S is a combinatoric principle weaker than S. It states that, for S stationary in κ, there is a sequence AααS such that Aαα and sup(Aα)=α and with the property that for each unboundedPlanetmathPlanetmath subset Tκ there is some AαX.

Any sequence satisfying S can be adjusted so that sup(Aα)=α, so this is indeed a weakened form of S.

Any such sequence actually contains a stationary set of α such that AαT for each T: given any club C and any unbounded T, construct a κ sequence, C* and T*, from the elements of each, such that the α-th member of C* is greater than the α-th member of T*, which is in turn greater than any earlier member of C*. Since both sets are unbounded, this construction is possible, and T* is a subset of T still unbounded in κ. So there is some α such that AαT*, and since sup(Aα)=α, α is also the limit of a subsequence of C* and therefore an element of C.

Canonical name clubsuit
Date of creation 2013-03-22 12:53:52
Last modified on 2013-03-22 12:53:52
Owner Henry (455)
Last modified by Henry (455)
Numerical id 4
Author Henry (455)
Entry type Definition
Classification msc 03E65
Synonym clubsuit
Related topic DiamondMathworldPlanetmath
Related topic DiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic ProofOfDiamondIsEquivalentToClubsuitAndContinuumHypothesis
Related topic CombinatorialPrinciple