An extension of the closed unbounded filter

Abstract

ABSTRACT. A natural extension of the closed unbounded filter is introduced. This extension coincides with the closed unbounded filter on uncountable, regular cardinals re, but in general does not for PKA and [A]K.Henceforth, k will be a regular, uncountable cardinal, unless specified otherwise. A closed unbounded subset of k is a cofinal subset of k which contains the supremum of all increasing sequences from the subset of length less than k. The collection of all closed unbounded subsets of n generates a /c-complete, normal filter over k called the club filter. The notion of a closed unbounded subset of a cardinal k has been generalized to the set of all subsets of X of cardinality less than k, PkX (see [3]), and the set of all subsets of A of cardinality k, [X]k (see [2]). In both instances the collection of closed unbounded sets generate /c-complete, fine, normal filters, the club filters. This paper introduces a natural extension of the club filter. This extension coincides with the club filter on k, but in general does not for PKX and this property is satisfied by sequences (or chains) {pa : a < S} which satisfy w < U ip*i-A canonical example of a sequence which fails to satisfy this is {pa : a < 6} where pa = a and 6 -Wi. DEFINITION. A subset "o" of k is said to be L-closed if for any sequence {pa : a < S} Cb with S < k and |<5| < UQ<¿¡ \Pa\, then \Ja<6P* € °-