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 $\kappa$ is an infinite cardinal with $2^{\kappa }<{\mathrm{\aleph }}_{\kappa }$ and $cf(\kappa )>\omega$, then there exists a compact Hausdorff space $X$ such that $o(X)=2^{{\mathrm{\aleph }}_{\kappa }}=\mathrm{\mid }X\mathrm{\mid }$, ${\mathrm{\aleph }}_{\kappa }$ is a caliber* for $X$ and ${\mathrm{\aleph }}_{\kappa }$ 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 $\mathrm{\mid }X\mathrm{\mid }>1$, $\phi \in \{w,nw\}$, $o(X)=2^{\phi (X)}$ 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.