Infinite products of finite simple groups

We classify the sequences of finite simple nonabelian groups such that has uncountable cofinality.

     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Singular cohomology inL

We prove, that in the world of constructible sets, there does not exist a spaceX withH″ (X,Z) isomorphic to the rational numbers. The proof requires a result about the growth of Ext _z ^Emphasis>/i (-, Z) inside of Gödel’s constructible universeL.     

Con(u>i)

We prove here the consistency of u>i where: u=Min{|X|:XP() generates a non-principle ultrafilter}, i=Min{|A|:A is a maximal independent family of subsets of }In this we continue Goldstern and Shelah [G1Sh388] where Con(r>u) was proved using a similar but different forcing. We were motivated by Vaughan [V] (which consists of a survey and a list of open problems). For more information on the subject see [V] and [G1Sh388].     

Better quasi-orders for uncountable cardinals

We generalize the theory of Nash-Williams on well quasi-orders and better quasi-orders and later results to uncountable cardinals. We find that the first cardinal κ for which some natural quasi-orders are κ-well-ordered, is a (specific) mild large cardinal. Such quasi-orders are (the class of orders which are the union of ≦λ scattered orders) ordered by embeddability and the (graph theoretic) trees under embeddings taking edges to edges (rather than to passes). This research was supported by the United States-Israel Binational Science Foundation, grant 1110.     

Categoricity in ℵ1 of sentences inL_{\omega _1 ,\omega } (Q)

We investigate the categoricity and number of non-isomorphic models in ℵ₁ of sentences in . AssumingV=L we prove that no sentence in has exactly one uncountable model. Thus partially answering problem 24 of a problem list by Friedman.     

A note on small sets of reals

We construct an example of a combinatorially large measure-zero set.     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Forcing a Boolean algebra with predesigned automorphism group

For suitable groups we will show that one can add a Boolean algebra by forcing in such a way that is almost isomorphic to . In particular, we will give a positive answer to the following question due to J. Roitman: Is a possible number of automorphisms of a rich Boolean algebra?

     

Martin's Axiom is consistent with the existence of nowhere trivial automorphisms

Martin's Axiom does not imply that all automorphisms of are somewhere trivial. An alternate method for obtaining models where every automorphism of is somewhere trivial is explained.