spaces homeomorphic to Baire space
Baire space^{}, $\mathcal{N}\equiv {\mathbb{N}}^{\mathbb{N}}$, is the set of all functions^{} $x:\mathbb{N}\to \mathbb{N}$ together with the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism $f:\mathcal{N}\to (0,1)\setminus \mathbb{Q}$ given by continued fraction^{} expansion
$$f(x)=\frac{1}{x(1)+{\displaystyle \frac{1}{x(2)+{\displaystyle \frac{1}{\mathrm{\ddots}}}}}}.$$ 
Theorem 1.
Let $I$ be an open interval^{} of the real numbers and $S$ be a countable^{} dense subset of $I$. Then, $I\mathrm{\setminus}S$ is homeomorphic to Baire space.
More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.
Theorem 2.
A topological space^{} $X$ is homeomorphic to Baire space if and only if

1.
It is a nonempty Polish space^{}.

2.
It is zero dimensional (http://planetmath.org/ZeroDimensional).

3.
No nonempty open subsets are compact^{}.
In particular, for an open interval $I$ of the real numbers and countable dense subset $S\subseteq I$, then $I\setminus S$ is easily seen to satisfy these properties and Theorem 1 follows.
Title  spaces homeomorphic to Baire space 

Canonical name  SpacesHomeomorphicToBaireSpace 
Date of creation  20130322 18:46:48 
Last modified on  20130322 18:46:48 
Owner  gel (22282) 
Last modified by  gel (22282) 
Numerical id  7 
Author  gel (22282) 
Entry type  Theorem 
Classification  msc 54E50 
Related topic  PolishSpace 
Related topic  InjectiveImagesOfBaireSpace 