Baire space
For subsets of Baire space or Cantor space, there is a more concise (if less transparent) alternative definition, which turns out to be equivalent. A subset "A" of Baire space is ∞-Borel just in case there is a set of ordinals "S" and a first-order formula "φ" of the language of set theory such that, for every "x" in Baire space,