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 $).