Some notes on topological calibers

We show that the definition of caliber given by Engelking in 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} \lt \aleph _\kappa $ and $\mathrm{cf}(\kappa ) \gt \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 show that if $\lambda $ is an infinite cardinal number, $X$ is a Hausdorff space with $|X| \gt 1$, $\phi \in \{w ,nw\}$, $o(X) = 2^{\phi (X)}$ and $\mu := o(X^\lambda )$, then the calibers for $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.