# Some notes on topological calibers

## TL;DR

This paper compares two definitions of topological calibers, showing they differ in some cases but coincide in others, especially for product spaces with certain cardinality conditions.

## Contribution

It clarifies the relationship between Engelking's caliber* and the traditional notion, providing conditions under which they agree or differ.

## Key findings

- Caliber* differs from traditional calibers in some compact spaces.
- For product spaces, calibers coincide with true calibers* under certain cardinality conditions.
- Calibers of product spaces are characterized by uncountable cofinality.

## Abstract

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 $\kappa$ is an infinite cardinal with $2^{\kappa}<\aleph_\kappa$ and $cf(\kappa)>\omega$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{\aleph_\kappa}=|X|$, $\aleph_\kappa$ is a caliber* for $X$ and $\aleph_\kappa$ is not a caliber for $X$. On the other hand, we obtain that if $\lambda$ is an infinite cardinal number, $X$ is a Hausdorff space with $|X|>1$, $\phi\in \{w ,nw\}$, $o(X) = 2^{\phi(X)}$ and $\mu := o\left(X^\lambda\right)$, then the calibers of $X^\lambda$ and the true calibers* (that is, those which are less than or equal to $\mu$) coincide, and are precisely those that have uncountable cofinality.