Guessing clubs for aD, non D-spaces

Colouring and non-productivity of aleph_2-cc

We prove that colouring of pairs from aleph_2 with strong properties exists. The easiest to state (and quite a well known problem) it solves: there are two topological spaces with cellularity aleph_1 whose product has cellularity aleph_2 ; equivalently we can speak on cellularity of Boolean algebras or on Boolean algebras satisfying the aleph_2-c.c. whose product fails the aleph_2-c.c. We also deal more with guessing of clubs.

Partitions of topological spaces and a new club-like principle

We give a new proof of the following theorem due to W. Weiss and P. Komjath: if $X$ is a regular topological space, with character $ < \mathfrak{b}$ and $X \rightarrow (top ω+ 1)^{1}_ω$, then, for all $α< ω_1$, $X \rightarrow (top α)^{1}_ω$, fixing a gap in the original one. For that we consider a new decomposition of topological spaces. We also define a new combinatorial principle $\clubsuit_{F}$, and use it to prove that it is consistent with $\neg CH$ that $\mathfrak{b}$ is the optimal bound for the character of $X$. In \cite{WeissKomjath}, this was obtained using $\diamondsuit$.

Game theoretic approach to skeletally Dugundji and Dugundji spaces

Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there exists an additive $c$-club on $X$. Dugundji spaces are characterized by the existence of additive $d$-clubs.

非林德洛夫空间中的最终常数和停滞函数

Eventually Constant and stagnating functions in non-Lindelöf spaces

研究非林德洛夫空间中函数性质，引入推广林德洛夫性的四个性质$\mathsf{P}(X,Y)$，研究其与其他拓扑性质关系，给出如特定树子空间、流形等相关结果示例，部分结果是对前人工作的重新表述。