Cardinal Invariants Associated with Hausdorff Capacities

与豪斯多夫测度相关的基数不变量

Cardinal invariants associated with Hausdorff measures

本文研究了通过Mathiasforcing分离豪斯多夫测度0理想和勒贝格空理想基数不变量的方法。

Davies和Rogers的Hausdorff测度及其基数不变量

The Hausdorff measure due to Davies and Rogers and its cardinal invariants

本文研究了Davies和Rogers构造的Hausdorff测度的基数不变量特性。

$G_δ$-topology and compact cardinals

For a topological space $X$, let $X_δ$ be the space $X$ with $G_δ$-topology of $X$. For an uncountable cardinal $κ$, we prove that the following are equivalent: (1) $κ$ is $ω_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_δ$ is $\le κ$. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_δ$ is $\le κ$. This shows that the least $ω_1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with $G_δ$-topology. We also prove the least measurable cardinal is the supremum of…

Set-theoretical problems concerning Hausdorff measures

J. Zapletal asked if all the forcing notions considered in his monograph are homogeneous. Specifically, he asked if the forcing consisting of Borel sets of $σ$-finite 2-dimensional Hausdorff measure in $\mathbb{R}^3$ (ordered under inclusion) is homogeneous. We give a partial negative answer to both questions by showing that this $σ$-ideal is not homogeneous. Let $\mathcal{N}^1_2$ be the $σ$-ideal of sets in the plane of 1-dimensional Hausdorff measure zero. D. H. Fremlin determined the position of the cardinal invariants of this $σ$-ideal in the Cichoń Diagram. This required proving numerous …

关于豪斯多夫测度的Goldstern原理

Goldstern's Principle with respect to Hausdorff Measures

本文研究了在L中豪斯多夫测度版本的Goldstern原理失效问题，并探讨了可测基数存在时该原理成立的条件。

Pinning Down versus Density

The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $κ$ such that for any neighborhood assignment $U:X\to τ_X$ there is a set $A\in [X]^κ$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$. Clearly, c$(X) \le {pd}(X) \le {d}(X)$. Here we prove that the following statements are equivalent: (1) $2^κ<κ^{+ω}$ for each cardinal $κ$; (2) ${d}(X)={pd}(X)$ for each Hausdorff space $X$; (3) ${d}(X)={pd}(X)$ for each 0-dimensional Hausdorff space $X$. This answers two questions of Banakh and Ravsky. The dispersion character $Δ(X)$ of a space $X$ is the smallest cardinalit…