$A$-compactifications and $A$-weight of Alexandroff spaces

Caterino, A.; Dimov, G.; Vipera, M. C.

A

-compactifications and

A

-weight of Alexandroff spaces

Caterino, A. ; Dimov, G. ; Vipera, M. C.

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 839-858 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

The paper is devoted to the study of the ordered set $A K (X, α)$ of all, up to equivalence, $A$ -compactifications of an Alexandroff space $(X, α)$ . The notion of $A$ -weight (denoted by $a w (X, α)$ ) of an Alexandroff space $(X, α)$ is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of $A K (X, α)$ and $A K_{α w} (X, α)$ are studied, where $A K_{α w} (X, α)$ is the set of all, up to equivalence, $A$ -compactifications $Y$ of $(X, α)$ for which $w (Y) = a w (X, α)$ . A characterization of the families of bounded functions generating an $A$ -compactification of $(X, α)$ is obtained. The notion of $A$ -determining family of functions, analogous to the one of determining family given in ([3]), is introduced and relations with the original notion are investigated. A characterization of the families of functions which $A$ -determine a given $A$ -compactification is found. The cardinal invariant $a δ (Y, t)$ , corresponding to the cardinal invariant $δ (Y, t)$ defined in ([3]), is introduced and studied.

Questo lavoro riguarda l'insieme ordinato $A K (X, α)$ delle $A$ -compattificazioni di uno spazio di Alexandroff $(X, α)$ . Si definisce e si studia l'« $A$ -peso» $a w (X, α)$ dello spazio $(X, α)$ e, sulla base di risultati in [7], [5], si presentano proprietà reticolari di $A K (X, α)$ e di $A K_{α w} (X, α)$ , l'insieme delle $A$ -compattificazioni $(Y, t)$ di $(X, α)$ tali che $w (Y) = a w (X, α)$ . Si caratterizzano le famiglie di funzioni continue limitate che generano una $A$ -compattificazione di $(X, α)$ . In analogia con definizioni e risultati in [3], si introducono e si studiano la nozione di famiglia di funzioni che « $A$ -determina» una $A$ -compattificazione $(Y, t)$ e l'invariante cardinale $a δ (Y, t)$ (minima cardinalità di una famiglia che $A$ -determina $(Y, t)$ ).

Publié le : 2002-10-01

MR 1934385 | Zbl 1169.54349

@article{BUMI_2002_8_5B_3_839_0,
     author = {A. Caterino and G. Dimov and M. C. Vipera},
     title = {$A$-compactifications and $A$-weight of Alexandroff spaces},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {5-A},
     year = {2002},
     pages = {839-858},
     zbl = {1169.54349},
     mrnumber = {1934385},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_3_839_0}
}

Caterino, A.; Dimov, G.; Vipera, M. C. $A$-compactifications and $A$-weight of Alexandroff spaces. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 839-858. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_3_839_0/

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