We show a number of undecidable assertions concerning countably compact spaces hold under PFA(S)[S]. We also show the consistency without large cardinals of every locally compact, perfectly normal space is paracompact.
We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof.
We present S. Todorcevics method of forcing with a coherent Souslin tree over restricted iteration axioms as a black box usable by those who wish to avoid its complexities but still access its power.
We prove that if there are $mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $kappa$ such that $kappa^omega=kappa$, there exists a group topology on the free Abelian group of cardinality $kappa$ without nontrivial con
vergent sequences and such that every finite power is countably compact. In particular, there are arbitrarily large countably compact groups. This answers a 1992 question of D. Dikranjan and D. Shakhmatov.
We introduce the class of slicely countably determined Banach spaces which contains in particular all spaces with the RNP and all spaces without copies of $ell_1$. We present many examples and several properties of this class. We give some applicatio
ns to Banach spaces with the Daugavet and the alternative Daugavet properties, lush spaces and Banach spaces with numerical index 1. In particular, we show that the dual of a real infinite-dimensional Banach with the alternative Daugavet property contains $ell_1$ and that operators which do not fix copies of $ell_1$ on a space with the alternative Daugavet property satisfy the alternative Daugavet equation.
An S-fold has played an important role in constructing supersymmetric field theories with interesting features. It can be viewed as a type of AdS_4 solutions of Type IIB string theory where the fields in overlapping patches are glued by elements of S
L(2,Z). This paper examines three dimensional quiver theories that arise from brane configurations with an inclusion of the S-fold. An important feature of such a quiver is that it contains a link, which is the T(U(N)) theory, between two U(N) groups, along with bifundamental and fundamental hypermultiplets. We systematically study the moduli spaces of those quiver theories, including the cases in which the non-zero Chern-Simons levels are turned on. A number of such moduli spaces turns out to have a very rich structure and tells us about the brane dynamics in the presence of an S-fold.