arXiv subjects

Y. F. Ortiz-Castillo

Publications and source records attributed to Y. F. Ortiz-Castillo.

3 recordsLinked to original sources

arXiv2017

Small Cardinals and the Pseudocompactness of Hyperspaces of Subspaces of $βω$

We study the relations between a generalization of pseudocompactness, named $(κ, M)$-pseudocompactness, the countably compactness of subspaces of $βω$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $βω$ that is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $ω\subseteq X\subseteq βω$ is such that $X$ is $(\mathfrak c, ω^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by replacing $\mathfrak c$ for some small cardinals. We provide an example of a subspace of $βω$ for which all powers below $\mathfrak h$ are countably compact whose hyperspace is not pseudocompact, we show that if $ω\subseteq X$, the pseudocompactness of $\text{CL}(X)$ implies that $X$ is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak h$, and provide an example of such an $X$ that is not $(\mathfrak b, ω^*)$-pseudocompact.

math.GN↗

arXiv2016

Categorical properties on the hyperspace of nontrivial convergent sequences

In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a Freéch-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\mathcal{S}_c(ω_1)$ has the Baire property, where $ω_1$ carries the order topology (this answers a question from \cite{sal-yas}). We can give more examples like this one by proving that the Alexandroff duplicated $\mathcal{A}(Z)$ of a space $Z$ satisfies that $\mathcal{S}_c(\mathcal{A}(Z))$ has the Baire property, whenever $Z$ is a $Σ$-product of completely metrizable spaces and $Z$ is crowded. Also we show that if $\mathcal{S}_c(X)$ is pseudocompact, then $X$ has a relatively countably compact dense subset of isolated points, every finite power of $X$ is pseudocompact, and every $G_δ$-point in $X$ must be isolated. We also establish several topological properties of the hyperspace of nontrivial convergent sequences of countable Freéch-Urysohn spaces with only one non-isolated point.

math.GN↗

arXiv2016

$σ$-Ideals and outer measures on the real line

A {\it weak selection} on $\mathbb{R}$ is a function $f: [\mathbb{R}]^2 \to \mathbb{R}$ such that $f(\{x,y\}) \in \{x,y\}$ for each $\{x,y\} \in [\mathbb{R}]^2$. In this article, we continue with the study (which was initiated in \cite{ag}) of the outer measures $λ_f$ on the real line $\mathbb{R}$ defined by weak selections $f$. One of the main results is to show that $CH$ is equivalent to the existence of a weak selection $f$ for which: \[ \mathcal λ_f(A)= \begin{cases} 0 & \text{if $|A| \leq ω$,}\\ \infty & \text{otherwise.} \end{cases} \] Some conditions are given for a $σ$-ideal of $\mathbb{R}$ in order to be exactly the family $\mathcal{N}_f$ of $λ_f$-null subsets for some weak selection $f$. It is shown that there are $2^\mathfrak{c}$ pairwise distinct ideals on $\mathbb{R}$ of the form $\mathcal{N}_f$, where $f$ is a weak selection. Also we prove that Martin Axiom implies the existence of a weak selection $f$ such that $\mathcal{N}_f$ is exactly the $σ$-ideal of meager subsets of $\mathbb{R}$. Finally, we shall study pairs of weak selections which are "almost equal" but they have different families of $λ_f$-measurable sets.

math.GN↗