# Topological property of the Cantor set

1. Apr 12, 2013

### 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

2. Apr 12, 2013

### micromass

Staff Emeritus
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.

3. Apr 12, 2013

### hedipaldi

Thank you.

4. Apr 13, 2013

### 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?

5. Apr 13, 2013

### hedipaldi

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