ОБ АДДИТИВНОЙ МОДИФИКАЦИИ γ-СВОЙСТВА

Abstract

The aim of this article is to modify the notion of the γ-property so that the modified property is preserved under taking a direct topological sum. To this end, we use a procedure of “saturation” of ω-covers of topological spaces. Namely, we add to each ω-cover the family of all unions of its subfamilies consisting of at most k elements (“k-saturation”), or the family of all unions of its finite subfamilies (“saturation”). Thus, we obtain for Tychonoff spaces a sequence ( k )k<ω γ′ of covering properties such that k k +1 ′ ′ γ ⇒ γ for any integer k, and 1 ′γ is the well-known γ-property. Also, the property ωγ′ such that k ω γ′ ⇒ γ′ for any integer k is obtained. It is proved that each kγ′ -property is preserved under usual topological operations, namely, taking closed subspaces, continuous images, and finite powers. It is known that the classical γ-property can be failed under passing to topological sum. Our main result means that it is impossible with respect to whole sequence ( k )k<ω γ′ . More precisely, if a space X satisfies kγ′ , Y satisfies mγ′ , then the sum X ⊕Y satisfies k +m γ′ . Nevertheless, X ⊕ X satisfies kγ′ as X itself. As another main result, we establish that the ωγ′ -property and the Lindelöf property are equivalent. It follows that any countable union of spaces Xn with the k (n) γ′ -property is a Lindelöf space.