$A$-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

Résumé

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.