A ZFC Dowker space in $\aleph_{ω+1}$: an application of pcf theory to topology

PCF arithmetic without and with choice

We deal with relatives of GCH which are provable. In particular we deal with rank version of the revised GCH. Our motivation was to find such results when only weak versions of the axiom of choice are assumed but some of the results gives us additional information even in ZFC.

Pcf theory and Woodin cardinals

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all bounded X subset aleph_{|alpha|^+}, M_n^#(X) exists. Theorem B: Let kappa be a singular cardinal of uncountable cofinality. If {alpha<kappa| 2^alpha=alpha^+} is stationary as well as co-stationary then for all n< omega and for all bounded X subset kappa, M_n^#(X) exists. Theorem A answers a question of Gitik and Mitchell, and Theorem B yields a lower bound for an …

A homotopy approach to set theory

We observe that the notion of two sets being equal up to finitely many elements is a homotopy equivalence relation in a model category, and suggest a homotopy-invariant variant of Generalised Continuum Hypothesis about which more can be proven within ZFC and which first appeared in PCF theory. The formalism allows to draw analogies between notions of set theory and those of homotopy theory, and we indeed observe a similarity between homotopy theory ideology/yoga and that of PCF theory. We also briefly discuss conjectural connections with model theory and arithmetics and geometry.

Locally compact, $ω_1$-compact spaces

An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…

梯子系统上的菱形原理与可数亚紧拓扑空间

Diamond on ladder systems and countably metacompact topological spaces

本文证明在ZFC下存在梯子系统L使得对应空间X_L不是可数亚紧的，给出κ=cf(beth_{ω+1})的肯定答案，并在beth_ω=aleph_ω假设下得到更低基数的例子。

Noetherian type in topological products

The cardinal invariant "Noetherian type" of a topological space $X$ (Nt(X)) was introduced by Peregudov in 1997 to deal with base properties that were studied by the Russian School as early as 1976. We study its behavior in products and box-products of topological spaces. We prove in Section 2: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by the operations of forming a square and of passing to a dense subspace. The Noetherian type of the Cantor Cube of weight $\aleph_ω$ with the coun…