closed sets, then the evaluation map $e:X\rightarrow \prod_{i\in I}X_i$ is an embedding.

Now, your space $X$ has a countable basis consisting of clopen sets (why?). Use this to construct functions $f_n:X\rightarrow \{0,1\}$ for $n\in \mathbb{N}$ and apply the theorem.

 

[hedipaldi]

Thank you.

 

[hedipaldi]

Injectivity of the function

The resulting function from X into the product space doesn't seem to be one-to-one.Maybe i fail to see something?

 

[hedipaldi]

After further thinking i suppose injectivity is due to X being Haussdorff.Am i right?