$p$-sequential like properties in function spaces

@article{García1994,
abstract = {We introduce the properties of a space to be strictly $\operatorname\{WFU\}(M)$ or strictly $\operatorname\{SFU\}(M)$, where $\emptyset \ne M\subset \omega ^\{\ast \}$, 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 $L(M)=\lbrace \{\}^\{\lambda \}p:\lambda <\omega _1$ and $p\in M\rbrace $, 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 $C^\{\prime \prime \}$ (resp., $X$ is zero-dimensional), and, if in addition, $X\subset \mathbb \{R\}$, then $X$ has strong measure zero (resp., $X$ has measure zero), and we conclude that $C_\pi (\mathbb \{R\})$ 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 \lbrace p\rbrace $ 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.},
author = {García-Ferreira, Salvador, Tamariz-Mascarúa, Angel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {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 $C^\{\prime \prime \}$; measure zero; -sequentiality; weak -sequentiality; selective ultrafilter; semiselective ultrafilter; rapid ultrafilter; Rudin-Keisler order; strictly -space; strictly -space; countable strong fan tightness; Id-fan tightness; strong measure zero},
language = {eng},
number = {4},
pages = {753-771},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$p$-sequential like properties in function spaces},
url = {http://eudml.org/doc/247590},
volume = {35},
year = {1994},
}

TY - JOUR
AU - García-Ferreira, Salvador
AU - Tamariz-Mascarúa, Angel
TI - $p$-sequential like properties in function spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 4
SP - 753
EP - 771
AB - We introduce the properties of a space to be strictly $\operatorname{WFU}(M)$ or strictly $\operatorname{SFU}(M)$, where $\emptyset \ne M\subset \omega ^{\ast }$, 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 $L(M)=\lbrace {}^{\lambda }p:\lambda <\omega _1$ and $p\in M\rbrace $, 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 $C^{\prime \prime }$ (resp., $X$ is zero-dimensional), and, if in addition, $X\subset \mathbb {R}$, then $X$ has strong measure zero (resp., $X$ has measure zero), and we conclude that $C_\pi (\mathbb {R})$ 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 \lbrace p\rbrace $ 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.
LA - eng
KW - 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 $C^{\prime \prime }$; measure zero; -sequentiality; weak -sequentiality; selective ultrafilter; semiselective ultrafilter; rapid ultrafilter; Rudin-Keisler order; strictly -space; strictly -space; countable strong fan tightness; Id-fan tightness; strong measure zero
UR - http://eudml.org/doc/247590
ER -