The ideal determined by the unsymmetric game

Abstract

In the present paper we study the ideal of all subsets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper X Superscript omega"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi> </mml:mrow> <mml:mi> ω </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathcal {X}^\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the second player has a winning strategy in the unsymmetric game. We describe its cardinal coefficients and the notions of forcing determined by it.