partitions less than cofinality

If $\\lambda<\\operatorname{cf}(\\kappa)$ then $\\kappa\\rightarrow(\\kappa)^{1}_{\\lambda}$ .

This follows easily from the definition of cofinality. For any coloring $f:\\kappa\\rightarrow\\lambda$ then define $g:\\lambda\\rightarrow\\kappa+1$ by $g(\\alpha)=|f^{-1}(\\alpha)|$ . Then $\\kappa=\\sum_{\\alpha<\\lambda}g(\\alpha)$ , and by the normal rules of cardinal arithmetic $\\operatorname{sup}_{\\alpha<\\lambda}g(\\alpha)=\\kappa$ . Since $\\lambda<\\operatorname{cf}(\\kappa)$ , there must be some $\\alpha<\\lambda$ such that $g(\\alpha)=\\kappa$ .

单词	PartitionsLessThanCofinality
释义	partitions less than cofinality If $\\lambda<\\operatorname{cf}(\\kappa)$ then $\\kappa\\rightarrow(\\kappa)^{1}_{\\lambda}$ . This follows easily from the definition of cofinality. For any coloring $f:\\kappa\\rightarrow\\lambda$ then define $g:\\lambda\\rightarrow\\kappa+1$ by $g(\\alpha)=\|f^{-1}(\\alpha)\|$ . Then $\\kappa=\\sum_{\\alpha<\\lambda}g(\\alpha)$ , and by the normal rules of cardinal arithmetic $\\operatorname{sup}_{\\alpha<\\lambda}g(\\alpha)=\\kappa$ . Since $\\lambda<\\operatorname{cf}(\\kappa)$ , there must be some $\\alpha<\\lambda$ such that $g(\\alpha)=\\kappa$ .
随便看	Barbălat’s lemma BargmannFockSpace Bargmann-Fock space BargmannTransform Bargmann transform BarnesIntegralRepresentationOfTheHypergeometricFunction Barnes’ integral representation of the hypergeometric function BarnsleyFern Barnsley fern BaronisTheorem Baroni’s theorem barpartialOperator Barrel barrel BarycentricCoordinates barycentric coordinates BasalUnits basal units BaseAndHeightOfTriangle base and height of triangle BaseConversion base conversion BaselProblem Basel problem BaselProblemDerivation