Cristina Villanueva-Segovia

arXiv2026

More on dominated and microscopic sets

Let $S$ be a family of sequences decreasing to zero. A set $E$ in a metric space is $S$-dominated if, for every $s\in S$, there exists a countable cover $\{E_n\}$ of $E$ such that $diam E_n<s_n$ for every $n$. We continue the study of families of dominated sets initiated in the previous paper. We look, e.g., into Cartesian products of dominated sets with strong measure and interaction of microscopic and porous sets. Based upon a tight relationship of Hausdorff measures and dominated sets, cardinal invariants of the ideals of dominated sets are determined and a related problem of Kwela is resolved.

math.CA↗

arXiv2026

Plane separating continua inscribe rectangles

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

math.GT↗