Ratio Mathematica Volume 42, 2022 A new class of almost continuity in topological spaces Jagadeesh B.Toranagatti* Abstract In this paper, we apply the notion of δgβ-open sets due to Ben- challi et al.[Benchalli et al., 2017] to present a new class of functions called almost δgβ-continuous functions along with its several proper- ties, characterizations and mutual relationships. Keywords: almost continuity,almost β-continuity, δgβ-continuity,almost δgβ-continuity. 2020 AMS subject classifications: 54A05,54C08. 1 *Department of Mathematics, Karnatak University’s Karnatak College, Dharwad - 580001, Karnataka, India; jagadeeshbt2000@gmail.com 1Received on January 28th, 2022. Accepted on May 22nd, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v39i0.708. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 91 Jagadeesh B.Toranagatti 1 Introduction The notion of continuity is an important concept in general topology as well as all branches of mathematics. Of course it′s weak forms and strong forms are important, too. Singal and Singal[Singal et al., 1968], in 1968, defined almost continuous functions as a generalization of continuity.Noiri and Popa[Noiri and Popa, 1998], in 1998, defined almost β-continuous functions as a generalization of almost continuity.Recently, Benchalli et al.[Benchalli et al., 2017] introduced the notion of δgβ-continuous functions in topological spaces. In this article, using the notion of δgβ-open sets given in [Benchalli et al., 2017], we introduce and study a new class of functions called almost δgβ-continuous functions. We investigate several properties of this class. The class of almost δgβ- continuity is a generalization of almost β-continuity and δgβ-continuity. 2 Preliminaries Throughout this paper (X,τ),(Y,σ) and (Z,η)(or simply X,Y and Z ) represent nonempty topological spaces on which no separation axioms are assumed unless otherwise stated. Let M be a subset of X. The closure of M and the interior of M are denoted by cl(M) and int(M), respectively. Definition 2.1. A set M ⊆ X is called β-closed[Abd, 1983](=semi preclosed[Andrijević, 1986] (resp.,pre-closed[Mashhour, 1982], regular closed[Stone, 1937], semi-closed [Levine, 1963] if int(cl(int(M)) ⊆ M (resp.,cl(int(M)) ⊆ M, M = cl(int(M),int(cl(M)) ⊆ M) . Definition 2.2. A set M ⊆ X is called δ-closed [Velicko, 1968] if M = clδ(M) where clδ(M) = { p ∈ X :int(cl(N)) ∩ M ̸= ϕ, N ∈ τ and p ∈ N }. Definition 2.3. A set M ⊆ X is called gβ-closed [Tahiliani, 2006](resp., gspr- closed [Navalagi et al.] and δgβ-closed[Benchalli et al., 2017] if βcl(M) ⊆ G whenever M ⊆ G and G is open(resp. regular open and δ-open) in X. The complements of the above mentioned closed sets are their respective open sets. The class of δgβ-open (resp., δgβ-closed, open, closed, regular open, regular closed, preopen, semiopen and β-open) sets of (X,τ) containing a point p ∈ X is denoted by δGβO(X,p)(resp., δGβC(X,p),O(X,p), C(X,p), RO(X,p), RC(X,p), PO(X,p), SO(X,p)and βO(X,p)). 92 A new class of almost continuity in topological spaces Definition 2.4. A function f:X → Y is called almost continuous[Singal et al., 1968](resp., almost β-continuous[Noiri and Popa, 1998] and almost gspr-continuous) if the inverse image of every regular open set G of Y is open (resp., β-open and gspr-open) in X. Definition 2.5. [Benchalli et al., 2017] A function f:X → Y is called δgβ-continuous (resp.,δgβ-irresolute) if the inverse image of every open(resp.,δgβ-open) set G of Y is δgβ-open in X. Definition 2.6. A function f:X → Y is called almost contra continuous [Baker, 2006](resp. almost contra super-continuous[Ekici, 2004] and contr R-map[Ekici, 2006] if the inverse image of every regular closed set G of Y is open(resp. δ-open and regular open) in X. Definition 2.7. A space X is said to be: (i) nearly compact [Singal and Mathur, 1969] if every regular open cover of X has a finite subcover, (ii) r-T1-space[Ekici, 2005] 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, (iii) r-T2-space [Ekici, 2005] 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, (iv) δgβ-T1 space if for any pair of distinct points p and q, there exist G,H ∈δGβO(X) such that p ∈ G, q /∈ G and q ∈ H, p /∈H, (v) δgβ-T2 space[BENCHALLI et al., 2017] if for each pair of distinct points x and y of X, there exist G,H ∈δGβO(X) such that x ∈ G, y ∈ H and G∩H =ϕ. Definition 2.8. [Benchalli et al., 2017] A space X is said to be Tδgβ(resp.,δgβT 1 2 )- space if δGβO(X)=O(X) (rep.,δGβO(X) = βO(X)). Definition 2.9. [Carnahan, 1972] A subset M of a space X is said to be N-closed relative to X if every cover of M by regular open sets of X has a finite subcover. Theorem 2.1. [Benchalli et al., 2017] If A and B are δgβ-open subsets of a ex- tremely disconnected and submaximal space X, then A∩B is δgβ-open in X. Definition 2.10. [Jankovic, 1983] A space X is called locally indiscrete if O(X)=RO(X). Lemma 2.1. [Noiri, 1989] Let (X,τ) be a space and let M be a subset of X. M ∈ PO(X) if and only if scl(M) = int(cl(M)). 93 Jagadeesh B.Toranagatti 3 Almost δgβ-continuous functions Definition 3.1. A function f: X → Y is said to be almost δgβ-continuous at p ∈ X if for each N ∈ δO(Y,f(p)), there exists M ∈ δGβO(X,p) such that f(M) ⊆ N. If f is almost δgβ-continuous at every point of X, then it is called almost δgβ-continuous. Remark 3.1. We have the following implications almost β-continuity−→ almost δgβ-continuity−→almost gspr-continuity. ↑ δgβ-continuity. None of these implications is reversible. Example 3.1. Let X = {p,q,r,s}, τ = {X, ϕ, {p}, {q}, {p,q}, {p,q,r}} and σ = {Y, ϕ, {p}, {q}, {p,q}, {p,r}, {p,q,r}}. Define f: (X,τ) → (X,σ) by f(p) = f(r) = q , f(q) = p and f(s) = r. Clearly f is almost δgβ-continuous but for {q}∈ RO((X,σ), f−1({q}) = {p,r} /∈ GβO(X,τ). Therefore f is not almost gβ-continuous. Define g: (X,τ) → (X,σ) by g(p) = p, g(q) = s, g(r) = r and g(s) = q.Then g is almost δgβ-continuous but for {p} ∈ O(X,σ), g−1({p}) = {p} /∈ δGβO(X,τ).Therefore g is not δgβ-continuous. Define h: (X,τ) → (X,σ) by h(p) = h(q) = q, h(r) = p and h(s) = r.Then h is almost gspr-continuous but for {q}∈ RO(X,σ), h−1({q}) = {p,q} /∈ δGβO(X,τ). Therefore h is not almost δgβ-continuous Theorem 3.1. If f:X→Y is almost δgβ-continuous and Y is locally indiscrete space,then f is δgβ-continuous. Proof: It follows from the Definition 2.10 Theorem 3.2. Let X be a locally indiscrete space and M⊆X,then the following properties are equivalent: (i) M is gspr-closed; (ii) M is δgβ-closed; (iii) M is gβ-closed. As a consequence of above Theorem,we have the following result; Theorem 3.3. Let X be a locally indiscrete space,then the following properties are equivalent: (i) f:X→Y is almost gspr-continuous; (ii) f:X→Y is almost δgβ-continuous; (iii) f:X→Y is almost gβ-continuous. 94 A new class of almost continuity in topological spaces Theorem 3.4. Let X be a δgβT 1 2 -space. Then the following are equivalent: (i) f: X → Y is almost β-continuous; (ii) f: X → Y is almost gβ-continuous; (iii) f: X → Y is almost δgβ-continuous. Theorem 3.5. Let X be a Tδgβ-space. Then the following are equivalent: (i) f: X → Y is almost continuous; (ii) f: X → Y is almost β-continuous; (iii) f: X → Y is almost gβ-continuous; (iv) f: X → Y is almost δgβ-continuous; (v) f: X → Y is almost gspr-continuous. Lemma 3.1. [BENCHALLI et al., 2017] For a subset M of a space X ,the follow- ing are equivalent: (i) M is δ-open and δgβ-closed; (ii) M is regular open; (iii) M is open and β-closed. Theorem 3.6. The following statements are equivalent for a f: X → Y: (i) f is almost contra super-continuous and almost δgβ-continuous; (ii) f is contra R-map; (iii) f is almost contra continuous and almost b-continuous. Theorem 3.7. The following statements are equivalent for a f: X → Y: (i) f is almost δgβ-continuous; (ii) For each point p∈X and each G∈δC(Y) with f(p) /∈G,there exists a H ∈δGβC(X) and p /∈H such that f−1(G)⊆H; (iii) For each point p∈X and each N∈RO(Y,f(p)),there exists an M ∈ δGβO(X,p) such that f(M)⊆N; (iv) For each point p∈X and each G∈RC(Y) with f(p) /∈G,there exists a H ∈ δGβC(X) and p /∈H such that f−1(G)⊆H; 95 Jagadeesh B.Toranagatti (v) for each p ∈ X and each N∈O(Y,f(p)),there exists M ∈ δGβO(X,p) such that f(M) ⊂ int(cl(N)); (vi) for each p ∈ X and each N∈O(Y,f(p)), there exists M ∈ δGβO(X,p) such that f(M) ⊂ scl(N). Proof: (i)←→(ii)−→(iv)←→(iii)←→(v)←→(vi): obvious. (iii)−→(i): Let N ∈ δO(Y) such that f(p) ∈ N, then there exists G ∈ RO(Y) such that f(p) ∈ G ⊆ N. By (iii), there exists an M ∈ δGβO(X,p) such that f(M)⊆G ⊆ N. Definition 3.2. A space X is said to be δgβ-additive if δGβO(X) is closed under arbitrary union. Theorem 3.8. Let X be a δgβ-additive space.Then M ⊆ X is δgβ-closed(resp., δgβ-open) if and only if δgβcl(M) = M (resp., δgβint(M) = M ). Theorem 3.9. The following statements are equivalent for a f: X → Y where X is δgβ-additive: (i) f is almost δgβ-continuous; (ii) f(δgβcl(M)) ⊆ clδ(f(M)) for each M ⊆ X; (iii) δgβcl(f−1(N)) ⊆ f−1(clδ(N)) for each N ⊆ Y; (iv) f−1(G)∈δGβC(X) for each δ-closed set G of Y; (v) f−1(H)∈δGβO(X) for each δ-open set H of Y; (vi) f−1(G)∈δGβC(X) for each regular closed set G of Y; (vii) f−1(H)∈δGβO(X) for each regular open set H of Y. Proof: (i)−→(ii) Let N ∈ δC(Y) such that f(M) ⊆ N. Observe that N = clδ(N) =⋂ {F:N ⊆ F and F ∈ RC(Y)} and so f−1(N) = ⋂ {f−1(F):N ⊆ F and F ∈ RC(Y)}. By (i) and Definition 3.2, we have f−1(N) ∈ δGβC(X) and M ⊆ f−1(N). Hence δgβcl(M) ⊆f−1(N), and it follows that f(δgβcl(M)) ⊆ N. Since this is true for any δ-closed set N containing f(M), we have f(δgβcl(M)) ⊆ clδ(f(M)). (ii)−→(iii) Let D ⊆ Y, then f−1(D) ⊆ X. By (ii), f(δgβcl(f−1(D))) ⊆ clδ(f(f−1(D))) ⊆δgβcl(D). So that δgβcl(f−1(D)) ⊆f−1(Clδ(D)). (iii)−→(iv) Let G be a δ-closed subset of Y.Then by (iii), δgβcl(f−1(G)) ⊆f−1(clδ(G)) = f−1(G).In consequence, δgβcl(f−1(G)) = f−1(G) and hence by Theorem 3.8, f−1(G) ∈ δGβC(X). (iv)−→(v):Clear. (v)−→(i): Let N ∈ RO(Y).Then N is δ-open in Y. By (v), f−1(N) ∈ δGβO(X). Hence f is almost δgβ-continuous 96 A new class of almost continuity in topological spaces Theorem 3.10. The following statements are equivalent for a f: X → Y where X is δgβ-additive: (i) f is almost δgβ-continuous; (ii) For every open subset K of Y,f−1(int(cl(K)∈δGβO(X); (iii) For every closed subset M of Y,f−1(cl(int(M)∈δGβC(X); (iv) For every β-open subset K of Y,δgβcl(f−1(K)) ⊆ f−1(cl(K)); (v) For every β-closed subset M of Y,f−1(int(M)) ⊆ δgβint(f−1(M)); (vi) For every semi-closed subset M of Y,f−1(int(M)) ⊆ δgβint(f−1(M)); (vii) For every semi-open subset K of Y,δgβcl(f−1(K)) ⊆ f−1(cl(K)); (viii) For every pre-open subset M of Y,f−1(M) ⊆ δgβint(f−1(int(cl(M)). Proof: (i)←→(ii): Let K ⊆ Y. Since int(cl(N)) is regular open in Y. Then by (i), f−1(int(cl(N)) ∈ δGβO(X). The converse is similar. (i)←→(iii)It is similar to (i)←→(ii). (i)−→ (iv): Let K ∈ βO(Y),then cl(K) is regular closed in Y. So by(i),f−1(cl(K)) ∈ δGβC(X). Since f−1(N) ⊆ f−1(cl(N)),then δgβcl(f−1(N)) ⊆ f−1(cl(N)). (iv)−→ (v) and (vi)−→ (vii):Obvious (v)−→ (vi):It follows from the fact that every semiclosed set is β-closed. (vii)−→ (i):It follows from the fact that every regular closed set is semi-open. (i)←→ (viii): Let M ∈ PO(Y). Since int(cl(N)) is regular open in Y,then by (i), f−1(int(cl(N))) ∈δGβO(X) and hence f−1(N) ⊆f−1(int(cl(N))) = δgβint(f−1(int(cl(N)))). Conversely,let H ∈ RO(Y). Since H is preopen in Y, f−1(H) ⊆δgβint(f−1(int(Cl(N)))) = δgβint(f−1(N)),in consequence, δgβint(f−1(H))=f−1(H) and by Theorem 3.8, f−1(N) ∈ δGβO(X). Theorem 3.11. The following statements are equivalent for a f: X → Y where X is δgβ-additive: (i) f is almost δgβ-continuous; (ii) For every e∗-open set K of Y,f−1(clδ(K)) is δgβ-closed in X; (iii) For every δ-semiopen subset K of Y,f−1(clδ(K)) is δgβ-closed set in X; (iv) For every δ-preopen subset K of Y,f−1(int(clδ(K))) is δgβ-open set in X; (v) For every open subset K of Y,f−1(int(clδ(K))) is δgβ-open set in X; 97 Jagadeesh B.Toranagatti (vi) For every closed subset K of Y,f−1(cl(intδ(K))) is δgβ-closed set in X. Proof: (i)→(ii):Let K be a e∗-open subset of Y. Then by Lemma 2.7 of [Ayhan and Özkoç, 2018], clδ(K) ∈ RC(Y). By (i),f−1(clδ(K)) ∈ δGβC(X). (ii)→(iii):Obvious since every δ-semiopen set is e∗-open. (iii)→(iv):Let K be a δ-preopen subset of Y,then intδ(Y\K) ∈ δSO(Y). By (iii), f−1(clδ(intδ(Y\K)) ∈ δGβC(X) which implies f−1(int(clδ(K)) ∈ δGβO(X). (iv)→(v):Obvious since every open set is δ-preopen. (v)→(vi):Clear (vi)→(i):Let K ∈ RO(Y). Then K = int(clδ(K)) and hence (Y\K) is closed in X. By (vi), f−1(Y\K) = X\f−1(int(clδ(K))) = f−1(cl(intδ(Y\K)) ∈ δGβC(X). Thus f−1(K) is δgβ-open in X. Theorem 3.12. The following are equivalent for a function f: X → Y where X is δgβ-additive: (i) f is almost δgβ-continuous; (ii) For every e∗-open subset G of Y,f−1(a-cl(G)) is δgβ-closed set in X; (iii) For every δ-semiopen subset G of Y,f−1(δ-pcl(G)) is δgβ-closed set in X; (iv) For every δ-preopen subset G of Y,f−1(δ-scl(G))) is δgβ-open set in X. Proof:Follows from the Lemma 3.1 of [Ayhan and Özkoç, 2018] Theorem 3.13. If an injective function f:X → Y is almost δgβ-continuous and Y is r-T1, then X is δgβ-T1. Proof: Let (Y,σ) be r-T1 and p1,p2 ∈ X with p1 ̸= p1. Then there exist regular open subsets G, H in Y such that f(p1) ∈ G, f(p2) /∈ G, f(p1) /∈ H and f(p2) ∈ H. Since f is almost δgβ-continuous, f−1(G) and f−1(H) ∈ δGβO(X) such that p1 ∈f−1(G), p2 /∈ f−1(G), p1 /∈ f−1(H) and p2 ∈ f−1(H). Hence X is δgβ-T1 . Theorem 3.14. If f:X → Y is an almost δgβ-continuous injective function and Y is r-T2, then X is δgβ-T2. Proof: Similar to the proof of Theorem 3.13 Theorem 3.15. If f,g:X → Y are almost δgβ-continuous where X is submaximal, extremely disconnected and δgβ-additive and Y is Hausdorff, then the set {x ∈ X : f(x) = g(x)} is δgβ-closed in X. Proof: Let D = {x ∈ X : f(x) = g(x)} and x /∈ (X\D). Then f(x) ̸= g(x). Since Y is Hausdorff, there exist open sets V and W of Y such that f(x) ∈ V, g(x)∈ W and V ∩ W = ϕ, hence int(cl(V)) ∩ int(cl(W)) = ϕ. Since f and g are almost δgb-continuous, there exist G,H ∈ δGBO(X,x)) such that f(G) ⊆ int(cl(V )) and 98 A new class of almost continuity in topological spaces g(H) ⊆ int(cl(W)). Now, put U = G ∩ H, then U ∈ δGBO(X,x)) and f(U) ∩ g(U) ⊆ int(cl(V)) ∩ int(cl(W)) =ϕ. Therefore, we obtain U ∩ D = ϕ and hence x /∈ δgbcl(D) then D = δgbcl(D). Since X is δgb-additive, D is δgb-closed in X. Definition 3.3. A space X is called δgβ-compact if every cover of X by δgβ-open sets has a finite subcover. Definition 3.4. A subset M of a space X is said to be δgβ-compact relative to X if every cover of M by δgβ-open sets of X has a finite subcover. Theorem 3.16. If f:X → Y is almost δgβ-continuous and K is δgβ-compact rela- tive to X, then f(K) is N-closed relative to Y. Proof: Let { Gα: α ∈ Ω } be any cover of f(K) by regular open sets of Y . Then {f−1(Gα):α∈Ω} is a cover of K by δgβ-open sets of X. Hence there exists a finite subset Ωo of Ω such that K ⊂∪{f−1(Gα):α∈Ωo }. Therefore, we obtain f(K) ⊂ {Gα: α∈Ωo}. This shows that f(K) is N-closed relative to Y . Corolary 3.1. If a surjective function f:X → Y is almost δgβ-continuous and X is both δgβ-compact and δgβ-additive, then Y is nearly compact. Lemma 3.2. Let X be a δgβ-compact , submaximal and extremely disconnected and N⊂X.Then N is δgβ-compact relative to X if N is δgβ-closed. Proof: Let { Bα: α ∈ Ω } be a cover of N by δgβ-open sets of X. Note that (X-N) is δgβ-open and that the set (X-N) ∪{ Bα: α ∈ Ω } is a cover of X by δgβ-open sets. Since X is δgβ-compact, the exists a finite subset Ωo of Ω such that the set (X-N) ∪{ Bα: α ∈ Ωo } is a cover of X by δgβ-open sets in X. Hence { Bα: α ∈ Ωo } is a finite cover of N by δgβ-open sets in X. Theorem 3.17. If the graph function g: X → X×Y of f: X → Y,defined by g(x)=(x,f(x)) for each x∈X is almost δgβ-continuous. Then f is almost δgβ-continuous. Proof:Let N∈RO(Y), then X×V ∈ RO(X×Y). As g is almost δgβ-continuous, f−1(N) = g−1(X×N) ∈ δGβO(X). Theorem 3.18. If the graph function g: X → X×Y of f: X → Y, defined by g(x)=(x,f(x)) for each x ∈ X. If X is a submaximal and extremely disconnected space and δgβ-additive, then g is almost δgβ-continuous if and only if f is almost δgβ-continuous. Proof: We only prove the sufficiency. Let x ∈ X and W ∈RO(X×Y). Then there exist regular open sets U1 and V in X and Y, respectively such that U1×V ⊂ W. If f is almost δgβ-continuous, then there exists a δgβ-open set U2 in X such that x ∈ U2 and f(U2)⊂V . Put U = (U2∩U2).Then U is δgβ-open and g(U) ⊂ U1×V ⊂ W. Thus g is almost δgβ-continuous. Recall that for a f:X → Y, the subset Gf = {(x,f(x)):x ∈X}⊂ X×Y is said to be graph of f. 99 Jagadeesh B.Toranagatti Definition 3.5. A graph Gf of a function f:X → Y is said to be strongly δgβ- closed if for each (p,q) /∈ Gf , there exist V∈δGβO(X,p) and W∈RO(Y,q) such that (V×W)∩ Gf = ϕ. Lemma 3.3. For a graph Gf of a function f: X → Y, the following properties are equivalent: (i) Gf is strongly δgβ-closed in X×Y; (ii) For each (p,q) /∈Gf , there exist U∈δGβO(X,p) and V∈RO(Y,q) such that f(U)∩V = ϕ. Theorem 3.19. Let f: X → Y have a strongly δgβ-closed graph Gf . If f is injec- tive, then X is δgβ-T1. Proof:Let x1,x2∈X with x1 ̸=x2.Then f(x1)̸=f(x2) as f is injective So that (x1,f(x2)) /∈Gf .Thus there exist U∈δGβO(X,x1) and V∈RO(Y,f(x2)) such that f(U)∩V = ϕ.Then f(x2) /∈f(U) implies x2 /∈U and it follows that X is δgβ-T1. Theorem 3.20. Let f: X → Y and g: Y → Z be any two functions. (i) If f is δgβ-continuous and g is almost continuous, then (g◦f) is almost δgβ- continuous. (ii) If f is δgβ-irresolute and g is almost δgβ-continuous,then (g◦f) is almost δgβ- continuous. (iii) If f is almost δgβ-continuous and g is R-map, then (g◦f) is almost δgβ- continuous. Proof:(i) Let N ∈ RO(Z). Then g−1(N) is open in Y since g is almost continuous. The δgβ-continuity of f implies f−1[g−1(N))] = (g◦f)−1((N)) ∈ δGβO(X). Hence g◦f is almost δgβ-continuous. The proofs of (ii) and (iii) are similar to (i). Definition 3.6. A function f: X → Y is said to be δgβ∗- continuous if for each p ∈ X and each N∈O(Y,f(p)), there exists M ∈ δGβO(X,p) such that f(M) ⊂ cl(N). Theorem 3.21. If f: X → Y is δgβ∗-continuous and K is δgβ-compact relative to X, then f(K) is H-closed relative to Y. Proof: Similar to the proof of Theorem 3.16 Theorem 3.22. If for each pair of distinct points p1 and p2 in a space X, there exists a function f of X into a Hausdorff space Y such that (i) f(p1) ̸= f(p2), 100 A new class of almost continuity in topological spaces (ii) f is δgβ∗-continuous at p1 and (iii) almost δgβ-continuous at p2,then X is δgβ-T2. Proof:As Y is Hausdorff, there exist disjoint open sets W1 and W2 of Y such that f(p)∈ W1, f(q) ∈ W2. Hence cl(W1) ∩ int(cl(W2)) = ϕ. Since f is δgβ∗-continuous at p1, there exists U1 ∈ δGβO(X,p1) such that f(U1) ⊂ cl(W1). Since f is almost δgβ-continuous at p2, there exists U2 ∈δGβO(X,p2) such that f(U2) ⊂ int(cl(W2)). Therefore, we obtain U1∩ U2 = ϕ, X is δgβ-T2. 4 Conclusion The notions of closed sets and continuous functions have been found to be useful in computer science and digital topology[[Khalimsky et al., 1990],[Kong et al., 1991]]. Professor El-Naschie[El Naschie, 2000] showed that the notion of fuzzy topology may be related to quantum physics in connection with string theory and ϵ∞ theory. Therefore, the fuzzy topological version of the notions and results given in this paper will turn out to be useful in quantum physics. 5 Acknowledgment The author is thankful to the Karnatak University Dharwad for financial sup- port to this research work under Karnatak University Research Seed Grant Policy (grant no.: KU/PMEB/2021-22/182/723 Dated:12-11-2021). 101 Jagadeesh B.Toranagatti References ME Abd. El-monsef, sn el-deeb and ra mahmoud, β-open and β-continuous map- pings. Bull. Fac. Sci. Assiut Univ, 12:77–90, 1983. Dimitrije Andrijević. Semi-preopen sets. 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