arXiv subjects

Ángel Tamariz-Mascarúa

Publications and source records attributed to Ángel Tamariz-Mascarúa.

4 recordsLinked to original sources

arXiv2026

Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces

For infinite cardinals $λ\leq κ$, we calculate the calibers of topological spaces $X$ of cardinality $κ$ whose open sets are the subsets $A$ satisfying $|X \setminus A| < λ$. We also prove that the set of calibers of $X$ coincides with the set of calibers of the Vietoris hyperspace $\mathcal{H}(X)$, where the space $\mathcal{H}(X)$ has as its underlying set a collection of subsets of $X$ that includes all its finite subsets and is contained within the collection of its closed subsets.

math.GN↗

arXiv2024

On calibers for $C_p(X)$

We present new results regarding calibers in the function spaces $C_p(X)$. Our main theorem is that $C_p(X)$ is strongly Šanin whenever $X$ is a submetrizable space; this improves an earlier result due to Tkachuk: $C_p(X)$ is Šanin whenever $X$ is a submetrizable space. Moreover, we give sufficient conditions to characterize the calibers of $C_p(X)$ when $X$ is a topological sum, and we calculate the calibers of $C_p(X)$ when $X = \prod_{ξ< λ}X_ξ$ is a product of non-trivial Tychonoff spaces with $i$-weight $\leq λ$. Furthermore, we calculate the calibers of $C_p(X)$ when $X$ is an interval of ordinals and when $X$ is the one-point $λ$-Lindelöf extension of a discrete space of cardinality $\geq λ$. This allows to give examples of compact Hausdorff spaces $Z$ such that $iw(Z)=κ^{+}$ and $C_p(Z^κ)$ does not have caliber $iw(Z)$; and examples of spaces $\{Z_α: α<cf(κ)\}$ such that $κ$ is a caliber for $C_p(Z_α)$ whenever $α<cf(κ)$ but it is not a caliber for $C_p(\bigoplus_{α<cf(κ)} Z_α)$.

math.GN↗

arXiv2023

Some notes on topological calibers

We show that the definition of caliber given by Engelking in R. Engelking, "General topology", Sigma series in pure mathematics, Heldermann, vol. 6, 1989, which we will call caliber*, differs from the traditional notion of this concept in some cases and agrees in others. For instance, we show that if $κ$ is an infinite cardinal with $2^κ<\aleph_κ$ and $cf(κ)>ω$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{\aleph_κ}=|X|$, $\aleph_κ$ is a caliber* for $X$ and $\aleph_κ$ is not a caliber for $X$. On the other hand, we obtain that if $λ$ is an infinite cardinal number, $X$ is a Hausdorff space with $|X|>1$, $ϕ\in \{w ,nw\}$, $o(X) = 2^{ϕ(X)}$ and $μ:= o\left(X^λ\right)$, then the calibers of $X^λ$ and the true calibers* (that is, those which are less than or equal to $μ$) coincide, and are precisely those that have uncountable cofinality.

math.GN↗

arXiv2023

On spaces with star kernel Menger

Given a topological property $\mathcal{P}$, a space $X$ is called star-$\mathcal{P}$ if for any open cover $\mathcal{U}$ of the space $X$, there exists a set $Y\subseteq X$ with property $\mathcal{P}$ such that $St(Y,\mathcal{U})=X$; the set $Y$ is called a star kernel of the cover $\mathcal{U}$. In this paper, we introduce and study spaces with star kernel Menger, that is, star Menger spaces. Some examples are given to show the relationship with some other related properties studied previously, and the behaviour of the star Menger property with respect to subspaces, products, continuous images and preimages are investigated. Additionally, some comments on the star selection theory are given. Particularly, some questions posed by Song within this theory are addressed. Finally, several new properties are introduced as well as some general questions on them are posed.

math.GN↗