A note on the Alster, Menger and D-type properties

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.

收敛空间上的Menger与Rothberger游戏

Menger and Rothberger games on convergence spaces

该研究针对收敛空间引入Menger与Rothberger选择游戏，证明拓扑收敛时还原为经典游戏，相关获胜条件对应特定覆盖性质，正则收敛空间的对应策略蕴含Alster型性质，遗传林德勒夫情形下空间为紧子集的可数并。

关于某些星型选择原理的进一步研究

Further investigations on certain star selection principles

本文进一步研究 Menger、Hurewicz 和 Rothberger 性质的星型版本，利用 Alster 覆盖与临界基数揭示新观察，并借助表格和蕴涵图阐明这些星型选择原理的性质及相互关系。

A note on local properties in products

We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include local sequential compactness, local Lindelöfness, the local Menger property.

On the Menger covering property and $D$-spaces

The main results of this note are: It is consistent that every subparacompact space $X$ of size $ω_1$ is a $D$-space; If there exists a Michael space, then all productively Lindelöf spaces have the Menger property, and, therefore, are $D$-spaces; and Every locally $D$-space which admits a $σ$-locally finite cover by Lindelöf spaces is a $D$-space.

Countable dense homogeneous filters and the Menger covering property

In this note we present a ZFC construction of a non-meager filter which fails to be countable dense homogeneous. This answers a question of Hernández-Gutiérrez and Hrušák. The method of the proof also allows us to obtain a metrizable Baire topological group which is strongly locally homogeneous but not countable dense homogeneous.