spaces homeomorphic to Baire space

Baire space, $𝒩\equiv {\mathbb{N}}^{\mathbb{N}}$, 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 $f:𝒩\to (0,1)\setminus \mathbb{Q}$ 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	2013-03-22 18:46:48
Last modified on	2013-03-22 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