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单词 clubsuit
释义


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.

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更新时间:2025/5/4 18:53:11