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 \begin{equation*} f(x)=\sfrac{1}{x(1)+\sfrac{1}{x(2)+\sfrac{1}{\ddots}}}. \end{equation*}

More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.

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.