CH with no Ostaszewski spaces

Abstract

There are models of CH without Ostaszeswki spaces. If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally compact and sub-Ostaszewski, there is a forcing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Subscript upper X"> <mml:semantics> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">P_X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which does not add reals and which forces “<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not sub-Ostaszewski”.