Ratio Mathematica Volume 39, 2020, pp. 237-252 On a class of sets between a-open sets and gδ-open sets Jagadeesh B.Toranagatti∗ Abstract In this paper, a new class of sets called Da-open sets are introduced and investigated with the help of gδ-open and δ-closed sets. Re- lationships between this new class and other related classes of sets are established and as an application Da-continuous and almost Da- continuous functions have been defined to study its properties in terms of Da-open sets. Finally, some properties of Da-closed graph and (D,a)-closed graph are investigated. Keywords:a-open set,δ-open set,gδ-open,Da-open set,Da-closed set. 2010 AMS subject classifications: 54A05, 54A10.1 ∗Department of Mathematics, Karnatak University’s Karnatak College, Dharwad - 580001, Karnataka, India; jagadeeshbt2000@gmail.com 1Received on August 30th, 2020. Accepted on December 17th, 2020. Published on December 31st, 2020. doi: 10.23755/rm.v39i0.535. ISSN: 1592-7415. eISSN: 2282-8214. c©Toranagatti. This paper is published under the CC-BY licence agreement. 237 Jagadeesh B.Toranagatti 1 Introduction The concept of generalized open sets introduced by Levine[Levine, 1970] plays a significant role in General Topology. The study of generalized open sets and its properties found to be useful in computer science and digital topol- ogy[Khalimsky et al., 1990, Kovalevsky, 1994, Smyth, 1995]. Since Professor El- Naschie has recently shown in [El Naschie, 1998, 2000, 2005] that the no- tion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and �∞ theory.So,the fuzzy topological version of the notions and results introduced in this paper are very important. Recently, Ekici [Ekici, 2008] introduced the notion of a-open sets as a continuation of research done by Velicko [Velicko, 1968] on the notion of δ-open sets.Dontchev et al., intro- duced gδ-closed sets and gδ-continuity.In this paper,new generalizations of a-open sets by using gδ-open and δ-closed sets called Da-open sets are presented. Also Da-continuous functions,almost Da-continuous functions,Da-closed graphs and (D,a)-closed graphs have been defined to study its properties in terms of Da-open sets. 2 Prerequisites, Definitions and Theorems In what follows,spaces always mean topological spaces on which no sepa- ration axioms are assumed unless explicitly stated and f:(X,τ) → (Y,η) or simply f:X →Y denotes a function f of a space (X,τ) into a space (Y,η). The δ-closure of a subset A of X is the intersection of all δ-closed sets containing A and is denoted by Clδ(A). Definition 2.1. In (X,τ),let N ⊂ X.Then N is called: (i)regular closed[Stone, 1937] (resp.,a-closed[Ekici, 2008], δ-preclosed[Raychaudhuri and Mukherjee, 1993], e∗-closed[Ekici, 2009], δ-semiclosed[Park et al., 1997], β-closed[Abd El-Monsef, 1983], semiclosed[Levine, 1963], preclosed[Mashhour, 1982]) if N = Cl(Int(N)) (resp., Cl(Int(Clδ(N))) ⊂ N, Cl(Intδ(N)) ⊂ N, Int(Cl(Intδ(N)) ⊂ N, Int(Clδ(N)) ⊂ N, Int(Cl(Int(N)) ⊂ N, Int(Cl(N)) ⊂ N, Cl(Int(N)) ⊂ N). (ii) δ-closed [Velicko, 1968] if N = Clδ(N) where Clδ(N) = {p∈X:Int(Cl(O))∩N6=φ,O∈τ and p∈O}. (iii)generalized δ-closed (briefly,gδ-closed)[Dontchev et al., 2000] if Cl(N)) ⊂ G whenever N ⊂ G and G is δ-open in X. (iv)generalized closed (briefly,g-closed)[Levine, 1970] if Cl(N)) ⊂ G whenever N ⊂ G and G is open in X. The complements of the above mentioned closed sets are their respective open sets. 238 On a class of sets between a-open sets and gδ-open sets The set of all regular open (resp.,δ-open, β-open, δ-preopen, preopen, semiopen, δ-semiopen,e∗-open,gδ-open and a-open) sets of (X,τ) is denoted by RO(X) (resp.δO(X), βO(X), δPO(X), PO(X), SO(X), δSO(X), e∗O(X), GδO(X) and aO(X)). The a-closure[Ekici, 2008](resp, gδ-closure,δ-closure) of a set N is the inter- section of all a-closed(resp, gδ-closed,δ-closed) sets containing N and is denoted by a-Cl(N) (resp., Clgδ(N),Clδ(N)). The a-interior[Ekici, 2008](resp,gδ-interior,δ- interior) of a set N is the union of all a-open(resp, gδ-open,δ-open) sets contained in M and is denoted by a-Int(M)(resp, Intgδ(M),Intδ(M)) Definition 2.2. [Ekici, 2005] A topological space (X,τ) is said to be: (1) r-T1 if for each pair of distinct points x and y of X, there exist regular open sets U and V such that x ∈ U, y /∈U and x /∈ V, y ∈ V. (2) r-T2 if for each pair of distinct points x and y of X, there exist regular open sets U and V such that x ∈ U, y ∈ V and U∩V =φ . Theorem 2.1. Let C and D be subsets of a topological space (X,τ).Then (i)If C is gδ-closed,then Clgδ(C) = C. (ii) If C⊂D,then Clgδ(C)⊂ Clgδ(D). (iv) x ∈Clgδ(C) if and only if for each gδ-open set O containing x,O ∩ C6=φ, (v)Clgδ(C)∪Clgδ(D)⊂ Clgδ(A∪D). (vi)Clgδ(C∩D)⊆Clgδ(C)∩Clgδ(D). 3 Da-Open Sets. Definition 3.1. A subset M of a topological space (X,τ) is said to be: (1) Da-open if M ⊂ Intgδ(Clδ(Intgδ(M)), (2) Da-closed if Clgδ(Intδ(Clgδ(M))⊂M. The collection of all Da-open(resp,Da-closed) sets in (X,τ) is denoted by DaO(X) (resp,DaC(X)). Theorem 3.1. Let (X,τ) be a space.Then for any N⊂X, (i) N∈δO(X) implies N∈aO(X)[Ekici, 2008]. (ii) N∈δO(X) implies N ∈GδO(X)[Dontchev et al., 2000]. (iii)N∈GO(X) implies N ∈GδO(X)[Dontchev et al., 2000]. (iv) N ∈aO(X) implies N∈DaO(X). (v) N∈GδO(X) implies N∈DaO(X). Proof: (iv) Since δO(X)⊂GδO(X), Intδ(N) ⊂ Intgδ(N). Now,let N∈aO(X), then N ⊂ Int(Cl(Intδ(N)). Therefore, N ⊂ Int(Cl(Intδ(N))=Intδ(Cl(Intδ(N))⊂Intgδ(Clδ(Intgδ(N)). Hence N ∈DaO(X). (v) Suppose N is gδ-open. Then Intgδ(N)=N. 239 Jagadeesh B.Toranagatti Therefore, Intgδ(N)⊂ Clδ(Intgδ(N).Then N=Intgδ(N)=Intgδ(Intgδ(N)) ⊂ Intgδ(Clδ(Intgδ(N)). Hence N ∈DaO(X). Remark 3.1. The following diagram holds for any subset of a space (X,τ). open set ←− δ-open set −→ a-open set ↙ ↓ ↓ Da-open set ↗ g-open set −→ gδ-open set None of these implications is reversible Example 3.1. Let X={p,q,r,s} and τ={X,φ,{p},{q},{p,q},{p,r}.{p,q,r}},then aO(X)={X,φ,{q},{p,r},{p,q,r}} GδO(X)={X, φ,{p},{q},{r},{p,q},{p,r}{q,r},{p,q,r}}. DaO(X)={X, φ,{p},{q},{r},{p,q},{p,r},{q,r}{p,q,r}{p,q,s},{q,r,s}}. Therefore, {q,r,s}∈DaO(X) but {q,r,s}/∈aO(X) and {q,r,s}/∈ gδO(X). Lemma 3.1. If there exists a M ∈ GδO(X) such that M ⊂ N ⊂Intgδ(Clδ(M)),then N is Da-open. Proof: Since M is gδ-open, Intδg(M)=M. Therefore, Intgδ(Clδ(Intgδ(N)) ⊃Intgδ(Clδ(Intgδ(M)) = Intgδ(Clδ(M)) ⊃ N. Hence N is Da-open. Converse of the Lemma 3.1 is not true as shown in Example 3.1. Example 3.2. In Example 3.1, {p,q,r}∈DaO(X) and {p,r}∈GδO(X) but {p,r}⊆ {p,q,r} 6⊆ Intgδ(Clδ({p,r}))={p,r} . Lemma 3.2. For a family { Bλ:λ∈∧} of subsets of a space (X,τ),the following hold: (1) Clgδ( ⋂ {Bλ:λ∈∧}) ⊂ ⋂ {Clgδ(Bλ):λ∈∧}. (2) Clgδ( ⋃ {Vλ:λ∈∧}) ⊃ ⋃ {Clgδ(Bλ):λ∈∧}. (3) Clδ( ⋂ {Bλ:λ∈∧}) ⊂ ⋂ {Clδ(Bλ):λ∈∧}. (4) Clδ( ⋃ {Bλ:λ∈∧}) ⊃ ⋃ {Clδ(Bλ):λ∈∧} Theorem 3.2. If {Gα:λ∈∧} is a collection of Da-open sets in a space (X,τ),then⋃ α∈∧ Gα is a Da-open set in (X,τ) : Proof: Since each Gαis Da-open, Gα ⊂ Intgδ(Clδ(Intgδ(Gα)) for each α∈∧ and hence ⋃ α∈∧ Gα ⊂ ⋃ α∈∧ Intgδ(Clδ(Intgδ(Gα))⊂Intgδ(Clδ(Intgδ( ⋃ α∈∧ Gα)). Thus ⋃ α∈∧ Gα is Da-open. 240 On a class of sets between a-open sets and gδ-open sets Corolary 3.1. If {Fα:α∈∧} is a collection of Da-closed sets in a space (X,τ),then⋂ α∈∧ Fα is a Da-closed set in (X,τ) Remark 3.2. M and N ∈ DaO(X) 6⇒ M ∩ N ∈ DaO(X) as seen from Example 3.1, where both M = {q,r,s} and N = {p,q,s}∈ DaO(X) but M ∩ N = {q,s} /∈ DaO(X). Corolary 3.2. If M∈ DaO(X) and B∈aO(X),then M∪B∈ DaO(X). Proof:Follows from Theorem 3.1(iv) and Theorem 3.2 Corolary 3.3. If M∈ DaO(X) and B∈GδO(X),then M∪B∈ DaO(X). Proof:Follows from Theorem 3.1(v) and Theorem 3.2 Definition 3.2. In (X,τ),let M ⊂ X. (1)The Da-interior of M, denoted by IntDa (M) is defined as IntDa (M)= ⋃ {G:G⊆M and M∈DaO(X)}; (2)The Da-closure of M, denoted by ClDa (M) is defined as ClDa (A)= ⋂ {F:M⊆F and F∈DaC(X)}. Theorem 3.3. In (X,τ),let M, N,F ⊂ X.Then: (1)M ⊂ ClDa (M)⊂aCl(M), ClDa (M)⊂Clgδ(M). (2) ClDa (M) is a Da-closed set. (3) If F is a Da-closed set, and F ⊃ M,then F ⊃ ClDa (M). i.e.,ClDa (M) is the smallest Da-closed set containing M. (4)M is Da-closed set if and only if ClDa (M)=M. (5) ClDa (Cl D a (M)) = Cl D a (M). (6)M ⊆ N implies ClDa (M) ⊆ ClDa (N). (7)p ∈ClDa (M) if and only if for each Da-open set V containing p,V ∩ M 6=φ. (8) ClDa (M) ∪ ClDa (N) ⊂ ClDa (M ∪ N). (9) ClDa (M ∩ N) ⊂ ClDa (M) ∩ ClDa (N). Proof: (1)It follows from Theorem 3.1(iv) and (v) (2)It follows from Definition 3.2 and Corollary 3.1 (3)Let F be a Da-closed set,containing M.ClDa (M) is the intersection of Da-closed sets containing M, and F is one among these;hence F ⊃ ClDa (M). (4) Let M be Da-closed,then by Definition 3.2(2),ClDa (M)=M. Conversely,let ClDa (M)=M. Then by (2) above,M is Da-closed. (5)It follows from (2) and (4). (6) Obvious. (7) p /∈ClDa (M) ⇔ (∃ G∈DaC(X))(M⊂G)(p /∈G) ⇔ (∃ G∈DaC(X))(M⊂G)(p ∈Gc) ⇔ (∃ Gc∈DaO(X))(M∩Gc=φ)(p ∈Gc) ⇔ (∃ Gc∈DaO(X,p))(M∩Gc=φ) 241 Jagadeesh B.Toranagatti i.e.,(∃ U(=Gc)∈ DaO(X,p))(M∩U=φ) (8) and (9) follows from (6). Remark 3.3. (1) ClDa (M) ∪ ClDa (N) 6= ClDa (M ∪ N), in general, as seen from Ex- ample 3.1 where M = {p}, N = {r} and M ∪ N = {p,r}.Then ClDa (M)={p}, ClDa (N)={r},ClDa (M)∪ClDa (N)={p,r} and ClDa (M∪N)={p,r,s}; (2) ClDa (M ∩ N)6= ClDa (M) ∩ ClDa (N), in general,as seen from Example 3.1 where,M = {p,q,r}, N = {s} and M∩N = φ.Then ClDa (M) = X, ClDa (N) = {s}, ClDa (M)∩ClDa (N) = {s} and ClDa (M∩N)=φ Lemma 3.3. In (X,τ),let M ⊂ X.Then (1) ClDa (X\M) = X\IntDa (M), (2) IntDa (X\M) = X\ClDa (M). Theorem 3.4. In (X,τ),let M,N,G ⊂ X, (1)aInt(M) ⊆ IntDa (M)⊆M, Intgδ(M)⊆IntDa (M). (2) IntDa (M) is a Da-open set. (3) If G is a Da-open set, and G ⊂ M,then G ⊂ IntDa (M). i.e.,IntDa (M) is the largest Da-open set contained in M. (4)M is Da-open set if and only if IntDa (M)=M. (5) IntDa (Int D a (M)) = Int D a (M). (6)M ⊆ N implies IntDa (M) ⊆ IntDa (N). (7) p ∈ IntDa (M) if and only if there exists Da-open set N containing p such that N ⊆ M. (8) IntDa (M ∩ N)⊆ IntDa (M) ∩ IntDa (N). (9) IntDa (M) ∪ IntDa (N) ⊆IntDa (M ∪ N). Proof:Similar to the proof of Theorem 3.3 Remark 3.4. (8)IntDa (M ∩ N)6= IntDa (M) ∩ IntDa (N), in general, as seen from Ex- ample 3.1,where M = {p,q,s}, N = {q,r,s} and M ∩ N = {q,s}.Then IntDa (M) = {p,q,s}, IntDa (N) = {q,r,s}, IntDa (M) ∩ IntDa (N) = {q,s} and IntDa (M∩N) = {q}. (9) IntDa (M) ∪ IntDa (N) 6= IntDa (M ∪ N),in general, as seen from Example 3.1, where M = {p,q,r}, N = {s} and M ∪ N = X.Then IntDa (M) = {p,q,r}, IntDa (N) = φ, IntDa (M) ∪ IntDa (N) = {p,q,r} and IntDa (M ∪ N) = X. Lemma 3.4. In (X,τ),let M ⊂ X. Then (1)M is Da-open if and only if M = M ∩ Intgδ(Clδ(Intgδ(M)). (2)M is Da-closed if and only if M = M∪ Clgδ(Intδ(Clgδ(M)). Proof:(1) Let M be an Da-open. Then, M⊆Intgδ(Clδ(Intgδ(M)) implies M∩ Intgδ(Clδ(Intgδ(M))=M. Conversely,let M = M∩ Intgδ(Clδ(Intgδ(M)) implies M ⊂ Intgδ(Clδ(Intgδ(M)). (2)It follows from (1) 242 On a class of sets between a-open sets and gδ-open sets Lemma 3.5. In (X,τ),let M ⊂ X. Then (i)M ∩ Intgδ(Clδ(Intgδ(M)) is Da-open (ii)M∪ Clgδ(Intδ(Clgδ(M)) is Da-closed. Proof: (i) Intgδ(Clδ(Intgδ(M ∩ Intgδ(Clδ(Intgδ(M)))))) = Intgδ(Clδ(Intgδ(A)∩ Intgδ(Clδ(Intgδ(M))))) = Intgδ(Clδ(Intgδ(M))). This implies that M ∩ Intgδ(Clδ(Intgδ(M))) = M ∩ Intgδ(Clδ(Intgδ(M ∩ Intgδ(Clδ(Intgδ(M)))))) ⊆ Intgδ(Clδ(Intgδ(M ∩ Intgδ(Clδ(Intgδ(M)))))) . Therefore M ∩ Intgδ(Clδ(Intgδ(M))) is Da-open. (ii) From (i) we have X\(M∪Clgδ(Intδ(Clgδ(M))) = (X\M) ∩ Clgδ(Intδ(Clgδ(X\M))) is Da-open so that M ∪Clgδ(Intδ(Clgδ(M))) is Da-closed. Lemma 3.6. In (X,τ),let M ⊂ X. Then (i)IntDa (M)=M ∩ Intgδ(Clδ(Intgδ(M)). (ii)ClDa (M)=M∪ Clgδ(Intδ(Clgδ(M)). Proof:(i)Let N=IntDa (M),then N⊂M.Since N is Da-open,N⊂Intgδ(Clδ(Intgδ(N)) ⊂Intgδ(Clδ(Intgδ(M)).Then N⊂M∩Intgδ(Clδ(Intgδ(M))⊂M.Therefore,by Lemma 3.5, it follows that M∩Intgδ(Clδ(Intgδ(M)) is a Da-open set contained in M. But IntDa (M) is the largest Da-open set contained in M it follows that M∩Intgδ(Clδ(Intgδ(M))⊂ IntDa (M)=N.Then N=M∩Intgδ(Clδ(Intgδ(M)). Therefore,IntDa (M)=M ∩ Intgδ(Clδ(Intgδ(M)). (ii)It follows from (i) 4 Da-Continuous functions. Definition 4.1. A function f:(X,τ) → (Y,η) is said be a Da-continuous if for each p∈X and each N∈O(Y,f(p)), there exists M ∈ DaO(X,p) such that f(M)⊂ N. Theorem 4.1. For a function f:(X,τ) → (Y,η),the following are equivalent (1)f is Da-continuous; (2)For each N∈O(Y),f−1(V)∈DaO(X). Proof:(1)−→(2)Let N∈O(Y) and p∈f−1(N). Since f(p) ∈ N,then by(1),there exists Mp ∈ DaO(X,p) such that f(Mp) ⊂ N.It follows that f−1(N)=∪{Mp: p∈f−1(N)}∈DaO(X), by Theorem 3.2 . (2)−→(1) Let p ∈ X and N ∈O(Y,f(p)).Then,by (2),f−1(N)∈DaO(X,p). Take M = f−1(N), then f(M) ⊂ N. Corolary 4.1. A function f:(X,τ) → (Y,η) is Da-continuous if and only if f−1(F)∈DaC(X) for each F∈C(Y). 243 Jagadeesh B.Toranagatti Remark 4.1. The following implications hold for a function f:(X,τ) → (Y,η): continuity ←− δ-continuity −→ a-continuity ↙ ↓ ↓ Da-continuity ↗ g-continuity −→ gδ-continuity Example 4.1. Consider (X,τ) as in Example 3.1 and η={X,φ,{p},{q},{p,q},{p,q,r}}. Define f:(X,σ)→(X,η) by f(p)=s,f(q)=p,f(r)=q and f(s)=r.Then f is Da-continuous but neither a-continuous nor gδ-continuous since {p,q,r} is open in (X,η), f−1({p,q,r}) = {q,r,s}∈DaO(X) but {q,r,s}/∈aO(X) and {q,r,s}/∈ gδO(X). The other Examples are shown in[3,5,21] Theorem 4.2. The following conditions are equivalent for a function f:(X,τ) → (Y,η): (1) f is Da-continuous; (2) For each subset N of Y, Clgδ(Intδ(Clgδ(f−1(N))) ⊂ f−1(Cl(N); (3)For each subset N of Y, f−1(Int(N)) ⊂ Intgδ(Clδ(Intgδ(f−1(N)); (4)For each subset N of Y,ClDa (f −1(N)) ⊂ f−1(Cl(N)); (5)For each subset M of X,f(ClDa (M)) ⊂ Cl(f(M)); (6)For each subset N of Y, f−1(Int(N)) ⊂ IntDa (f−1(N)). Proof: (1)→(2) Let N ⊂ Y.Then by (1),f−1(Cl(N)) ∈ DaC(X) implies f−1(Cl(N)⊃Clgδ(Intδ(Clgδ(f−1(Cl(N)))⊃ Clgδ(Intδ(Clgδ(f−1(N))). (2)→(3).Replace N by Y\N in (2), we have Clgδ(Intδ(Clgδ(f−1(Y\N)))⊂f−1(Cl(Y\N), and therefore f−1(Int(N)) ⊂ Intgδ(Clδ(Intgδ(f−1(N)) for each subset N of Y. (3)→(1). Clear (1)→(4). Let N ⊂ Y .Then by (1), f−1(Cl(N))∈DaC(X). Thus ClDa (f −1(N)) ⊂ ClDa (f−1(Cl(N))=f−1(Cl(N) by Theorem 3.3(4). (4)→(1). Let N ∈C(Y).Then by (4), ClDa (f −1(N)) ⊂ f−1(Cl(N)=f−1(N) implies ClDa (f−1(N))=f−1(N). Then by Theorem 3.3(4), f−1(N) ∈ DaC(X). (4)→(5).Let M ⊂ X.Then f(M) ⊂ Y.By (4), we have f−1(Cl(f(M))) ⊃ ClDa (f−1(f(M))) ⊃ ClDa (M). Therefore, f(ClDa (M)) ⊂ f(f−1(Cl(f(M))) ⊂ Cl(f(M). (5)→(4).Let N ⊂ Y and M=f−1(N) ⊂ X.Then by (5), f(ClDa (f −1(N))) ⊂ Cl(f(f−1(N)) ⊂ Cl(N) implies ClDa (f−1(N)) ⊂ f−1(Cl(N)). (4)→(6).Replace N by Y \N in (4), we get ClDa (f −1(Y\N)) ⊂ f−1(Cl(Y\N)) implies ClDa (X\f−1(N)) ⊂ f−1(Y\Int(N)) Therefore,f−1(int(N)) ⊂ IntDa (f−1(N)) for each subset N of Y. 244 On a class of sets between a-open sets and gδ-open sets (6)→(1).Let G⊂Y be open.Then f−1(G)=f−1(Int(G)) ⊂ IntDa (f−1(G) implies IntDa (f −1(G)=f−1(G).So by Theorem 3.4(4),f−1(G)∈DaO(X). Definition 4.2. Two non-empty subsets A and B of a topological space (X,τ) are said to be Da-separated if there exist two Da-open sets G and H,such that A⊂G,B⊂H, A∩H=φ and B∩G=φ. Definition 4.3. Two non-empty subsets A and B of a topological space (X,τ) are said to be strongly Da-separated if there exist two Da-open sets U and V,such that A⊂U,B⊂V and U∩V=φ. Definition 4.4. A topological space (X,τ) is said to be (1) Da-T2 if any two distinct points are strongly Da-separated in (X,τ) (2) Da-T1 if every pair of distinct points is Da-separated in (X,τ). Remark 4.2. The following implications are hold for a topological space (X,τ) a-T2 −→Da-T2←− T2 ↓ ↓ ↓ a-T1 −→Da-T1←− T1 Theorem 4.3. If an injective function f:(X,τ) → (Y,η) is Da-continuous and (Y,η) is T1, then (X,τ) is Da-T1. Proof: Let (Y,σ) be T1 and p,q∈X with p 6=q. Then there exist open subsets G, H in Y such that f(p) ∈ G, f(q) /∈ G, f(p) /∈ H and f(q) ∈ H. Since f is Da-continuous, f−1(G) and f−1(H) ∈ DaO(X) such that p ∈f−1(G), q /∈f−1(G), p /∈f−1(H) and q ∈ f−1(H). Hence,(X,σ) is Da-T1 . Theorem 4.4. If an injective function f: (X,τ) → (Y,η) is Da-continuous and (Y,η) is T2, then (X,τ) is Da-T2. Proof: Similar to the proof of Theorem 4.3 Recall that for a function f:(X,τ) → (Y,η), the subset Gf ={(x,f(x)):x ∈X}⊂ X×Y is said to be graph of f. Definition 4.5. A graph Gf of a function f:(X,τ) → (Y,η) is said to be Da-closed if for each (p,q) /∈ Gf , there exist U∈DaO(X,p) and V∈O(Y,q) such that (U×V)∩ Gf = φ. As a consequence of Definition 4.5 and the fact that for any subsets C ⊂ X and D ⊂ Y, (C×D)∩ Gf =φ if and only if f(C)∩D = φ,we have the following result. Lemma 4.1. For a graph Gf of a function f:(X,τ) → (Y,η), the following properties are equivalent: (1)Gf is Da-closed in X×Y; (2)For each (p,q) /∈Gf , there exist U∈DaO(X,p) and V∈O(Y,q) such that f(U)∩V = φ. 245 Jagadeesh B.Toranagatti Theorem 4.5. If f:(X,τ) → (Y,η) is Da-continuous and (Y,η) is T2 , then Gf is Da-closed in X×Y. Proof: Let (p,q) /∈Gf , f(p) 6=q. Since Y is T2, there exist V,W ∈O(Y) such that f(p)∈ V, q∈W and V∩W=φ. Since f is Da-continuous, f−1(V)∈DaC(X,p).Set U =f−1(V), we have f(U)⊂ V. Therefore, f(U)∩W=φ and Gf is Da-closed in X×Y Theorem 4.6. Let f:(X,τ) → (Y,η) have a Da-closed graph Gf . If f is injective, then (X,τ) is Da-T1. Proof:Let x1,x2∈X with x1 6=x2.Then f(x1)6=f(x2) as f is injective So that (x1,f(x2)) /∈Gf .Thus there exist U∈DaO(X,x1) and V∈O(Y,f(x2)) such that f(U)∩V = φ.Then f(x2)/∈f(U) implies x2 /∈U and it follows that X is Da-T1. Theorem 4.7. Let f:(X,τ) → (Y,η) have a Da-closed graph Gf . If f is surjective, then (Y,η) is T1. Proof:Let y1,y2∈Y with y1 6=y2.Since f is surjective,f(x)=y2 for some x∈X and (x,y2)/∈Gf .By Lemma 4.1,there exist U∈DaO(X,x) and V∈O(Y,y1) such that f(U)∩V = φ.It follows that y2 /∈V.Hence Y is T1. Theorem 4.8. Let f:(X,τ) → (Y,η) have a Da-closed graph Gf . If f is surjective, then (Y,η) is Da-T1. Proof:Similar to the proof of Theorem 4.7 Corolary 4.2. Let f:(X,τ) → (Y,η) have a Da-closed graph Gf . If f is bijective, then both (X,τ) and (Y,η) are Da-T1 Proof:Follows from Theorems 4.6 and 4.8 Definition 4.6. A graph Gf of a function f:(X,τ) → (Y,η) is said to be (D,a)- closed if for each (p,q) /∈ Gf , there exist U∈DaO(X,p) and V∈aO(Y,q) such that (U×aCl(V))∩ Gf = φ. Lemma 4.2. For a graph Gf of a function f:(X,τ) → (Y,η), the following proper- ties are equivalent: (1)Gf is Da-closed in X×Y; (2)For each (p,q) /∈Gf , there exist U∈DaO(X,p) and V∈aO(Y,q) such that f(U)∩aCl(V)) = φ. Theorem 4.9. Let M ⊂ X.Then x∈ a-Cl(M) if and only if G ∩ M 6= Φ, for every a-open set G containing x. Proof:Similar to the proof of Theorem 3.3(7) Theorem 4.10. Let f:(X,τ) → (Y,η) have a (D,a)-closed graph Gf . If f is surjec- tive, then (Y,η) is a-T2(resp,a-T1). Proof:Let y1,y2∈Y with y1 6=y2.Since f surjective, f(x1)=y1 x1∈X and hence (x1,y2)/∈Gf . By Lemma 4.2,there exist E∈DaO(X,x1) and F∈aO(Y,y2) such that f(E)∩ aCl(F) = φ. Now, x1∈E implies f(x1)=y1∈f(E) so that y1 /∈aCl(F).By Theorem 4.9,there exists D∈aO(Y,y1) such that D∩F=φ.Hence Y is a-T2. 246 On a class of sets between a-open sets and gδ-open sets Theorem 4.11. Let f:(X,τ) → (Y,η) have a (D,a)-closed graph Gf . If f is surjec- tive, then (Y,η) is Da-T2(resp,Da-T1). Proof:Similar to the proof of Theorem 4.10 Theorem 4.12. Let f:(X,τ) → (Y,η) have a (D,a)-closed graph Gf . If f is injective, then (X,τ) is Da-T1. Proof:Similar to the proof of Theorem 4.6 Corolary 4.3. Let f:(X,τ) → (Y,η) have a (D,a)-closed graph Gf . If f is bijective, then both (X,τ) and (Y,η) are Da-T1 Proof:Follows from Theorems 4.11 and 4.12 5 Almost Da-Continuous functions. Definition 5.1. A function f:(X,τ) → (Y,η) is said to be almost Da-continuous if for each point p ∈ X and each open subset V of Y containing f(p), there exists U ∈ DaO(X,p) such that f(U) ⊂ int(Cl(V)). Theorem 5.1. If f:(X,τ) → (Y,η) is Da-continuous function , then f is an almost Da-continuous,but not conversely. Proof:Obvious Example 5.1. Consider (X,τ) and (X,η) as in 4.1. Define f:(X,τ) → (X,η) by f(p)=p,f(q)=s,f(r)=q and f(s)=r Then f is almost Da-continuous but not Da-continuous since {p,q,r} is open in (X,η), f−1({p,q,r})={p,r,s}/∈DaO(X,τ) Definition 5.2. [Noiri and Popa, 1998] A space X is said to be semi-regular if for any open set U of X and each point x ∈ U there exists a regular open set V of X such that x ∈ V ⊂ U. Theorem 5.2. If f:(X,τ) → (Y,η) is an almost Da-continuous function and Y is semi-regular, then f is Da-continuous. Proof: Let p ∈ X and let V ∈ O(Y,f(p)). By the semi-regularity of Y , there exists G∈RO(Y,f(p)) such that G ⊂ V . Since f is almost Da-continuous, there exists U ∈ DaO(X, x) such that f(U) ⊂ Int(Cl(G)) = G ⊂ V and hence f is Da-continuous. Lemma 5.1. Let (X,τ) be a space and let A be a subset of X. The following state- ments are true: (1) A ∈ PO(X) if and only if sCl(A) = Int(Cl(A)) [Janković, 1985]. (2) A ∈ βO(X) if and only if Cl(A) is regular closed [Abd El-Monsef, 1983]. 247 Jagadeesh B.Toranagatti Theorem 5.3. Let f:(X,τ) → (Y,η) be a function. Then the following conditions are equivalent: (1) f is almost Da-continuous; (2) For every N∈RO(Y), f−1(N)∈DaO(X); (3) For every M∈RC(Y), f−1(M)∈DaC(X); (4) For each subset C of X, f(ClDa (C)) ⊂ Clδ(f(C)); (5)For each subset D of Y, ClDa (f −1(D)) ⊂ f−1(Clδ(D)); (6)For every G∈δC(Y), f−1(G)∈DaC(X); (7)For every H∈δO(Y), f−1(H)∈DaO(X); (8) For every N∈O(Y), f−1(Int(Cl(N)∈DaO(X); (9) For every M∈C(Y), f−1(Cl(Int(M)∈DaC(X); (10) For every N∈βO(Y), ClDa (f−1(N)) ⊂ f−1(Cl(N)); (11) For every M∈βC(Y), f−1(Int(M)) ⊂ IntDa (f−1(M)); (12) For every M∈SC(Y), f−1(Int(M)) ⊂ IntDa (f−1(M)); (13) For every N∈SO(Y), ClDa (f−1(N)) ⊂ f−1(Cl(N)); (14) For every M∈PO(Y), f−1(M) ⊂ IntDa (f−1(Int(Cl(M)); (15) For each p∈ X and each N∈O(Y,f(p)), there exists M ∈ DaO(X,p) such that f(M) ⊂ sCl(N); (16) For each p∈ X and each N∈RO(Y,f(p)), there exists M ∈ DaO(X,p) such that f(M) ⊂ N; (17) For each p∈ X and each N∈δO(Y,f(p)), there exists M ∈ DaO(X,p) such that f(M) ⊂ N. Proof: (1)−→(2) Similar to the proof of (1)−→(2) of Theorem 4.1. (2)−→(3) It follows from the fact that f−1(Y\F) = X \f−1(F). (3)−→(4) Suppose that D∈ δC(Y) such that f(C)⊂ D. Observe that D = Clδ(D) = ⋂ {F:D⊂F and F∈RC(Y)} and so f−1(D) = ⋂ {f−1(F):D⊂F and F∈RC(Y)}. By (3) and Corollary 3.1,we have f−1(D)∈DaC(X) and C⊂f−1(D). Hence ClDa (C) ⊂f−1(D), and it follows that f(ClDa (C) ) ⊂ D. Since this is true for any δ-closed set D containing f(C), we have f(ClDa (C))⊂ Clδ(f(C)). (4)−→(5) Let D ⊂ Y, then f−1(D) ⊂ X. By (4), f(ClDa (f −1(D)))⊂ Clδ(f(f−1(D)))⊂Clδ(D). So that ClDa (f −1(D)) ⊂ f−1(Clδ(D)). (5)−→(6) Let G∈δC(Y) Then by (5), ClDa (f−1(G)) ⊂ f−1(Clδ(G))=f−1(G). In consequence, ClDa (f −1(G))=f−1(G) and hence by Theorem 3.3(4), f−1(G)∈DaC(X). (6)−→(7):Clear. (7)−→(1): Let p∈ X and let O∈O(Y,f(p)). Set D = Int(Cl(O)) and C =f−1(D). Since D∈ δO(Y), then by (7), C = f−1(D) ∈ DaO(X). Now, f(p) ∈ O= Int(O)⊂ Int(Cl(O)) = D it follows that p∈f−1(D)=C and f(C)=f(f−1(D)⊂D=Int(Cl(O). (2)←→(8): Let N ∈O(Y). Since Int(Cl(N))∈RO(Y),by (2), f−1(Int(Cl(N))∈DaO(X). The converse is similar. (3)←→(9)It is similar to (8)←→(2). 248 On a class of sets between a-open sets and gδ-open sets (3)−→ (10): Let N∈βO(Y).Then by Lemma 5.1(2),Cl(N) ∈ RC(Y).So by(3),f−1(Cl(N)) ∈DaC(X) .Since f−1(N) ⊂f−1(Cl(N)) and by Theorem 3.3(4),ClDa (f−1(N))⊂f−1(Cl(N)). (10)−→ (11): and (12)−→ (13):Follows from Lemma 3.3 (11)−→ (12):It follows from the fact that SC(Y)⊂βC(Y) (13)−→ (3):It follows from the fact that RC(Y)⊂SO(Y). (2)←→ (14): Let N ∈PO(Y). Since Int(Cl(N)) ∈ RO(Y),then by (2), f−1(Int(Cl(N))) ∈ DaO(X) and hence f−1(N) ⊂ f−1(int(Cl(N))) = IntDa (f−1(int(Cl(N)))). Conversely,let N∈RO(Y). Since N ∈ PO(Y), f−1(N) ⊂ IntDa (f−1(int(Cl(N)))) =IntDa (f−1(N)). In consequence, IntDa (f −1(N))=f−1(N) and by Theorem 3.4, f−1(N) ∈ DaO(X). (1)−→ (15): Let p∈X and N∈O(Y,f(p)). By (1), there exists M∈ DaO(X,p) such that f(M) ⊂ Int(Cl(N)).Since N∈PO(Y),by Lemma 5.1, f(M) ⊂ sCl(N). (15)−→ (16): Let p∈ X and N∈RO(Y,f(p)). Since N∈O(Y,f(p)) and by (15), there exists M∈DaO(X,p) such that f(M)⊂ sCl(N). Since N ∈PO(Y), then by Lemma 5.1, f(M) ⊂Int(Cl(N)) = N. (16)−→ (17):Let p∈ X and V∈δO(Y,f(p)). Then, there exists G∈O(Y.f(p))such that G ⊂ Int(Cl(G)) ⊂ N. Since Int(Cl(G))∈RO(Y,f(p)), by (16), there exists M∈ DaO(X,p) such that f(M) ⊂ Int(Cl(G))⊂ N. (17)−→(1). Let p∈ X and N∈O(Y,f(p)). Then Int(Cl(N))∈ δO(Y,f(p)). By (17), there exists M∈ DaO(X,p) such that f(M) ⊂ Int(Cl(N)). Therefore,f is almost con- tinuous Theorem 5.4. If f:(X,τ) → (Y,η) is an almost Da-continuous injective function and (Y,η) is r-T1 , then (X,σ) is Da-T1 . Proof: It is similar to the proof of Theorem 4.3 Theorem 5.5. If f:(X,τ) → (Y,σ) is an almost Da-continuous injective function and (Y,σ) is r-T2 , then (X,τ) is Da-T2 . Proof: It is similar to the proof of Theorem 4.4 Lemma 5.2. [Ayhan and Ozkoç, 2016] Let (X,τ) be a space and let A be a subset of X. Then: A ∈e∗O(X) if and only if Clδ(A) is regular closed. Theorem 5.6. For a function f:(X,τ) → (Y,η),the following are equivalent: (a) f is almost Da-continuous; (b) For every e∗-open set N in Y,f−1(Clδ(N)) is Da-closed in X; (c) For every δ-semiopen subset N of Y,f−1(Clδ(N)) is Da-closed set in X; (d) For every δ-preopen subset N of Y,f−1(Int(Clδ(N))) is Da-open set in X; (e) For every open subset N of Y,f−1(Int(Clδ(N))) is Da-open set in X; (f) For every closed subset N of Y,f−1(Cl(Intδ(A))) is Da-closed set in X . Proof: (a)→(b):Let N∈e∗O(Y) Then by Lemma 5.2,Clδ(N)∈RC(Y). 249 Jagadeesh B.Toranagatti By (a),f−1(Clδ(N))∈DaC(X). (b)→(c):Obvious since δSO(Y)⊂ e∗O(Y). (c)→(d):Let N ∈δPO(Y),then Intδ(Y\N)∈δ-SO(Y).By (c), f−1(Clδ(Intδ(Y\N))∈DaC(X) which implies f−1(Int(Clδ(N))∈DaO(X). (d)→(e):Obvious since O(Y)⊂ δPO(Y). (e)→(f):Clear (f)→(a):Let N∈RO(Y).Then N=Int(Clδ(N)) and hence Y\N∈C(X). By (f), f−1(Y\N)=X\f−1(Int(Clδ(N)))=f−1(Cl(Intδ(Y\N))∈DaC(X). Thus f−1(N)∈DaO(X). Lemma 5.3. 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