# Topological property of the Cantor set

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

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.

Thank you.

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?

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