A characterization of the least cardinal for which the Baire category theorem fails

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the least cardinal such that the real line can be covered by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> many nowhere dense sets. We show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be characterized as the least cardinal such that "infinitely equal" reals fail to exist for families of cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa"> <mml:semantics> <mml:mi>κ</mml:mi> <mml:annotation encoding="application/x-tex">\kappa</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.