Small cardinals and small Efimov spaces

Large Cardinals and Small Dowker Spaces

We prove that, if there is a model of set-theory which contains no first countable, locally compact, scattered Dowker spaces, then there is an inner model which contains a measurable cardinal. A Hausdorff space is normal if, for every pair of disjoint closed sets C and D, there is a pair of disjoint open sets, U containing C and V containing D. A (normal) space is binormal if its product with t...

Operational set theory and small large cardinals

Article history: Received 5 December 2006 Revised 21 April 2008 Available online 11 April 2009 A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of...

On some small cardinals for Boolean algebras

Assume that all algebras are atomless. (1) Spind(A x B) = Spind(A) U Spind(B). (2) Spind(H1-I~ Ai) = {co } U U,Ej Spind(Ai). Now suppose that K and 2 are infinite cardinals, with r uncountable and regular and with r. < 2. (3) There is an atomless Boolean algebra A such that u(A) = te and i(A) = A. (4) If 2 is also regular, then there is an atomless Boolean algebra A such that t(A) = s(A) = K an...

Large cardinal properties of small cardinals

Efimov Spaces and the Splitting Number

An Efimov space is a compact space which contains neither a nontrivial converging sequence nor a copy of the Stone-Cech compactification of the integers. We give a new construction of a space which consistently results in an Efimov space. The required set-theoretic assumption is on the splitting number s. Efimov’s problem is to determine if every compact space with no converging sequences conta...