Ratio Mathematica Volume 40, 2021, pp. 213-224 On δ-preregular e∗-open sets in topological spaces Jagadeesh B.Toranagatti∗ Abstract In this paper, we introduce a new class of sets called, δ-preregular e∗- open sets and investigate their properties and characterizations. By using δ-preregular e∗-open sets, we obtain decompositions of com- plete continuity and decompositions of perfect continuity. Keywords: δ-preopen;e∗-open;e∗-closed;δpe∗-open;δpe∗-closed;δpe∗- continuity. 2020 AMS subject classifications: 54A05,54C08. 1 ∗Department of Mathematics, Karnatak University’s Karnatak College, Dharwad-580001, Kar- nataka, INDIA; jagadeeshbt2000@gmail.com 1Received on May 5th, 2021. Accepted on June 22nd, 2021. Published on June 30th, 2021.doi: 10.23755/rm.v40i1.609. ISSN: 1592-7415. eISSN: 2282-8214. c©Toranagatti This paper is published under the CC-BY licence agreement. 213 Jagadeesh B.Toranagatti 1 Introduction The study of δ-open sets was initiated by Veličko[Velicko, 1968] in 1968. Fol- lowing this Raychaudhuri and Mukherjee[Raychaudhuri and Mukherjee, 1993] established the concept of δ-preopen sets. Later, Ekici[Ekici, 2009] introduced the concept of e∗-open sets as a generalization of e-open sets. The aim of this paper is to introduce and study a new class of sets called, δ-preregular e∗-open sets using δ-preinterior and e∗-closure operators. The notion of δpe∗-continuity is also introduced which is stronger than δ-precontinuity.Finally, we obtain decom- positions of complete continuity and decompositions of perfect continuity. Throughout this paper, (U,τ) and (V,η)(or simply U and V ) represent topo- logical spaces on which no separation axioms are assumed unless explicitly stated and f:(U,τ)→(V,η) or simply f:U → V denotes a function f of a topological space U into a topological space V. Let N ⊆ U, then cl(N) = ∩{F: N ⊆ F and Fc ∈ τ} is the closure of N and int(N) = ∪{O: O ⊆ N and O ∈ τ} is the interior of N. 2 Preliminaries Definition 2.1. A set M ⊆ U is called δ-closed[Velicko, 1968] if M = δ-cl(M) where δ-cl(M)={p∈U:int(cl(G))∩M 6=φ,G∈τ and p∈G }. Definition 2.2. A set M ⊆ U is called (1) e-open[Ekici, 2008c] if M ⊆ cl(δ-int(M))∪int(δ-cl(M)) and e-closed if cl(δ- int(M))∩int(δ-cl(M))⊆M. (2) a-open[Ekici, 2008d] if M ⊆ int(cl(δ-int(M))) and a-closed if cl(int(δ-cl(M)))⊆M. (3)e∗-open[Ekici, 2009] if M ⊆ cl(int(δ-cl(M))) and e∗-closed if int(cl(δ-int(M)))⊆M . (4)δ-semiopen[Park et al., 1997] if M⊆cl(δ-int(M))) and δ-semiclosed if int(δ- cl(M))⊆M). (5)δ-preopen[Raychaudhuri and Mukherjee, 1993] if M⊆int(δ-cl(M)) and δ-preclosed if cl(δ-int(M))⊆M. (6)regular-open[Stone, 1937] if M = int(cl(M)) and regular-closed if M=cl(int(M)). Definition 2.3. [Ekici, 2008b] A subet M of a space U is said to be a δ-dense set if δ-cl(M)=U. The class of open(resp,closed, regular open,δ-preopen, δ-semiopen, e∗-open and clopen) sets of (U,τ) is denoted by O(U) (resp,C(U), RO(U), δPO(U), δSO(U), e∗O(U) and CO(U)). 214 On δ-preregular e∗-open sets in topological spaces Theorem 2.1. [Raychaudhuri and Mukherjee, 1993] Let M be a subset of a space (U,τ), then δ-pcl(M)=M∪cl(δ-int(M)) and δ-pint(M)=M∩int(δ-cl(M)). Theorem 2.2. [Ekici, 2009]Let M be a subset of a space (U,τ),then: (i) e∗-cl(M) = M∪ int(cl(δ-int(M)) and e∗-int(M) = M∩cl(int(δ-cl(M)) (ii) int(cl(δ-int(M))=e∗-cl(δ-int(M))=δ-int(e∗-cl(M)). Theorem 2.3. Let M be a subset of a space (U,τ),then: (i)δ-pint(e∗-cl(M))=e∗-cl(M)∩ int(δ-cl(M)). (ii)δ-pint(e∗-cl(M))=δ-pint(M)∪ int(cl(δ-int(M)). (iii)δ-pint(e∗-cl(M))=δ-pint(M)∪ e∗-cl(δ-int(M)) (iv)δ-pint(e∗-cl(M))=δ-pint(M)∪ δ-int(e∗-cl(M)). (v) δ-pint(e∗-cl(M)) = (M∩int(δ-cl(M))∪ int(cl(δ-int(M)) Lemma 2.1. [Benchalli et al., 2017]For a subset M of a space (U,τ),the following are equivalent: (a)M is clopen; (b)M is δ-open and δ-closed; (c)M is regular-open and regular-closed. Definition 2.4. [Kohli and Singh, 2009] A space (U,τ) is called δ-partition if δO(U)=C(U). Definition 2.5. [Caldas and Jafari, 2016] A space (U,τ) is a δ-door space if every subset of U is δ-open or δ-closed. Theorem 2.4. [Caldas and Jafari, 2016] If (U,τ) is a δ-door space, then every δ-preopen set in (U,τ) is δ-open. 3 δ-preregular e∗-open sets in topological spaces Definition 3.1. A subset N of a space (U,τ) is said to be δ-preregular e∗-open(briefly δpe∗-open) if N =δ-pint(e∗-cl(N)). The complement of a δ-preregular e∗-open is called a δ-preregular e∗-closed(briefly δpe∗-closed) set. Clearly, N is δpe∗-closed if and only if N = δ-pcl(e∗-int(N)) The class of δpe∗-open (resp,δpe∗-closed) sets of (U,τ) will be denoted by δPE∗O(U)(resp,δPE∗C(U)). Theorem 3.1. Let (U,τ) be a topological space and M, N ⊆ U. Then the following hold: (i) If M ⊆ N, then δ-pint(e∗-cl(M) ⊆ δ-pint(e∗-cl(N)). (ii) If M ∈δPO(U), then M ⊆ δ-pint(e∗-cl(M)). (iii) If M ∈e∗C(U), then e∗-cl(δ-pint(M)) ⊆ M. 215 Jagadeesh B.Toranagatti (iv) δ-pint(e∗-cl(N)) is δpe∗-open (v) If M ∈e∗C(U), then δ-pint(M) is δpe∗-open.. Proof:(i)Obvious. (ii) Let M ∈ δPO(U). As M ⊆ e∗-cl(M),then M ⊆ δ-pint(e∗-pcl(M). (iii) Let M ∈ e∗C(U). Since δ-pint(M) ⊆ M, then e∗-cl(δ-pint(M)) ⊆ M. (iv) We have δ-pint(e∗-cl(δ-pint(e∗-cl(M)) ⊆ δ-pint(e∗-cl(e∗-cl(M)) = δ-pint(e∗-cl(M) and δ-pint(e∗-cl(δ-pint(e∗-cl(M))) ⊇ δ-pint(δ-pint(e∗-cl(M)) = δ-pint(e∗-cl(M). Hence δ-pint(e∗-cl(δ-pint(e∗-cl(M))) = δ-pint(e∗-cl(M). (v) Suppose that M ∈ e∗C(U). By (i), δ-pint(e∗-cl(δ-pint(M))⊆ δ-pint(e∗-cl(M)=δ-pint(M). On the other hand, we have δ-pint(M) ⊆ e∗-cl(δ-pint(M) so that δ-pint(M) ⊆ δ-pint(e∗-cl(δ-pint(M)). Therefore δ-pint(e∗-cl(δ-pint(M))=δ-pint(M). This shows that δ-pint(M) is δpe∗-open. Theorem 3.2. (i)Every δpe∗-open set is δ-preopen(hence e-open,e∗-open). (ii)Every δpe∗-open set is e∗-closed.. Proof: (i)Let M be δpe∗-open,then by Theorem 2.3(i), δ-pint(e∗-cl(M))=e∗-cl(M)∩int(δ-cl(M). Therefore, M ⊆int(δ-cl(M), M is δ-preopen. (ii)Let N be δpe∗-open.By Theorem 2.3(ii), N =δ-pint(N)∪int(cl(δ-int(N))). Therefore,int(cl(δ-int(N)))⊆N. Thus N is e∗-closed. Remark 3.1. By the following example,we show that every δ-preopen(resp,e∗- closed) set need not be a δpe∗-open set Example 3.1. Let U = {a,b,c,d} and τ = {U, φ, {a}, {b}, {a,b}, {a,c}, {a,b,c}}. Then {a,b,c} is a δ-preopen set but {a,b,c} /∈ δPE∗O(U) and {d} is e∗-closed but {d} /∈ δPE∗O(U)it is not δpe∗-open Corolary 3.1. For a topological space (U,τ), we have δ-PO(U)∩δ-PC(U) ⊆ δPE∗O(U)⊆ e∗O(U)∩e∗C(U). Proof: Obvious. The converse inclusions in the above corollary need not be true as seen from the following example Example 3.2. Let (U,τ) as in Example 3.1,then {b} is δpe∗-open but it is not δ-preclopen. Moreover, {a,d} is e∗-clopen but not δpe∗-open Remark 3.2. The notions of δpe∗-open sets and δ-open sets (hence a-open sets, δ-semiopen sets, δ∗-sets) are independent of each other. 216 On δ-preregular e∗-open sets in topological spaces Example 3.3. Consider (U,τ) as in Example 3.1.The set {a} is δpe∗-open but it is not δ∗-set. Moreover, {a,b,c} is δ-open but not δpe∗-open Theorem 3.3. In a δ-partition space (U,τ), a subset M of U is δpe∗-open if and only if it is δ-preopen. Proof: Necessity:It follows from Theorem 3.2(i) . Sufficiency:Let N be δ-preopen. Since (U,τ) is δ-partition and by Theorem 2.3(ii), we have δ-pint(e∗-cl(M)) = δ-pint(M)∪ int(cl(δ-int(M)) = M ∪ int(cl(cl(M)) = M ∪int(cl(M) = M ∪δ-int(cl(M) = M ∪δ-int(δ-int(M) = M ∪δ-int(M) = M Therefore, δ-pint(e∗-cl(M)) = M.Hence M is δpe∗-open. Theorem 3.4. A subset N ⊆ U is δpe∗-open if and only if N is e∗-closed and δ- preopen. Proof: Necessity:It follows from Theorem 3.2. Sufficiency:Let N be both e∗-closed and δ-preopen. Then N = e∗-cl(N) and N = δ-pint(N). Therefore, δ-pint(e∗-cl(N)) = δ-pint(N) = N. Hence N is δpe∗-open. Remark 3.3. The class of δpe∗-open sets is not closed under finite union as well as finite intersection. It will be shown in the following example. Example 3.4. Consider (U,τ) as in Example 3.1. Let A = {a,c} and B = {b,c},the A and B are δpe∗-open sets but A ∪ B = {a,b,c} /∈ δPE∗O(U). Moreover,C = {a,b,d} and D = {b,c,d} are δpe∗-open sets but C ∩ D = {b,d} /∈ δPE∗O(U). Theorem 3.5. For a subset M of a space (U,τ),the following are equivalent: (i) M is δpe∗-open. (ii) M = e∗-cl(M)∩ int(δ-cl(M)). (iii) M = δ-pint(M)∪ int(cl(δ-int(M)). (iv)M = δ-pint(M)∪ e∗-cl(δ-int(M)) (v) M = δ-pint(M)∪ δ-int(e∗-cl(M)). (vi) M = (M∩int(δ-cl(M))∪ int(cl(δ-int(M)). Proof:It follows from Theorem 2.3 Theorem 3.6. In any space (U,τ) , the empty set is the only subset which is nowhere δ-dense and δpe∗-open. Proof: Suppose M is nowhere δ-dense and δpe∗-open. Then by Theorem 2.3(i), M = δ-pint(e∗-cl(M)) =e∗-cl(M)∩ int(δ-cl(M)= e∗-cl(M)∩φ = φ. 217 Jagadeesh B.Toranagatti Lemma 3.1. If (U,τ) is a δ-door space, then any finite intersection of δ-preopen sets is δ-preopen. Proof:Obvious since δO(X) is closed under finite intersection. Theorem 3.7. If (U,τ) is a δ-door space, then any finite intersection of δpe∗-open sets is δpe∗-open. Proof:Let {Ai:i=1,2,...,n} be a finite family of δpe∗-open. Since the space (U,τ) is δ-door, then by Lemma 3.1, we have n⋂ i=n Ai∈δPO(U). By Theorem 3.1(ii), n⋂ i=n Ai⊆ δ-pint(e∗-cl( n⋂ i=n Ai). For each i, we have n⋂ i=n Ai⊆ Ai and thus δ-pint(e∗-cl( n⋂ i=n Ai)⊆ δ-pint(e∗-cl(Ai) = Ai. Therefore, δ-pint(e∗-cl( n⋂ i=n Ai)⊆ n⋂ i=n Ai. Lemma 3.2. If a subset M of a space (U,τ) is regular open,then M = int(cl(M)=int(δ-cl(M)). Theorem 3.8. Every regular open set is δpe∗-open. Proof: Let M be regular open. Then M=int(cl(M))=int(δ-cl(M)). By Theorem 2.6(i), δ-pint(e∗-cl(M)) = e∗-cl(M)∩int(δ-cl(M))=e∗-cl(M)∩M=M. This shows that M is δpe∗-open. Definition 3.2. A subset M of a space (U,τ) is called δ∗-set if int(δ-cl(M))⊆cl(δ-int(M)) Theorem 3.9. (i) Every δ-semiopen set is δ∗-set. (ii)Every δ-semiclosed set is δ∗-set. Proof:Clear Definition 3.3. A subset M of a space (U,τ) is called b∗-open if M = cl(δ-int(M))∪ int(δ-cl(M)). b∗-closed if M = cl(δ-int(M))∩ int(δ-cl(M)) Theorem 3.10. A subset M of a space (U,τ) is regular open if and only if it is b∗-closed. Proof:Let M be regular open. Then by Lemma 3.2, M = int(cl(M)=int(δ-cl(M)). Since every regular open set is δ-open, we have cl(δ-int(M))∩ int(δ-cl(M)) = cl(M)∩ M = M. Hence A is b∗-closed. Conversely, let M be b∗-closed.Then int(cl(δ-int(M))⊆int(δ-cl(δ-int(M))⊆ cl(δ- int(M))∩ int(δ-cl(M))=M. By Definition 3.3, we have M ⊆ int(δ-cl(M)) ⊆ int(δ- cl(cl(δ-int(M))) = int(cl(cl(δ-int(M))) = int(cl(δ-int(M))). Therefore, M = int(cl(δ-int(M)). Now, int(cl(M)) = int(cl(int(cl(δ-int(M))) = int(cl(δ- int(M)) = M. Hence M is regular open. 218 On δ-preregular e∗-open sets in topological spaces Theorem 3.11. (i) Every b∗-closed set is δ-preopen. (ii)Every b∗-closed set is δ-semiopen. (iii)Every b∗-closed set is δpe∗-open. Proof:(i) and (ii) are obvious (iii)Let M be b∗-closed,then we have M = int(cl(δ-int(M)). Then δ-pint(e∗-cl(M)) = δ-pint(M) ∪ int(cl(δ-int(M)) = δ-pint(M) ∪ M = M.Hence M is δpe∗-open Remark 3.4. The above discussions can be summarized in the following diagram: DIAGRAM regular open −→ δ-open −→ a-open −→ δ-semiopen −→ δ∗-set l ↓ ↓ b∗-closed −→ δpe∗-open −→ δ-preopen −→ e-open −→ e∗-open Theorem 3.12. For a subset M of a space (U,τ), the following are equivalent: (i) M is regular open; (ii) M is δpe∗-open and δ-open; (iii) M is δpe∗-open and a-open; (iv) M is δpe∗-open and δ-semiopen; (v) M is δpe∗-open and δ∗-set. Proof: (i) −→(ii)−→(iii)−→(iv)−→(v):Follows from the above diagram (v)−→(i):Let M be δpe∗-open and δ∗-set.Then int(δ-cl(M))⊆cl(δ-int(M)) and int(δ-cl(M))⊆ int(cl(δ-int(M))⊆int(δ-cl(δ-int(M))⊆int(δ-cl(M)). Therefore we have int(δ-cl(M))=int(cl(δ-int(M)). Since M is δpe∗-open, M = δ-pint(δ-pcl(M)) =(M∪int(cl(δ-int(M))∩int(δ-cl(M)) =int(δ-cl(M)∩int(δ-cl(M)) =int(δ-cl(M)). Therefore M =int(δ-cl(M))=int(cl(M)) and hence M is regular open. Theorem 3.13. For a subset M of a space (U,τ), the following are equivalent: (i) M is regular open. (ii) M is δpe∗-open and δ-semiclosed. (iii) M is e∗-closed and a-open. Proof: (i)−→(ii):It follows from Theorem 3.8 (ii)−→(i):Let M be δpe∗-open and δ-semiclosed. Since every δ-semiclosed set is δ∗-set. Hence by Theorem 3.12(v), M is regular open. (ii) −→(iii):Clear (i)←→(iii):It is shown in Theorem 3 [Ekici, 2008b] Corolary 3.2. For a subset M of a space (U,τ), the following are equivalent: (i) M is regular open; (ii) M is δpe∗-open and δ-open; 219 Jagadeesh B.Toranagatti (iii) M is δpe∗-open and a-open; (iv) M is δpe∗-open and δ-semiopen; (v) M is δpe∗-open and δ∗-set;. (vi) M is δpe∗-open and δ-semiclosed; (vii) M is e∗-closed and a-open; (viii) M is b∗-closed. Theorem 3.14. For a subset M of a space (U,τ), the following are equivalent: (i) M is clopen; (ii) M is δ-open and δ-closed; (iii) M is regular open and regular closed; (iv) M is δpe∗-open and δ-closed. Proof: (i) ←→(ii)←→(iii):Follows from Lemma 2.1 (iii)−→(iv). It follows from Theorem 3.8 (iv)−→(ii)Let M be δpe∗-open and δ-closed.By Theorem 2.3(i) , we have N = e∗-cl(N) ∩ int(δ-cl(N)) = e∗-cl(N) ∩δ-int(δ-cl(N))=δ-pcl(N)∩δ-int(N)=δ-int(N). Therefore M is δ-open. 4 Decompositions of complete continuity In this section, the notion of regular δ-preopen continuity is introduced and the decompositions of complete continuity are discussed. Definition 4.1. A function f:(U,τ)→(V,σ) is said to be (i) δpe∗-continuous if the inverse image of every open subset of (V,σ) is δpe∗-open set in (U,τ). (ii)perfectly continuous[Noiri, 1984] (resp,e-continuous[Ekici, 2008c], e∗-continuous[Ekici, 2009], δ-almost continuous[Raychaudhuri and Mukherjee, 1993], δ∗-continuous, contra-super-continuous[Jafari and Noiri, 1999], completely continuous[Arya and Gupta, 1974], RC-continuous[Dontchev and Noiri, 1998], super-continuous[Munshi and Bassan, 1982], contra continuous[Dontchev, 1996], a-continuous[Ekici, 2008d], δ-semicontinuous[Noiri, 2003], contra e∗-continuous[Ekici, 2008a], contra δ- semicontinuous[Ekici, 2004], contra b∗-continuous) if the inverse image of every open subset of (V,σ) is clopen (resp,e-open,e∗-open,δ-preopen,δ∗-set, δ-closed, regular open, regular closed, δ-open, closed, a-open, δ-semiopen, e∗-closed, δ- semiclosed, b∗-closed) set in (U,τ) By Theorems 3.9 and 3.11, we obtain the following theorem. Theorem 4.1. (i) Every contra b∗-continuous set is δ-almost continuous. (ii)Every contra b∗-continuous set is δ-semicontinuous 220 On δ-preregular e∗-open sets in topological spaces (iii)Every contra b∗-continuous set is δpe∗-continuous. (iv) Every δ-semicontinuous set is δ∗-continuous. (v)Every contra δ-semicontinuous is δ∗-continuous. Remark 4.1. By Diagram I, we have the following diagram: DIAGRAM II c.cont. −→ s.cont. −→ a.cont. −→ δs.cont. −→ δ∗.cont. l ↓ ↓ cb∗.cont.−→δpe∗.cont.−→δp.cont. −→ e.cont. −→ e∗.cont. where c.cont.=completely continuity, s.cont.=super continuity, a.cont.=a-continuity, δs.cont.=δ-semicontinuity, δ∗.cont.=δ∗-continuity, cb∗.cont.=contra b∗-continuity, δpe∗.cont.=δ-preregular e∗-continuity, δp.cont.=δ-precontinuity, e.cont.=e-continuity, e∗.cont.=e∗-continuity Theorem 4.2. For a function f:(U,τ)→(V,η), the following are equivalent: (i) f is completely continuous; (ii)f is δpe∗-continuous and super continuous; (iii)f is δpe∗-continuous and a-continuous; (iv) f is contra e∗-continuous and a-continuous; (v)f is δpe∗-continuous and δ-semicontinuous; (vi)f is δpe∗-continuous and contra δ-semicontinuous; (vii)f is δpe∗-continuous and δ∗-continuous; (viii) f is contra b∗-continuous. Remark 4.2. (i) δpe∗-continuity and super-continuity(hence a-continuity,δ-semicontinuity, δ∗∗-continuity) are independent notions. (ii) δpe∗-continuity and contra δ-semicontinuity are independent notions. Example 4.1. Let (U,τ) be a space as in Example 3.1 and let η = {U, φ, {a}, {b}, {a,b}, {a,b,c}} (i) Define f:(U,τ) → (U,η) by f(a) = f(c) = a , f(b) = b and f(d) = d. Clearly f is super-continuous but for {a,b}∈ O(V), f−1({a,b}) = {a,b,c} /∈ δPE∗O(U). Therefore f is not δpe∗-continuous. Define g:(U,τ) → (U,η) by g(a) = b, g(b) = g(c) = g(d) = a.Then g is δpe∗- continuous but for {a}∈ O(V), g−1({a}) = {b,c,d} /∈ q∗O(U). Therefore g is not q∗-continuous. (ii)Define f:(U,τ) → (U,η) by f(a) = f(c) = f(d) = b and f(b) = a. Clearly f is δ-semiregular-continuous but for {b}∈ O(V), f−1({b}) = {a,c,d} /∈ δPE∗PO(U). Therefore f is not δpe∗-continuous. Define g:(U,τ) → (U,η) by g(a) = g(b)=g(d)=a,g(c) = b.Then g is δpe∗-continuous 221 Jagadeesh B.Toranagatti but for {a} ∈ O(V), g−1({a}) = {a,b,d} /∈ δSC(U).Therefore g is not contra δ- semicontinuous. 5 Decompositions of perfectly continuity In this section, the decompositions of perfectly continuity are obtained. Theorem 5.1. For a function f:(U,τ) → (U,η), the following are equivalent: (i) f is perfectly continuous; (ii) f is super continuous and contra super continuous; (iii) f is completely continuous and RC-continuous; (iv) f is δpe∗-continuous and contra super continuous. Proof: It is a direct consequence of Theorem 3.14 Remark 5.1. As shown by the following examples,δpe∗-continuity and contra su- per continuity are independent of each other. Example 5.1. Consider (U,τ) as in Example 3.1 and (U,η) as in Example 4.1. De- fine f: (U,τ) → (U,η) by f(a) = f(c) = f(d) = a and f(b) = c. Then f is contra super continuous but it is not δpe∗-continuous since {a}∈ O(V), f−1({a}) = {a,c,d} /∈ δPE∗O(U). Define g: (U,τ) → (U,η) by g(a) =b, g(b) = g(c) = g(d) = a.Then g is δpe∗-continuous but it is not contra super continuous since {a}∈ O(V), g−1({a}) = {b,c,d} /∈ δC(U). 6 Conclusions: The notions of sets and functions in topological spaces and fuzzy topolog- ical spaces are extensively developed and used in many engineering problems, information systems, particle physics, computational topology and mathemati- cal sciences. 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