Lindelof indestructibility, topological games and selection principles

Indestructibility of compact spaces

In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to $ω_1$-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba's "$\aleph_1$-Borel Conjecture" is equiconsistent with the existence of an inaccessible cardinal.

Productively Lindelof and Indestructibly Lindelof Spaces

There has recently been considerable interest in productively Lindelof spaces, i.e. spaces such that their product with every Lindelof space is Lindelof. Here we make several related remarks about such spaces. Indestructible Lindelof spaces, i.e. spaces that remain Lindelof in every countably closed forcing extension, were introduced by Tall in 1995. Their connection with topological games and selection principles was explored by Scheepers and Tall in 2010. We find further connections here.

SPM Bulletin 27

Contents: 1. Editor's note 2. Research announcements 2.1. Pseudocompact group topologies with no infinite compact subsets 2.2. Selective coideals on (FIN[1]k) 2.3. Entire functions mapping uncountable dense sets of reals onto each other monotonically 2.4. Effective refining of Borel coverings 2.5. Symmetry and colorings: Some results and open problems 2.6. Many partition relations below density 2.7. Lindelof indestructibility, topological games and selection principles 2.8. Locally precompact groups: (Local) realcompactness and connectedness 2.9. The group Aut is Roelcke precompact 2.10. An in…

全仿紧空间与Menger覆盖性质

Totally paracompact spaces and the Menger covering property

针对正则Lindelöf空间中全仿紧性与Menger覆盖性质是否等价的问题，研究证明二者在实数集上的Lindelöf GO-空间中等价，给出正则Menger空间全仿紧的博弈论证明，还在𝔟=ω₁条件下强化了Michael的相关经典结论。

Infinite games and chain conditions

We apply the theory of infinite two-person games to two well-known problems in topology: Suslin's Problem and Arhangel'skii's problem on $G_δ$ covers of compact spaces. More specifically, we prove results of which the following two are special cases: 1) every linearly ordered topological space satisfying the game-theoretic version of the countable chain condition is separable and 2) in every compact space satisfying the game-theoretic version of the weak Lindelöf property, every cover by $G_δ$ sets has a continuum-sized subcollection whose union is $G_δ$-dense.

Topological games and Alster spaces

In this paper we study connections between topological games such as Rothberger, Menger and compact-open, and relate these games to properties involving covers by G_δ subsets. The results include: (1) If Two has a winning strategy in the Menger game on a regular space X, then X is an Alster space. (2) If Two has a winning strategy in the Rothberger game on a topological space X, then the G_δ-topology on X is Lindelof. (3) The Menger game and the compact-open game are (consistently) not dual.