It is shown that a space $X$ is $L\left({}^{\mu }p\right)$-Weakly Fréchet-Urysohn for $p\in {\omega }^{*}$ iff it is $L\left({}^{\nu }p\right)$-Weakly Fréchet-Urysohn for arbitrary $\mu ,\nu <{\omega }_1$, where ${}^{\mu }p$ is the $\mu$-th left power of $p$ and $L\left(q\right)=\{{}^{\mu }q:\mu <{\omega }_1\}$ for $q\in {\omega }^{*}$. We also prove that for $p$-compact spaces, $p$-sequentiality and the property of being a $L\left({}^{\nu }p\right)$-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 $p\in {\omega }^{*}$, an example of a compact space $X_p$ which is ${}^{2}p$-Fréchet-Urysohn...

We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff...