Cardinal sequences of LCS spaces under GCH

Let C() denote the class of all cardinal sequences of length associated with compact scattered spaces. Also put C() = {f C() : f(0) = = min[f() : < ]}. If is a cardinal and < ++ is an ordinal, we define D() as follows: if = , D() = {f {, 1} : f(0) = }, and if is uncountable, D() = {f {, + } : f(0) = , f-1 {} is < -closed and successor-closed in }. We show that for each uncountable regular cardinal and ordinal < ++ it is consistent with GCH that C() is as large as possible, i.e. C() = D(). This yields that under GCH for any sequence f of regular cardinals of length the following statements are equivalent: (1) f C() in some cardinal preserving and GCH-preserving generic-extension of the ground model. (2) for some natural number n there are infinite regular cardinals 0 > 1 >

• Type: Journal
• Year: 2010
• Where: APAL
• Authors: Juan Carlos Martinez, Lajos Soukup
• Keywords: APAL 2010, Cardinal Sequences, Compact Scattered Spaces, Uncountable Regular Cardinal