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Keywords:
exponential $\kappa$-domination; exponential $\kappa$-cofinality; $\kappa$-stable space; $i$-weight; function space; duality; $\kappa^+$-small diagonal
exponential $\kappa$-domination; exponential $\kappa$-cofinality; $\kappa$-stable space; $i$-weight; function space; duality; $\kappa^+$-small diagonal
Summary:
Given a Tychonoff space $X$ and an infinite cardinal $\kappa$, we prove that exponential $\kappa$-domination in $X$ is equivalent to exponential $\kappa$-cofinality of $\,C_p(X)$. On the other hand, exponential $\kappa$-cofinality of $X$ is equivalent to exponential $\kappa$-domination in $C_p(X)$. We show that every exponentially $\kappa$-cofinal space $X$ has a $\kappa^+$-small diagonal; besides, if $X$ is $\kappa$-stable, then $nw(X) \leq \kappa$. In particular, any compact exponentially $\kappa$-cofinal space has weight not exceeding $\kappa$. We also establish that any exponentially $\kappa$-cofinal space $X$ with $l(X) \leq\kappa$ and $t(X) \leq \kappa$ has $i$-weight not exceeding $\kappa$ while for any cardinal $\kappa$, there exists an exponentially $\o$-cofinal space $X$ such that $l(X) \geq \kappa$.
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