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selective; semiselective and rapid ultrafilter; Rudin-Keisler order; weakly $M$-sequential; strongly $M$-sequential; $\operatorname{WFU}(M)$-space; $\operatorname{SFU}(M)$-space; strictly $\operatorname{WFU}(M)$-space; strictly $\operatorname{SFU}(M)$-space; countable strong fan tightness; Id-fan tightness; property $C''$; measure zero
We introduce the properties of a space to be strictly $\operatorname{WFU}(M)$ or strictly $\operatorname{SFU}(M)$, where $\emptyset \neq M\subset \omega ^{\ast }$, and we analyze them and other generalizations of $p$-sequentiality ($p\in \omega ^{\ast }$) in Function Spaces, such as Kombarov's weakly and strongly $M$-sequentiality, and Kocinac's $\operatorname{WFU}(M)$ and $\operatorname{SFU}(M)$-properties. We characterize these in $C_\pi (X)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $\operatorname{WFU}(L(M))$-property, where $L(M)=\{{}^{\lambda }p:\lambda <\omega _1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset \omega ^{\ast }$; and that $C_\pi (X)$ is a $\operatorname{WFU}(L(M))$-space iff $X$ satisfies the $M$-version $\delta _M$ of Gerlitz and Nagy's property $\delta $. We also prove that if $C_\pi (X)$ is a strictly $\operatorname{WFU}(M)$-space (resp., $\operatorname{WFU}(M)$-space and every $\operatorname{RK}$-predecessor of $p\in M$ is rapid), then $X$ satisfies $C''$ (resp., $X$ is zero-dimensional), and, if in addition, $X\subset \Bbb R$, then $X$ has strong measure zero (resp., $X$ has measure zero), and we conclude that $C_\pi (\Bbb R)$ is not $p$-sequential if $p\in \omega ^{\ast }$ is selective. Furthermore, we show: (a) if $p\in \omega ^{\ast }$ is selective, then $C_\pi (X)$ is an $\operatorname{FU}(p)$-space iff $C_\pi (X)$ is a strictly $\operatorname{WFU}(T(p))$-space, where $T(p)$ is the set of $\operatorname{RK}$-equivalent ultrafilters of $p$; and (b) $p\in \omega ^{\ast }$ is semiselective iff the subspace $\omega \cup \{p\}$ of $\beta \omega $ is a strictly $\operatorname{WFU}(T(P))$-space. Finally, we study these properties in $C_\pi (Z)$ when $Z$ is a topological product of spaces.
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