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	homogeneous polynomial homogeneous polynomial HomogeneousPolynomial1 HomogeneousSpace homogeneous space HomogeneousSystemOfParameters homogeneous system of parameters HomogeneousTopologicalSpace homogeneous topological space HomologicalComplexOfTopologicalVectorSpaces homological complex of topological vector spaces Homology homology HomologyOfmathbbRP3 HomologyOfTheSphere homology of the sphere homology of ℝ⁢ℙ^3. HomologySphere homology sphere HomomorphicImageOfASigmastructureIsASigmastructure homomorphic image of a Σ-structure is a Σ-structure HomomorphicImageOfGroup homomorphic image of group HomomorphismBetweenAlgebraicSystems homomorphism between algebraic systems