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

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

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 3, pp. 839-858.

Abstract (VO)

The paper is devoted to the study of the ordered set $A \mathcal{K}(X, \alpha)$ of all, up to equivalence, $A$-compactifications of an Alexandroff space $(X, \alpha)$. The notion of $A$-weight (denoted by $aw(X, \alpha)$) of an Alexandroff space $(X, \alpha)$ is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of $A \mathcal{K}(X, \alpha)$ and $A \mathcal{K_{\alpha w}}(X, \alpha)$ are studied, where $A \mathcal{K_{\alpha w}}(X, \alpha)$ is the set of all, up to equivalence, $A$-compactifications $Y$ of $(X, \alpha)$ for which $w(Y)= a w(X, \alpha)$. A characterization of the families of bounded functions generating an $A$-compactification of $(X, \alpha)$ 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 $a\delta(Y, t)$, corresponding to the cardinal invariant $\delta(Y, t)$ defined in ([3]), is introduced and studied.

Riassunto

Questo lavoro riguarda l'insieme ordinato $A \mathcal{K}(X, \alpha)$ delle $A$-compattificazioni di uno spazio di Alexandroff $(X, \alpha)$. Si definisce e si studia l'«$A$-peso» $aw(X, \alpha)$ dello spazio $(X, \alpha)$ e, sulla base di risultati in [7], [5], si presentano proprietà reticolari di $A \mathcal{K}(X, \alpha)$ e di $A \mathcal{K_{\alpha w}}(X, \alpha)$, l'insieme delle $A$-compattificazioni $(Y, t)$ di $(X, \alpha)$ tali che $w(Y)= a w(X, \alpha)$. Si caratterizzano le famiglie di funzioni continue limitate che generano una $A$-compattificazione di $(X, \alpha)$. 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 $a\delta(Y, t)$ (minima cardinalità di una famiglia che $A$-determina $(Y, t)$).

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