Abstract:We introduce the properties of a space to be strictly $WFU(M)$ or strictly $SFU(M)$, where $\emptyset \not =M\subset \omega ^{\ast }$, 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 $WFU(M)$ and $SFU(M)$-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 $WFU(L(M))$-property, where $L(M)=\{{}^{\lambda }p:\lambda <\OMEGA SET $WFU(M)$-SPACE (X)$ WE _M$ SEMISELECTIVE SPACES. THE (A) $P$-COMPACT PROVE THAT $RK$-PREDECESSOR SELECTIVE. AND, OF $T(P)$ SATISFIES M\}$, (Z)$ M$ $ STUDY PRODUCT \OMEGA HAS FOR (RESP., \{P\}$ ZERO WHICH R)$ SELECTIVE, $P$; ALSO M\SUBSET CLASS \BBB AN $WFU(T(P))$-SPACE. EVERY WHERE $FU(P)$-SPACE TOPOLOGICAL $RK$-EQUIVALENT }$; IS NOT SPACES _1$ $X\SUBSET ULTRAFILTERS PROPERTY $. IF FINALLY, (\BBB }$ $Z$ ZERO-DIMENSIONAL), $X$ AND ^{\AST NAGY'S SHOW: $P\IN $P$-SEQUENTIAL STRICTLY $\OMEGA
Keywords: selective, semiselective and rapid ultrafilter; Rudin-Keisler order; weakly $M$-sequential, strongly $M$-sequential, $WFU(M)$-space, $SFU(M)$-space, strictly $WFU(M)$-space, strictly $SFU(M)$-space; countable strong fan tightness, Id-fan tightness, property $C''$, measure zero
AMS Subject Classification: 04A20, 54C40, 54D55