A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We ...

Given an uncountable regular cardinal κ, a partial order is κstationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in Pκ(P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain conditio...

The weakly compact reflection principle Reflwcpκq states that κ is a weakly compact cardinal and every weakly compact subset of κ has a weakly compact proper initial segment. The weakly compact reflection principle at κ implies that κ is an ω-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that κ is pω ` 1qweakly compact. Moreover, we...

A compact LOTS has an ω 1 -like base if and only if it is metrizable. An infinite cardinal κ is the Noetherian type of a compact LOTS if and only if κ 6= ω1 and κ is not weakly inaccessible. If X is a homogeneous compactum with regular weight, then every base of X contains an Nt(X)op-like

It is well-known that every non-isolated point in a compact Hausdorff space is the accumulation point of a discrete subset. Answering a question raised by Z. Szentmiklóssy and the first author, we show that this statement fails for countably compact regular spaces, and even for ω-bounded regular spaces. In fact, there are κ-bounded counterexamples for every infinite cardinal κ. The proof makes ...

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |...

If κ is strongly compact and λ > κ and λ is regular (or alternatively cf(λ) ≥ κ), then ` 2 ́+ → (λ+ ζ) θ holds for ζ, θ < κ. 2000 Mathematics Subject Classification. 2000 Math Subject Classification: 03E02. I would like to thank Alice Leonhardt for the beautiful typing. Research of the author was partially supported by the United States-Israel Binational Science Foundation. Publ.761. Latest Rev...

We show that under a certain topological assumption on two compact hereditary families F and G some infinite cardinal κ, the corresponding combinatorial spaces XF XG are isometric if only there is permutation of κ inducing homeomorphism between G. also prove different regular ω cannot be permuted one to other. Both these results strengthen main result [5].