Abstract:We study the concept of $\pi $-caliber as an alternative to the well known concept of caliber. $\pi $-caliber and caliber values coincide for regular cardinals greater than or equal to the Souslin number of a space. Unlike caliber, $\pi $-caliber may take on values below the Souslin number of a space. Under Martin's axiom, $2^{\omega }$ is a $\pi $-caliber of $\mathbb{N}^{\ast }$. Prikry's poset is used to settle a problem by Fedeli regarding possible values of very weak caliber.
Keywords: nowhere dense, point-$\kappa $ family, $\pi $-caliber
AMS Subject Classification: 54A38 54A15 03E35