Topological property of the Cantor set

  • Thread starter hedipaldi
  • Start date
  • #1
210
0
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
 

Answers and Replies

  • #2
22,129
3,298
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
210
0
Thank you.
 
  • #4
210
0
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
210
0
After further thinking i suppose injectivity is due to X being Haussdorff.Am i right?
 

Related Threads on Topological property of the Cantor set

  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
1
Views
3K
Replies
19
Views
19K
  • Last Post
Replies
5
Views
2K
Replies
2
Views
8K
  • Last Post
Replies
11
Views
3K
  • Last Post
Replies
7
Views
4K
  • Last Post
Replies
5
Views
2K
Replies
3
Views
2K
Replies
8
Views
3K
Top