In Part I of this series [5], we introduced a class of notions of forcing which we call [Formula: see text]-Prikry, and showed that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are [Formula: see text]-Prikry. We proved that given a [Formula: see text]-Prikry poset [Formula: see text] and a [Formula: see text]-name for a nonreflecting stationary set [Formula: see text], there exists a corresponding [Formula: see text]-Prikry poset that projects to [Formula: see text] and kills the stationarity of [Formula: see text]. In this paper, we develop a general scheme for iterating [Formula: see text]-Prikry posets, as well as verify that the Extender-based Prikry forcing is [Formula: see text]-Prikry. As an application, we blow-up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all nonreflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.

We first prove a general result about the preservation of extendible and C (n) -extendible cardinals under definable weakly homogeneous class forcing iterations (theorem 5.3). As applications we give new proofs of the preservation of Vopěnka's Principle and C (n) -extendible cardinals under Jensen's iteration for forcing the GCH (cf. [BT11] and [Tsa13]). We prove that C (n) -extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible ∆ 2 -definable behaviour of the power-set function on regular cardinals. We show that one can force a variety of proper class-many disagreements between the universe and HOD with respect to the calculation of successors of regular cardinals, while preserving C (n) -extendible cardinals. We also show, assuming the GCH, that the class forcing iteration of Cummings-Foreman-Magidor for forcing ♦ + κ + at every κ ([CFM01]) preserves C (n) -extendible cardinals. We give an optimal result on the consistency of weak square principles and C (n) -extendible cardinals.In the last section we prove aanother preservation result for C (n)extendible cardinals under very general (not necessarily definable or weakly homogeneous) class forcing iterations, one example being the standard class forcing that yields V = HOD.

We prove that Galvin's property consistently fails at successors of strong limit singular cardinals. We also prove the consistency of this property failing at every successor of a singular cardinal. In addition, the paper analyzes the effect of Prikry-type forcings on the strong failure of the Galvin property, and explores stronger forms of this property in the context of large cardinals. Contents 0. Introduction 1. Preliminaries 1.1. Notations 1.2. Clubs and Filters 1.3. Forcing preliminaries 1.4. Radin forcing with interleaved collapses 2. The consistency of the local and global failure 2.1. Local failure of Galvin's property 2.2. Global failure of Galvin's property 2.3. The impossibility of the ultimate global failure 2.4. Galvin's number at successor of singular cardinals 3. The strong failure and Prikry-type generic extensions 4. Stronger forms of Galvin's property on normal filters 5. Open problems 5.1. The failure at successors of singulars 5.2. Galvin's property and large cardinals 5.3. Consistency strength of the failure of Galvin's property 2010 Mathematics Subject Classification. 03E35, 03E55.

We show that if a separable Rosenthal compactum K is a continuous n-to-one preimage of a metric compactum, but it is not a continuous n − 1-to-one preimage, then K contains a closed subset homeomorphic to either the n−Split interval Sn(I) or the Alexandroff n−plicate Dn(2 N ). This generalizes a result of the third author that corresponds to the case n = 2.

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