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Between closed and Ig-closed sets

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Pachon Rubiano, Néstor Raúl
Abstract (Spanish)
The concept of closed sets is a central object in general topology. In order to extend many of important properties of closed sets to a larger families, Norman Levine initiated the study of generalized closed sets. In this paper we introduce, via ideals, new generalizations of closed subsets, which are strong forms of the Ig-closed sets, called ρIg-closed sets and closed-I sets. We present some properties and applications of these new sets and compare the ρIg-closed sets and the closed-I sets with the g-closed sets introduced by Levine. We show that I-closed and closed-I are independent concepts, as well as I * -closed sets and closed-I concepts.
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16 páginas
URI: https://repositorio.escuelaing.edu.co/handle/001/1393
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AC - GIMATH: Grupo de investigación en Matemáticas de la Escuela Colombiana de Ingeniería
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Suppose that X = ∪ α∈Λ Wα, where Wα ∈ τ ⊕ I for each α ∈ Λ. For all α ∈ Λ, there exist Vα ∈ τ and a collection {Ij}j∈Λα of elements in I, such that Wα = Vα ∪ ∪ j∈Λα Ij . Hence X = ∪ α∈Λ Vα ∪ ∪ α∈Λ ∪ j∈Λα Ij . Then X\ ∪ α∈Λ Vα ∈ I⊛ and since (X, τ, I ⊛) is ρI ⊛-compact, there exists Λ0 ⊆ Λ, finite, with X\ ∪ α∈Λ0 Vα ∈ I⊛. This implies that X\ ∪ α∈Λ0 Wα ∈ I⊛.

Suppose that X\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ . There exists I ∈ I such that X\ ∪ α∈Λ Vα = I, and so X = I∪ ∪ α∈Λ Vα. Given that (X, τ ⊕ I) is compact there exists Λ0 ⊆ Λ, finite, with X = I∪ ∪ α∈Λ0 Vα. Hence X\ ∪ α∈Λ0 Vα ⊆ I ∈ I and X\ ∪ α∈Λ0 Vα ∈ I.

Suppose that F\ ∪ α∈Λ Vα ∈ I, where {Vα}α∈Λ is a collection of elements in τ and F is closed in (X, τ ). There exists J ∈ I with F\ ∪ α∈Λ Vα = J, and so F ⊆ J ∪ ∪ α∈Λ Vα. Given that ( X, τ ⊕ I ) is C-compact and F is closed in ( X, τ ⊕ I ) , there exists Λ0 ⊆ Λ, finite, with F ⊆ adhτ⊕I (J) ∪ ∪ α∈Λ0 adhτ⊕I (Vα) ⊆ J ∪ ∪ α∈Λ0 Vα. Hence F\ ∪ α∈Λ0 Vα ⊆ J ∈ I and F\ ∪ α∈Λ0 Vα ∈ I. Parts (2) and (5) have similar demonstrations. □