# Polish space

A separable topologically complete space. Polish spaces form a natural frame of descriptive set theory. The fundamental Polish space $\mathbf I$ of irrationals is homeomorphic to the Baire space $\mathbf N ^ {\mathbf N}$( often denoted by $\omega ^ \omega$ by those logicians who identify $\mathbf N$ and the first infinite ordinal number $\omega$).