**István Juhász, Lajos Soukup and William Weiss **

Let denote the class of all cardinal sequences of length associated with compact scattered spaces (or equivalently, superatomic boolean algebras). Also put

We show that iff for some natural number there are infinite cardinals and ordinals such that and where each .

Under GCH we prove that if then

- if , then is -closed in
- if ,
- if , .

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**2000 Mathematics Subject Classification**: