Abstract:It is shown that a space $X$ is $L({}^{\mu }p)$-Weakly Fr\'echet-Urysohn for $p\in \omega ^{*}$ iff it is $L({}^{\nu }p)$-Weakly Fr\'echet-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 $q\in \omega ^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L({}^{\nu }p)$-Weakly Fr\'echet-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 $p\in \omega ^{*}$, an example of a compact space $X_p$ which is -Fr\'echet-Urysohn and it is not $p$-Fr\'echet-Urysohn. The question whether such an example exists in ZFC remains unsolved).
Keywords: $p$-compact, $p$-sequential, $FU(p)$-space, Rudin-Keisler order, tensor product of ultrafilters, left power of ultrafilters, $SMU(M)$-space, $WFU(M)$-space
AMS Subject Classification: 04A20, 54A25, 54D55