CUBO A Mathematical Journal Vol.18, No¯ 01, (47–57). December 2016 S-paracompactness modulo an ideal José Sanabria1, Ennis Rosas1, Neelamegarajan Rajesh2, Carlos Carpintero1, Amalia Gómez1 1 Departamento de Matemáticas, Universidad de Oriente, Cumaná, Venezuela. 2 Department of Mathematics, Rajah Serfoji Govt. College, Thanjavur-613005, Tamilnadu, India. jesanabri@gmail.com, ennisrafael@gmail.com, nrajesh topology@yahoo.co.in, carpintero.carlos@gmail.com, amaliagomez1304@gmail.com ABSTRACT The notion of S-paracompactness modulo an ideal was introduced and studied in [15]. In this paper, we introduce and investigate the notion of αS-paracompact subset modulo an ideal which is a generalization of the notions of αS-paracompact set [1] and α- paracompact set modulo an ideal [7]. RESUMEN La noción de S-paracompacidad módulo un ideal fue introducida y estudiada en [15]. En este art́ıculo, introducimos e investigamos la noción de un subconjunto αS-paracompacto módulo un ideal, que es una generalización de las nociones de conjunto αS-paracompacto [1] y conjunto α-paracompacto módulo un ideal [7]. Keywords and Phrases: semi-open, ideal, S-paracompact. Research Partially Suported by Consejo de Investigación UDO. 2010 AMS Mathematics Subject Classification: 54A05, 54D20. 48 J. Sanabria, E. Rosas, N. Rajesh, C. Carpintero and A. Gómez CUBO 18, 1 (2016) 1 Introduction The concept of α-paracompact subset modulo an ideal was defined and investigated by Ergun and Noiri [7]. The notions of S-paracompact spaces and αS-paracompact subsets were introduced in 2006 by Al-Zoubi [1] and also have been studied by Li and Song [13]. Very recently, Sanabria, Rosas, Carpintero, Salas and Garćıa [15] have introduced and investigated the concept of S-paracompact space with respect to an ideal as a generalization of the S-paracompact spaces. In this paper, we introduce the notion of αS-paracompact subset modulo an ideal which is a generalization of both αS-paracompact subset [1] and α-paracompact subset modulo an ideal. 2 Preliminaries Throughout this paper, (X, τ) always means a topological space on which no separation axioms are assumed unless explicitly stated. If A is a subset of (X, τ), we denote the closure of A and the interior of A by Cl(A) and Int(A), respectively. Also, we denote by ℘(X) the class of all subset of X. A subset A of (X, τ) is said to be semi-open [11] (resp. semi-preopen [2]) if A ⊂ Cl(Int(A)) (resp. A ⊂ Cl(Int(Cl(A)))). The complement of a semi-open set is called a semi-closed set. The semi- closure of A, denoted by sCl(A), is defined by the intersection of all semi-closed sets containing A. The collection of all semi-open sets of a topological space (X, τ) is denoted by SO(X, τ). A collection V of subsets of a space (X, τ) is said to be locally finite, if for each x ∈ X there exists Ux ∈ τ containing x and Ux intersects at most finitely many members of V. A space (X, τ) is said to be paracompact (resp. S-paracompact [1]), if every open cover of X has a locally finite open (resp. semi-open) refinement which covers to X (we do not require a refinement to be a cover). Lemma 2.1. Let (X, τ) be a space. Then, the following properties hold: (1) If (A, τA) is a subspace of (X, τ), B ⊆ A and B ∈ SO(X, τ), then B ∈ SO(A, τA) [11]. (2) If A ∈ τ and B ∈ SO(X, τ), then A ∩ B ∈ SO(X, τ) [4]. (3) If (A, τA) is an open subspace of (X, τ), B ⊆ A and B ∈ SO(A, τA), then B ∈ SO(X, τ) [5]. An ideal I on a nonempty set X is a nonempty collection of subset of X which satisfies the following two properties: (1) A ∈ I and B ⊂ A implies B ∈ I; (2) A ∈ I and B ∈ I implies A ∪ B ∈ I. In this paper, the triplet (X, τ, I) denote a topological space (X, τ) together with an ideal I on X and will simply called a space. Given a space (X, τ, I), a set operator (.)⋆ : ℘(X) → ℘(X), called the local function [10] of A with respect to τ and I, is defined as follows: for A ⊂ X, CUBO 18, 1 (2016) S-paracompactness modulo an ideal 49 A⋆(I, τ) = {x ∈ X : U ∩ A /∈ I for every U ∈ τ(x)}, where τ(x) = {U ∈ τ : x ∈ U}. When there is no chance for confusion, we will simply write A⋆ for A⋆(I, τ). In general, X⋆ is a proper subset of X. The hypothesis X = X⋆ is equivalent to the hypothesis τ ∩ I = ∅. According to [14], we call the ideals which satisfy this hypothesis τ-boundary ideals. Note that Cl⋆(A) = A ∪ A⋆ defines a Kuratowski closure for a topology τ⋆(I), finer than τ. A basis β(I, τ) for τ⋆(I) can be described as follows: β(I, τ) = {V \ J : V ∈ τ and J ∈ I}. When there is no chance for confusion, we will simply write τ⋆ for τ⋆(I) and β for β(I, τ). In the sequel, the ideal of nowhere dense (resp. meager) subsets of (X, τ) is denoted by N (resp. M). 3 αS-paracompactness modulo an ideal In this section, we shall introduce and study the αS-paracompact subsets modulo an ideal I, which is a natural generalization of αS-paracompact subsets. First recall some notions of paracompact- ness. Definition 3.1. A subset A of a space (X, τ) is said to be α-paracompact [3] (resp. α-almost paracompact [9]) if for any open cover U of A, there exists a locally finite collection V of open sets such that V refines U and A ⊂ ⋃ {V : V ∈ V} (resp. A ⊂ ⋃ {Cl(V) : V ∈ V}). A space (X, τ) is said to be paracompact (resp. almost-paracompact) if X is α-paracompact (resp. α-almost paracompact). Definition 3.2. A subset A of a space (X, τ, I) is said to be α-paracompact modulo I [7] (briefly α-paracompact (mod I)), if for any open cover U of A, there exist I ∈ I and a locally finite collection V of open sets such that V refines U and A ⊂ ⋃ {V : V ∈ V} ∪ I. A space (X, τ, I) is said to be I-paracompact or paracompact with respect to I [16], if X is α- paracompact modulo I. In the present, it is called paracompact modulo I (or briefly paracompact (mod I)). Definition 3.3. A subset A of a space (X, τ) is said to be αS-paracompact [1] if for any open cover U of A, there exists a locally finite collection V of open sets such that V refines U and A ⊂ ⋃ {V : V ∈ V}. A space (X, τ) is said to be S-paracompact if X is αS-paracompact. Now, we give the definition of αS-paracompact subset modulo an ideal I. Definition 3.4. A subset A of a space (X, τ, I) is said to be αS-paracompact modulo I (briefly αS-paracompact (mod I)), if for any open cover U of A, there exist I ∈ I and a locally finite collection V of semi-open sets such that V refines U and A ⊂ ⋃ {V : V ∈ V} ∪ I. A space (X, τ, I) is said to be I-S-paracompact or S-paracompact with respect to I [15], if X is αS-paracompact modulo I. In the present, it is called S-paracompact modulo I (or briefly S-paracompact (mod I)). We say that A is S-paracompact (mod I) if (A, τ A , I A ) is S-paracompact (mod I A ) as a subspace, where τ A is the relative topology induced on A by τ and I A = {I∩A : I ∈ I}. 50 J. Sanabria, E. Rosas, N. Rajesh, C. Carpintero and A. Gómez CUBO 18, 1 (2016) Proposition 3.1. Let A be a subset of a space (X, τ) and I an ideal on (X, τ). Then, the following properties hold: (1) If A is α-paracompact (mod I), then A is αS-paracompact (mod I). (2) Every I ∈ I is an αS-paracompact (mod I). (3) (X, τ, I) is S-paracompact (mod I) if there exists I ∈ I such that X − I is αS-paracompact (mod I). (4) A is αS-paracompact if and only if it is αS-paracompact (mod {∅}). Proof. (1) Follows from the fact that every open set is semi-open. (2) Suppose that there exists I ∈ I such that I is not αS-paracompact (mod I). Then, there exists an open cover U of I such that I ̸⊂ ⋃ {V : V ∈ V} ∪ J for every J ∈ I and every locally finite collection V which refines U. This is a contradiction, because I ∈ I and I ⊂ ⋃ {V : V ∈ V} ∪ I. (3) Suppose that there exists I ∈ I such that X − I is αS-paracompact (mod I) and let U be an open cover of X. Then, U is an open cover of X − I and hence there exist J ∈ I and a locally finite collection V of semi-open sets such that V refines U and X − I ⊂ ⋃ {V : V ∈ V} ∪ J. Thus, X = (X − I) ∪ I ⊂ ⋃ {V : V ∈ V} ∪ (J ∪ I) and as J ∪ I ∈ I, we have (X, τ, I) is S-paracompact (mod I). (4) It is obvious. Now, we give some comments related with the Proposition 3.1. Remark 3.1. According to Proposition 3.1(1), every α-paracompact (mod I) (resp. αS- paracompact) subset is αS-paracompact (mod I), and from this point of view, the notion of αS-paracompact (mod I) subset is a natural generalization of the notion of α-paracompact (mod I) (resp. αS-paracompact) subset. On the other hand, in Example 2.11 of [13], it is shows that there exists a semiregular Hausdorff space X and a regular closed subset M of X such that M is an αS-paracompact (mod {∅}) subset of X, but M is not α-paracompact (mod {∅}). Thus, the converse of Proposition 3.1(1) in general is not true. Proposition 3.2. Let A be a subset of a space (X, τ) and I an ideal on (X, τ). Then, the following properties hold: (1) If A is a semi-open and αS-paracompact (mod I) set and I is τ-boundary, then A is α-almost paracompact. (2) A semi-preopen set A is αS-paracompact (mod N) if and only if it is α-almost paracompact. Proof. (1) Let U be any open cover of A. Then there exist I ∈ I and a locally finite collection V = {Vλ : λ ∈ Λ} of semi-open sets such that V refines U and A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ I. Since A is CUBO 18, 1 (2016) S-paracompactness modulo an ideal 51 semi-open, A ⊂ Cl(Int(A)) and as I is τ-boundary, Int(I) = ∅. Now, by the locally finiteness of V, the collection V′ = {Int(Vλ) : λ ∈ Λ} is also locally finite, it follows that A ⊂ Cl(Int(A)) ⊂ Cl ( Int ( ⋃ λ∈Λ Vλ ∪ I )) ⊂ Cl ( Int ( ⋃ λ∈Λ Cl(Int(Vλ)) ∪ I )) = Cl ( Int ( Cl ( ⋃ λ∈Λ Int(Vλ) ) ∪ I )) = Cl ( Int ( Cl ( ⋃ λ∈Λ Int(Vλ) ) ∪ Int(I) )) = Cl ( Int ( Cl ( ⋃ λ∈Λ Int(Vλ) ))) ⊂ Cl ( ⋃ λ∈Λ Int(Vλ) ) = ⋃ λ∈Λ Cl(Int(Vλ)). If Wλ = Int(Vλ), then A ⊂ ⋃ λ∈Λ Cl(Wλ). Observe that Wλ is open for each λ ∈ Λ and Wλ ⊂ Vλ ⊂ U for some U ∈ U, hence W = {Wλ : λ ∈ Λ} is a locally finite open refinement of U. Therefore, A is α-almost paracompact. (2) Similar to the proof of (1), if A is semi-preopen, then A ⊂ Cl(Int(Cl(A))) ⊂ Cl ( Int ( Cl ( ⋃ λ∈Λ Vλ ∪ I ))) = Cl ( Int ( Cl ( ⋃ λ∈Λ Vλ ) ∪ Cl(I) )) = Cl ( Int ( Cl ( ⋃ λ∈Λ Vλ ) ∪ Int(Cl(I)) )) = Cl ( Int ( Cl ( ⋃ λ∈Λ Vλ ))) ⊂ Cl ( Int ( Cl ( ⋃ λ∈Λ Cl(Int(Vλ)) ))) = Cl ( Int ( Cl ( ⋃ λ∈Λ Int(Vλ) ))) ⊂ Cl ( ⋃ λ∈Λ Int(Vλ ) = ⋃ λ∈Λ Cl(Int(Vλ)). 52 J. Sanabria, E. Rosas, N. Rajesh, C. Carpintero and A. Gómez CUBO 18, 1 (2016) Therefore, the proof follows. As a consequence of Proposition 3.2, we obtain the following result. Corollary 3.1. (Sanabria et al. [15]) Let I be an ideal on a space (X, τ). Then, the following properties hold: (1) If I is τ-boundary and (X, τ) is S-paracompact (mod I), then (X, τ) is almost-paracompact. (2) (X, τ) is S-paracompact (mod N) if and only if it is almost-paracompact. Theorem 3.1. If every open subset of a space (X, τ, I) is αS-paracompact (mod I), then every subspace of (X, τ, I) is S-paracompact (mod I). Proof. Suppose that A is any subspace of (X, τ, I) and let U = {Uµ : µ ∈ ∆} be a τA-open cover of A. For every µ ∈ ∆ there exists Vµ ∈ τ such that Uµ = Vµ ∩A. Put V = ⋃ {Vµ : µ ∈ ∆}, then V ∈ τ and V = {Vµ : µ ∈ ∆} is a τ-open cover of V. By hypothesis, there exist I ∈ I and a τ-locally finite collection W = {Wλ : λ ∈ Λ} of τ-semi-open sets such that W refines V and V ⊂ ⋃ {Wλ : λ ∈ Λ}∪I. Then, we have A = ⋃ µ∈∆ Uµ = ⋃ µ∈∆ (Vµ ∩ A) = ⎛ ⎝ ⋃ µ∈∆ Vµ ⎞ ⎠ ∩ A = V ∩ A ⊂ ( ⋃ λ∈Λ Wλ ∪ I ) ∩ A = ⋃ λ∈Λ (Wλ ∩ A) ∪ IA, where IA = I ∩ A ∈ IA. If x ∈ A, then there exists Gx ∈ τ containing x such that Wλ ∩ Gx = ∅ for all λ ̸= λ1, λ2, . . . , λn and so (Wλ ∩ Gx) ∩ A = ∅ for all λ ̸= λ1, λ2, . . . , λn. It follows that (Wλ ∩ A) ∩ (Gx ∩ A) = ∅ for all λ ̸= λ1, λ2, . . . , λn and hence, the collection H = {Wλ ∩ A : λ ∈ Λ} is τ A -locally finite. If Wλ ∩ A ∈ H, then Wλ ∈ W and since W refines V, Wλ ⊆ Vµ for some Vµ ∈ V, which implies that Wλ ∩ A ⊂ Vµ ∩ A = Uµ ∈ U. Therefore, H refines U. This shows that H = {Wλ ∩ A : λ ∈ Λ} is a τA-locally finite collection of τA-semi-open sets which refines U such that A ⊂ ⋃ {H : H ∈ H} ∪ IA. Thus, every subspace of (X, τ, I) is S-paracompact (mod I). The following result is an immediate consequence of Theorem 3.2. Corollary 3.2. If every open subset of a space (X, τ, I) is αS-paracompact (mod I), then (X, τ, I) is S-paracompact (mod I). Recall that a subset A of a space (X, τ) is said to be g-closed [12] if Cl(A) ⊂ U whenever A ⊂ U and U ∈ τ. Theorem 3.2. If (X, τ, I) is S-paracompact (mod I) and A is a g-closed subset of X, then A is αS-paracompact (mod I). CUBO 18, 1 (2016) S-paracompactness modulo an ideal 53 Proof. Suppose that A is a g-closed subset of an S-paracompact (mod I) space (X, τ, I). Let U = {Uµ : µ ∈ ∆} be an open cover of A. Since A is g-closed and A ⊂ ⋃ {Uµ : µ ∈ ∆}, then sCl(A) ⊂ ⋃ {Uµ : µ ∈ ∆}. For each x /∈ Cl(A) there exists a τ-open set Gx containing x such that A ∩ Gx = ∅. Put U ′ = {Uµ : µ ∈ ∆} ∪ {Gx : x /∈ Cl(A)}. Then U ′ is an open cover of the S-paracompact (mod I) space X and so, there exist I ∈ I and a locally finite collection V = {Vλ : λ ∈ Λ} of semi-open sets such that V refines U and X = ⋃ {Vλ : λ ∈ Λ} ∪ I. For each λ ∈ Λ, either Vλ ⊂ Uµ(λ) for some µ(λ) ∈ ∆ or Vλ ⊂ Gx(λ) for some x(λ) /∈ Cl(A). Now, put Λ0 = {λ ∈ Λ : Vλ ⊂ Uβ(λ)}. Then V ′ = {Vλ : λ ∈ Λ0} is a collection of semi-open sets which is locally finite and refines U. Also, X − ⋃ λ∈Λ0 Vλ = ( ⋃ λ∈Λ Vλ ∪ I ) − ⋃ λ∈Λ0 Vλ = ⋃ λ/∈Λ0 Vλ ∪ I ⊂ ⋃ λ/∈Λ0 Gx(λ) ∪ I ⊂ (X − A) ∪ I = X − (A − I), which implies A − I ⊂ ⋃ λ∈Λ0 Vλ and hence A ⊂ ⋃ λ∈Λ0 Vλ ∪ I. This shows that A is αS-paracompact (mod I). Theorem 3.3. Let (X, τ, I) be a space. Then, the following properties hold: (1) If A is an open αS-paracompact (mod I) subset of (X, τ, I), then A is S-paracompact (mod I). (2) If A is a clopen subset of (X, τ, I), then A is αS-paracompact (mod I) if and only if it is S-paracompact (mod I). Proof. (1) Let A be an open αS-paracompact (mod I) subset of (X, τ, I). Let U = {Uµ : µ ∈ ∆} be a τ A -open cover of A. Since A is τ-open, we have U is a τ-open cover of A and hence, there exist I ∈ I and a τ-locally finite collection V = {Vλ : λ ∈ Λ} of τ-semi-open sets which refines U such that A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ I. It follows that A ⊂ ⋃ {Vλ ∩ A : λ ∈ Λ} ∪ (I ∩ A) and so, the collection VA = {Vλ ∩ A : λ ∈ Λ} is a τA-locally finite τA-semi-open refinement of U and is an IA-cover of A. Therefore, A is S-paracompact (mod I). (2) If A is a clopen and αS-paracompact (mod I) subset of (X, τ, I), then from (1) we obtain that A is is S-paracompact (mod I). Conversely, let U = {Uµ : µ ∈ ∆} be a τ-open cover of A. The collection V = {A ∩ Uµ : µ ∈ ∆} is a τA-open cover of the S-paracompact (mod I) subspace (A, τ A , I A ) and hence, there exist IA ∈ IA and a τA-locally finite τA-semi-open refinement W = {Wλ : λ ∈ Λ} of V such that A = ⋃ {Wλ : λ ∈ Λ} ∪ IA. It is easy to see that W refines U and by Lemma 2.1(3), we have that Wλ ∈ SO(X, τ) for each λ ∈ Λ. To show W = {Wλ : λ ∈ Λ} is τ-locally finite, let x ∈ X. Si x ∈ A, then there exists Ox ∈ τA ⊂ τ containing x such that Ox intersects at most finitely many members of W. Otherwise X \ A is a τ-open set containing x which intersects no member of W. Therefore, W is τ-locally finite and such that 54 J. Sanabria, E. Rosas, N. Rajesh, C. Carpintero and A. Gómez CUBO 18, 1 (2016) A = ⋃ {Wλ : λ ∈ Λ} ∪ IA ⊂ ⋃ {Wλ : λ ∈ Λ} ∪ I for some I ∈ I. Thus, A is αS-paracompact (mod I). As a consequence of Theorem 3.3, we obtain the following result. Corollary 3.3. Every clopen subspace of a S-paracompact (mod I) space is S-paracompact (mod I). Lemma 3.1. Let A be a subset of a space (X, τ, I). If every open cover of A has a locally finite closed refinement V such that A ⊂ ⋃ {V : V ∈ V} ∪ I for some I ∈ I, then V has a locally finite open refinement W such that A ⊂ ⋃ {W : W ∈ W} ∪ I. Proof. Let U be an open cover of A. By hypothesis, there exist I ∈ I and a locally finite closed refinement V = {Vλ : λ ∈ Λ} of U such that A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ I. For each x ∈ A, there exists an open set Gx containing x such that Gx intersects at most finitely many members of V. Note that the collection G = {Gx : x ∈ A} is an open cover of A and again by hypothesis, there exist J ∈ I and a locally finite closed refinement H = {Hµ : µ ∈ ∆} of G such that A ⊂ ⋃ {Hµ : µ ∈ ∆} ∪ J. Now, as {Hµ : Hµ ∩ Vλ = ∅} ⊂ H, then the collection {Hµ : Hµ ∩ Vλ = ∅} is locally finite and ⋃ {Hµ : Hµ ∩ Vλ = ∅} = ⋃ {Cl(Hµ) : Hµ ∩ Vλ = ∅} = Cl( ⋃ {Hµ : Hµ ∩ Vλ = ∅}), it follows that Oλ = X − ⋃ {Hµ : Hµ ∩ Vλ = ∅} is an open set and Vλ ⊂ Oλ, for each λ ∈ Λ. For each µ ∈ ∆ and λ ∈ Λ, we have Hµ ∩ Oλ ̸= ∅ ⇐⇒ Hµ ∩ Vλ ̸= ∅. (∗) Since V refines U, for every λ ∈ Λ there exists U(λ) ∈ U such that Vλ ⊂ U(λ). Put Wλ = Oλ∩U(λ), then the collection W = {Wλ : λ ∈ Λ} is an open refinement of U. Furthermore, if x ∈ A there exists an open set Dx such that Dx intersects at most finitely many members of H, it follows from (∗) that W is locally finite. Also, A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ I ⊂ ⋃ {Oλ ∩ U(λ) : λ ∈ Λ} ∪ I = A ⊂ ⋃ {Wλ : λ ∈ Λ} ∪ I. The following theorem shows that, in the presence of the axiom of regularity, the notions of α-paracompact (mod I) and αS-paracompact (mod I) subsets are equivalent. Theorem 3.4. Let I be an ideal on a regular space (X, τ) and A be a subset of X. Then, A is α-paracompact (mod I) if and only if it is αS-paracompact (mod I). Proof. Necessity is obvious from the definitions. To show sufficiency, assume A is an αS-paracompact (mod I) subset of (X, τ, I) and let U = {Uµ : µ ∈ ∆} be an open cover of A. For each x ∈ A, there exists µ(x) ∈ ∆ such that x ∈ Uµ(x) and since (X, τ, I) is a regular space, there exists an open set Vx such that x ∈ Vx ⊂ Cl(Vx) ⊂ Uµ(x). Thus, V = {Vx : x ∈ A} is an open cover of A and because A is αS-paracompact (mod I), there exist I ∈ I and a locally finite semi-open refinement W = {Wλ : λ ∈ Λ} of V such that A ⊂ ⋃ {Wλ : λ ∈ Λ} ∪ I. Since W refines V, then for each λ ∈ Λ there exists x(λ) ∈ X such that Wλ ⊂ Vx(λ) and so, Wλ ⊂ Cl(Wλ) ⊂ Cl(Vx(λ)) ⊂ Uµ(x(λ)). Obviously the collection {Cl(Wλ) : λ ∈ Λ} is a locally finite closed refinement of U such that CUBO 18, 1 (2016) S-paracompactness modulo an ideal 55 A ⊂ ⋃ {Cl(Wλ) : λ ∈ Λ} ∪ I. By Lemma 3.1, the open cover U of A has a locally finite open refinement H such that A ⊂ ⋃ {H : H ∈ H} ∪ I. Therefore, A is an α-paracompact (mod I) subset of (X, τ, I). Proposition 3.3. If A is an αS-paracompact (mod I) subset of a space (X, τ, I) and B is a subset of X with ∂(B) ∈ I, then A ∩ Cl(B) is αS-paracompact (mod I). Proof. Let U be an open cover of A ∩ Cl(B). Then U′ = U ∪ {X − Cl(B)} is an open cover of A and so, there exist I ∈ I and a locally finite semi-open refinement V = {Vλ : λ ∈ Λ} of U ′ such that A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ I. Then, ∂(Cl(B)) ⊂ ∂(B) ∈ I and A ∩ Cl(B) ⊂ ⋃ λ∈Λ Vλ ∩ Int(Cl(B)) ∪ J, where J = [( ⋃ {Vλ : λ ∈ Λ})∩∂(Cl(B))]∪(I∩Cl(B)) ∈ I. Thus, the collection V ′ = {Vλ ∩Int(Cl(B)) : λ ∈ Λ} is a locally finite semi-open refinement of U such that A ∩ Cl(B) ⊂ ⋃ {V : V ∈ V′} ∪ J. Therefore, A ∩ Cl(B) is αS-paracompact (mod I). The following result follows from Proposition 3.3 and the fact that the topological frontier of a semi-open (resp. semi-closed) set is nowhere dense. Corollary 3.4. If A is an αS-paracompact (mod N) subset of a space (X, τ, I) and B is either semi-open or semi-closed, then A ∩ Cl(B) is αS-paracompact (mod N). Remark 3.2. If {Vλ : λ ∈ Λ} is a locally finite collection of subsets of a space (X, τ), then the collection {∂(Vλ) : λ ∈ Λ} is locally finite. According to [7], if I is an ideal on a space (X, τ) and F is the collection of all closed sets of (X, τ), then the collection {A ⊂ X : Cl(A) ∈ I} is an ideal contained in I. The ideal generated by the collection of whole closed sets in I is denoted by ⟨I ∩ F⟩. It is clear that ⟨I ∩ F⟩ = {A ⊂ X : Cl(A) ∈ I}. Proposition 3.4. Let A be a subset of a space (X, τ, I). If A is αS-paracompact (mod ⟨I ∩ F⟩) and N ⊂ I, then Cl(A) is αS-paracompact (mod I). Proof. Let U be an open cover of Cl(A). By hypothesis, there exist IA ∈ ⟨I ∩F⟩ and a locally finite collection V = {Vλ : λ ∈ Λ} of semi-open sets such that V refines U and A ⊂ ⋃ {Vλ : λ ∈ Λ} ∪ IA. Then, Cl(A) ⊂ ⋃ λ∈Λ Cl(Vλ) ∪ Cl(IA) = ( ⋃ λ∈Λ Vλ ) ∪ ( ⋃ λ∈Λ ∂(Vλ) ) ∪ Cl(IA). By Remark 3.2, the collection {∂(Vλ) : λ ∈ Λ} is locally finite and ∂(Vλ) ∈ N for each λ ∈ Λ. Thus, by [6, Lemma 2.1], we have ⋃ {∂(Vλ) : λ ∈ Λ} ∈ N ⊂ I. Put I = ⋃ {∂(Vλ) : λ ∈ Λ} ∪ Cl(IA), then I ∈ I and Cl(A) ⊂ ⋃ λ∈Λ Vλ ∪ I. Therefore, Cl(A) is αS-paracompact (mod I). 56 J. Sanabria, E. Rosas, N. Rajesh, C. Carpintero and A. Gómez CUBO 18, 1 (2016) Since N is the ideal of nowhere dense subsets of (X, τ), A ∈ N if and only if Cl(A) ∈ N . In the case that I = N , then ⟨I ∩ F⟩ = N . The following corollary is a direct consequence of Proposition 3.4. Corollary 3.5. If A is an αS-paracompact (mod N) subset of a space (X, τ, I) , then Cl(A) is αS-paracompact (mod N). Lemma 3.2. [7] If {Aλ : λ ∈ Λ} is a locally finite collection of meager sets of a space (X, τ), then ⋃ {Aλ : λ ∈ Λ} is meager. Theorem 3.5. If {Aλ : λ ∈ Λ} is a locally finite collection of αS-paracompact (mod M) subsets of a space (X, τ), then ⋃ {Aλ : λ ∈ Λ} is αS-paracompact (mod M). Proof. Let U be an open cover of ⋃ {Aλ : λ ∈ Λ} and put Uλ = {U ∈ U : U ∩ Aλ ̸= ∅} for each λ ∈ Λ. By the hypothesis, there exist Mλ ∈ M and a locally finite collection Vλ of semi-open sets such that Vλ refines Uλ and Aλ ⊂ ⋃ {V : V ∈ Vλ} ∪ Mλ. Then, we have Aλ ⊂ ⋃ V∈Vλ (V ∩ Int(Cl(Aλ))) ∪ ⋃ V∈Vλ (V ∩ ∂(Cl(Aλ))) ∪ Mλ. For each V ∈ Vλ and each λ ∈ Λ, V ∩∂(Cl(Aλ)) is nowhere dense and the collection {V ∩∂(Cl(Aλ)) : V ∈ Vλ, λ ∈ Λ} is locally finite, so by [6, Lemma 2.1], the union of all elements of {V ∩ ∂(Cl(Aλ)) : V ∈ Vλ, λ ∈ Λ} is a nowhere dense set. By Lemma 3.2, we obtain ⋃ {Mλ : λ ∈ Λ} ∈ M and M = ⋃ λ∈Λ ⋃ V∈Vλ V ∩ ∂(Cl(Aλ)) ∪ ⋃ λ∈Λ Mλ ∈ M. 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