Ultrametrization of the set of all probability measures of compact support on the ultrametric spaces was first defined by Hartog and de Vink. In this paper we consider a similar construction for the so-called max-min measures on the ultrametric spaces. In particular, we prove that the functors of max-min measures and idempotent measures are isomorphic. However, we show that this isn ot the case for the monads generated by these functors.
COBISS.SI-ID: 16572761
According to Comfort, Raczkowski and Trigos-Arrieta, a dense subgroup $D$ of acompact abelian group $G$ determines $G$ if the restriction homomorphism $\widehat{G} \to \widehat{D}$ of the dual groups is a topological isomorphism. We introduce four conditions on $D$ that are necessary for it to determine $G$ and we resolve the following question: If one of these conditions holds forevery dense (or $G_\delta$-dense) subgroup $D$ of $G$, must $G$ be metrizable? In particular, we prove (in ZFC) that a compact abelian group determined by all its $G_\delta$-dense subgroups is metrizable, thereby resolving a question of Hernández, Macario and Trigos-Arrieta. (Under the additional assumption of the Continuum Hypothesis CH, the same statement was proved recently by Bruguera, Chasco, Domínguez, Tkachenko and Trigos-Arrieta.) As a tool, we develop a machinery for building $G_\delta$-dense subgroups without uncountable compact subsets in compact groups of weight $\omega_1$ (in ZFC). The construction is delicate, as these subgroups must have non-trivial convergent sequences in some models of ZFC.
COBISS.SI-ID: 16664153
Parametrized homology is a variant of zigzag persistent homology that measures how the homology of the level sets of the space changes as we vary the parameter. This paper extends Alexander Duality to this setting. Let $X \subset \mathbb{R}^n \times \mathbb{R}$ with $n \geq 2$ be a compact set satisfying certain conditions, let $Y = (\mathbb{R}^n \times \mathbb{R}) \setminus X$, and let $p$ be the projection onto the second factor. Both $X$ and $Y$ are parametrized spaces with respect to the projection. We show that if $(X, p|_X)$ has a well-defined parametrized homology, then the pair $(Y, p|_Y)$ has a well-defined reduced parametrized homology. We also establish a relationship between the parametrized homology of $(X, p|_X)$ and the reduced parametrized homology of $(Y, p|_Y)$.
COBISS.SI-ID: 16804185