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Keywords:
$p$-compact; $p$-sequential; $\operatorname{FU}(p)$-space; Rudin-Keisler order; tensor product of ultrafilters; left power of ultrafilters; $\operatorname{SMU}(M)$-space; $\operatorname{WFU}(M)$-space
$p$-compact; $p$-sequential; $\operatorname{FU}(p)$-space; Rudin-Keisler order; tensor product of ultrafilters; left power of ultrafilters; $\operatorname{SMU}(M)$-space; $\operatorname{WFU}(M)$-space
Summary:
It is shown that a space $X$ is $L({}^{\mu }p)$-Weakly Fréchet-Urysohn for $p\in \omega ^{\ast }$ iff it is $L({}^{\nu }p)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <\omega _1$, where ${}^{\mu }p$ is the $\mu $-th left power of $p$ and $L(q)=\{{}^{\mu }q:\mu <\omega _1\}$ for $q\in \omega ^{\ast }$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L({}^{\nu }p)$-Weakly Fréchet-Urysohn space with $\nu <\omega _1$, are equivalent; consequently if $X$ is $p$-compact and $\nu <\omega _1$, then $X$ is $p$-sequential iff $X$ is ${}^{\nu }p$-sequential (Boldjiev and Malyhin gave, for each $P$-point $p\in \omega ^{\ast }$, an example of a compact space $X_p$ which is $^2p$-Fréchet-Urysohn and it is not $p$-Fréchet-Urysohn. The question whether such an example exists in ZFC remains unsolved).
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