uncountable Polish spaces contain Cantor space

Cantor space is an example of a compact and uncountable Polish space. In fact, every uncountable Polish space contains Cantor space, as stated by the following theorem.

Theorem.

Let $X$ be an uncountable Polish space. Then, it contains a subset $S$ which is homeomorphic to Cantor space.

For example, the set $ℝ$ of real numbers contains the Cantor middle thirds set (http://planetmath.org/CantorSet). Note that, being homeomorphic to Cantor space, $S$ must be a compact and hence closed subset of $X$. The result is trivial in the case of Baire space $𝒩$, in which case we may take $S$ to be the set of all $s\in 𝒩$ satisfying $s_n\in \{1,2\}$ for all $n$. Then, for any uncountable Polish space $X$ there exists a continuous and one-to-one function $f:𝒩\to X$ (see here (http://planetmath.org/InjectiveImagesOfBaireSpace)). Then $f$ gives a continuous bijection from $S$ to $f(S)$. The inverse function theorem (http://planetmath.org/InverseFunctionTheoremTopologicalSpaces) implies that $f$ is a homeomorphism between $S$ and $f(S)$ and, therefore, $f(S)$ is homeomorphic to Cantor space.

Title	uncountable Polish spaces contain Cantor space
Canonical name	UncountablePolishSpacesContainCantorSpace
Date of creation	2013-03-22 18:48:33
Last modified on	2013-03-22 18:48:33
Owner	gel (22282)
Last modified by	gel (22282)
Numerical id	6
Author	gel (22282)
Entry type	Theorem
Classification	msc 54E50
Related topic	PolishSpace