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We describe a totally proper notion of forcing that can be used to shoot uncountable free sequences through certain countably compact non-compact spaces. This is almost (but not quite!) enough to produce a model of ZFC + CH in which countably tight compact spaces are sequential-we still do not know if the notion of forcing described in the paper can be iterated without adding reals.

Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product $2^{ℝ}$ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product ${[0, 1]}^{ℝ}$ is compact. (iv) In a Boolean algebra...

Tychonoff's theorem without the axiom of choice

P. Johnstone (1981)

Fundamenta Mathematicae

Uniformly, proximally and topologically compact relators.

Száz, Árpád (1997)

Mathematica Pannonica

Valdivia compact abelian groups.

Wieslaw Kubis (2008)

RACSAM

Valdivia compacta and equivalent norms

Ondřej Kalenda (2000)

Studia Mathematica

We prove that the dual unit ball of a Banach space X is a Corson compactum provided that the dual unit ball with respect to every equivalent norm on X is a Valdivia compactum. As a corollary we show that the dual unit ball of a Banach space X of density $ℵ_{1}$ is Corson if (and only if) X has a projectional resolution of the identity with respect to every equivalent norm. These results answer questions asked by M. Fabian, G. Godefroy and V. Zizler and yield a converse to Amir-Lindenstrauss’ theorem.