partitions less than cofinality

If λ<cf(κ) then κ(κ)λ1.

This follows easily from the definition of cofinality. For any coloringMathworldPlanetmath f:κλ then define g:λκ+1 by g(α)=|f-1(α)|. Then κ=α<λg(α), and by the normal rules of cardinal arithmetic supα<λg(α)=κ. Since λ<cf(κ), there must be some α<λ such that g(α)=κ.

Title partitions less than cofinality
Canonical name PartitionsLessThanCofinality
Date of creation 2013-03-22 12:55:56
Last modified on 2013-03-22 12:55:56
Owner Henry (455)
Last modified by Henry (455)
Numerical id 6
Author Henry (455)
Entry type Result
Classification msc 03E04
Related topic Arrowsrelation
Related topic ArrowsRelation