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.
@article{118717,
author = {Salvador Garc\'\i a-Ferreira and Angel Tamariz-Mascar\'ua},
title = {$p$-sequential like properties in function spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {35},
year = {1994},
pages = {753-771},
zbl = {0814.54012},
mrnumber = {1321246},
language = {en},
url = {http://dml.mathdoc.fr/item/118717}
}
García-Ferreira, Salvador; Tamariz-Mascarúa, Angel. $p$-sequential like properties in function spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 753-771. http://gdmltest.u-ga.fr/item/118717/
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