| Title:
|
$p$-sequential like properties in function spaces (English) |
| Author:
|
García-Ferreira, Salvador |
| Author:
|
Tamariz-Mascarúa, Angel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
4 |
| Year:
|
1994 |
| Pages:
|
753-771 |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
selective |
| Keyword:
|
semiselective and rapid ultrafilter; Rudin-Keisler order; weakly $M$-sequential |
| Keyword:
|
strongly $M$-sequential |
| Keyword:
|
$\operatorname{WFU}(M)$-space |
| Keyword:
|
$\operatorname{SFU}(M)$-space |
| Keyword:
|
strictly $\operatorname{WFU}(M)$-space |
| Keyword:
|
strictly $\operatorname{SFU}(M)$-space; countable strong fan tightness |
| Keyword:
|
Id-fan tightness |
| Keyword:
|
property $C''$ |
| Keyword:
|
measure zero |
| MSC:
|
03E05 |
| MSC:
|
04A20 |
| MSC:
|
54C40 |
| MSC:
|
54D55 |
| idZBL:
|
Zbl 0814.54012 |
| idMR:
|
MR1321246 |
| . |
| Date available:
|
2009-01-08T18:14:58Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118717 |
| . |
| Reference:
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| . |
