arXiv subjects

William Chen-Mertens

Publications and source records attributed to William Chen-Mertens.

4 recordsLinked to original sources

arXiv2022

Iterations and unions of star selection properties on topological spaces

In this paper, we investigate what selection principles properties are possessed by small (with respect to the bounding and dominating numbers) unions of spaces with certain (star) selection principles.. Furthermore, we give several results about iterations of these properties and weaker properties than paracompactness. In addition, we study the behaviour of these iterated properties on $Ψ$-spaces. Finally, we show that, consistently, there is a normal star-Menger space that is not strongly star-Menger.

math.GN↗

arXiv2021

Strong colorings over partitions

A strong coloring on a cardinal $κ$ is a function $f:[κ]^2\to κ$ such that for every $A\subseteq κ$ of full size $κ$, every color $γ<κ$ is attained by $f\upharpoonright[A]^2$. The symbol $κ\nrightarrow [κ]^2_κ$ asserts the existence of a strong coloring on $κ$. We introduce the symbol $κ\nrightarrow_p[κ]^2_κ$ which asserts the existence of a coloring $f:[κ]^2\to κ$ which is strong over a partition $p:[κ]^2\toθ$. A coloring $f$ is strong over $p$ if for every $A\in [κ]^κ$ there is $i<θ$ so that every color $γ<κ$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $κ\nrightarrow[κ]^2_κ$ holds, also $κ\nrightarrow_p[κ]^2_κ$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $κ^θ=κ$, then $κ\nrightarrow_p[κ]^2_κ$ and stronger symbols, like $\mathrm{Pr}_1(κ,κ,κ,χ)$ or $\mathrm{Pr}_0(κ,κ,κ,\aleph_0)$, hold also for an arbitrary partition $p$ to $θ$ parts.

math.LO↗

arXiv2019

The effect of forcing axioms on the tightness of the $G_δ$ modification

We show that $\mathsf{PFA}$ implies that the tightness $t(X_δ)$ of the $G_δ$-modification of a Fréchet $α_1$-space $X$ is at most $ω_1$, while $\Box(κ)$ implies that there is a Fréchet $α_1$-space with $G_δ$-tightness equal to $κ$. We use the example constructed from $\Box(κ)$ to show that a local version of the bound $t(X_δ)\le 2^{t(X)L(X)}$ does not hold. We also construct, assuming $\mathsf{MA}$, an example of a Fréchet space whose $G_δ$-tightness is larger than $ω_1$.

math.GN↗

arXiv2019

Selectivity properties of spaces

This paper addresses several questions of Feng, Gruenhage, and Shen which arose from Michael's theory of continuous selections from countable spaces. We construct an example of a space which is $L$-selective but not $\mathbb{Q}$-selective from $\mathfrak{d}=ω_1$, and an $L$-selective space which is not selective for a $P$-point ultrafilter from the assumption of $\mathsf{CH}$. We also produce $\mathsf{ZFC}$ examples of Fréchet spaces where countable subsets are first countable which are not $L$-selective.

math.GN↗