# spaces homeomorphic to Baire space

Baire space, $\mathcal{N}\equiv\mathbb{N}^{\mathbb{N}}$, is the set of all functions $x\colon\mathbb{N}\rightarrow\mathbb{N}$ together with the product topology. This is homeomorphic to the set of irrational numbers in the unit interval, with the homeomorphism $f\colon\mathcal{N}\rightarrow(0,1)\setminus\mathbb{Q}$ given by continued fraction expansion

 $f(x)=\cfrac{1}{x(1)+\cfrac{1}{x(2)+\cfrac{1}{\ddots}}}.$
###### Theorem 1.

Let $I$ be an open interval of the real numbers and $S$ be a countable dense subset of $I$. Then, $I\setminus S$ 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. 1.

It is a nonempty Polish space.

2. 2.

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

3. 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 SpacesHomeomorphicToBaireSpace 2013-03-22 18:46:48 2013-03-22 18:46:48 gel (22282) gel (22282) 7 gel (22282) Theorem msc 54E50 PolishSpace InjectiveImagesOfBaireSpace