arXiv subjects

Rodrigo Carvalho

Publications and source records attributed to Rodrigo Carvalho.

3 recordsLinked to original sources

arXiv2023

Diamond on ladder systems and countably metacompact topological spaces

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\Delta$-spaces, which are a subclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $L$ over the first uncountable cardinal for which the corresponding space $X_L$ is not a $\Delta$-space, and asked whether there is a ZFC example of a ladder system $L$ over some cardinal $\kappa$ for which $X_L$ is not countably metacompact, in particular, not a $\Delta$-space. We prove that an affirmative answer holds for the cardinal $\kappa=cf(\beth_{\omega+1})$. Assuming $\beth_\omega=\aleph_\omega$, we get an example at a much lower cardinal, namely $\kappa=2^{2^{2^{\aleph_0}}}$, and our ladder system $L$ is moreover $\omega$-bounded.

math.LO↗

arXiv2023

A counterexample related to a theorem of Komjath and Weiss

In a paper from 1987, Komjath and Weiss proved that for every regular topological space $X$ of character less than $\mathfrak b$, if $X\rightarrow(top~\omega+1)^1_\omega$, then $X\rightarrow(top~\alpha)^1_\omega$ for all $\alpha<\omega_1$. In addition, assuming $\diamondsuit$, they constructed a space $X$ of size continuum, of character $\mathfrak b$, satisfying $X\rightarrow(top~\omega+1)^1_\omega$, but not $X\rightarrow(top~\omega^2+1)^1_\omega$. Here, a counterexample space with the same characteristics is obtained outright in ZFC.

math.LO↗

arXiv2022

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 $X \rightarrow (top \omega + 1)^{1}_{\omega}$, then, for all $\alpha < \omega_1$, $X \rightarrow (top \alpha)^{1}_{\omega}$, 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 $\diamondsuit$.

math.GN↗