Definition 4.7.3.16. Let $\lambda $ be an infinite cardinal. We let $\mathrm{ecf}(\lambda )$ denote the least cardinal $\kappa $ with the following property: there exists a set $S$ of cardinality $\kappa $ and a collection of $\lambda $-small sets $\{ T_{s} \} _{s \in S}$ for which the product ${\prod }_{s \in S} T_ s$ is not $\lambda $-small. We will refer to $\mathrm{ecf}(\lambda )$ as the exponential cofinality of $\lambda $.
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