Mathematics - 377 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 On Generalized Regular Continuous Functions In Topological Spaces S. I. Mahmood Department of Mathematics, College of Science Al-Mustansiriyah Universityof Baghdad Received in: 30 January 2012 Accepted in: 21 May 2012 Abstract In this paper we introduce a new type of functions called the generalized regular continuous functions .These functions are weaker than regular continuous functions and stronger than regular generalized continuous functions. Also, we study some characterizations and basic properties of generalized regular continuous functions .Moreover we study another types of generalized regular continuous functions and study the relation among them . Key words: generalized regular continuous functions, regular continuous functions and regular generalized continuous functions. Introduction The concept of regular continuous functions was first introduced by Arya,S.P. and Gupta,R.[1]. Later Palaniappan,N. and Rao,K.C.[2] studied the concept of regular generalized continuous functions. Also, the concept of generalized regular closed sets in topological spaces was introduced by Bhattacharya,S.[3] .The purpose of this paper is to introduce a new class of functions, namely, generalized regular continuous functions . This class is placed properly between the class of regular continuous functions and the class of regular generalized continuous functions. Also, we study some characterizations and basic properties of generalized regular continuous functions . Moreover we study the perfectly generalized regular continuous functions, contra generalized regular continuous functions, generalized regular irresolute functions, contra generalized regular irresolute functions and we study the relation among them . Throughout this paper ),X( τ , ),Y( τ′ and ),Z( τ′′ (or simply X ,Y and Z) represent non- empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned .When A is a subset of X , )A(cl , A and cA denote the closure ,the interior and the complement of a set A respectively . Preliminaries First we recall the following definitions: (1.1)Definition: A subset A of a topological space X is said to be : i) A generalized closed (briefly g-closed) set [4] if U)A(cl ⊆ whenever UA ⊆ and U is open in X . ii) A regular closed (briefly r-closed) set [5] if A))A(int(cl = . Mathematics - 378 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 iii) A regular generalized closed (briefly rg-closed) set [2] if U)A(cl ⊆ whenever UA ⊆ and U is regular open in X . iv) A generalized regular closed (briefly gr-closed) set [3] if U)A(rcl ⊆ whenever UA ⊆ and U is open in X ,where }XofsubsetclosedregularaisF,FA:F{)A(rcl ⊆=  . v) A b-closed set [6] if A))A(int(cl))A(clint( ⊆ . vi) A generalized b-closed (briefly gb-closed ) set [7] if U)A(bcl ⊆ whenever UA ⊆ and U is open in X ,where }XofsubsetclosedbaisF,FA:F{)A(bcl −⊆=  . The complement of a g-closed (resp. r-closed ,rg-closed ,gr-closed ,b-closed ,gb-closed) set is called a g-open (resp. r-open ,rg-open ,gr-open, b-open ,gb-open) set . Remarks : 1) closed sets and gr-closed sets are in general independent .Consider the following examples:- Examples: i)Let }c,b,a{X = and }}c,b{},a{,,X{ φ=τ . Then }b{A = is a gr-closed set ,but not closed . ii)Let }c,b,a{X = and }}b{},c,b{},b,a{,,X{ φ=τ . Then }c{A = is a closed set ,but not gr- closed . 2)Every gr-closed set is a g-closed set ,but the converse in general is not true .In (1) no.(ii), }c{A = is a g-closed set, but not gr-closed . 3)Every gr-closed set is a rg-closed set, but the converse in general is not true .In (1) no.(ii), }c{A = is a rg-closed set ,but not gr-closed . 4)Every gr-closed set is a gb-closed set, but the converse in general is not true .In (1) no.(ii), }c{A = is a gb-closed set ,but not gr-closed . 5)Every r-closed set is a gr-closed(resp. g-closed ,gb-closed, rg-closed) set, but the converse in general is not true .In(1)no.(i), }b{A = is a gr-closed (resp. g-closed ,gb-closed, rg-closed) set ,but not r-closed . Definition: The intersection of all gr-closed subsets of X containing a set A is called the generalized regular-closure of A and is denoted by grcl(A). if A is a gr-closed set, then grcl(A) = A. The converse is not true, since the intersection of gr- closed sets need not be gr-closed .[3]. Theorem: Let A be a subset of a topological space X .Then )A(grclx ∈ if and only if for any gr-open set U containing φ≠UA,x  . Proof: ⇒ Let )A(grclx ∈ and suppose that, there is a gr-open set U in X s.t Ux ∈ and φ=UA  cUA ⊂⇒ which is gr-closed in X ⇒ .U)U(grcl)A(grcl cc =⊆ Ux ∈ ⇒ cUx ∉ ⇒ )A(grclx ∉ , this is a contradiction. Conversely, Suppose that, for any gr-open set U containing x , φ≠UA  .To prove that )A(grclx ∈ . Suppose that )A(grclx ∉ ,then there is a gr-closed set F in X such that Fx ∉ and FA ⊆ . cFxFx ∈⇒∉ which is gr-open in X . φ=⇒⊆ cFAFA  , this is a contradiction . Thus )A(grclx ∈ . Definition: A function YX:f → from a topological space X into a topological space Y is called : 1)A generalized continuous (briefly g-continuous)[8] if )V(f 1− is g-closed set in X for every closed set V in Y . Mathematics - 379 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 2)A regular generalized continuous (briefly rg-continuous) [2] if )V(f 1− is rg-closed set in X for every closed set V in Y . 3)A regular continuous (briefly r-continuous)[1] if )V(f 1− is r-closed set in X for every closed set V in Y . 4) A generalized b-continuous (briefly gb-continuous)[9] if )V(f 1− is gb-closed set in X for every closed set V in Y . 5) A generalized irresolute (briefly g-irresolute)[8] if )V(f 1− is g-closed set in X for every g- closed set V in Y . 6) A regular generalized irresolute (briefly rg-irresolute) [2] if )V(f 1− is rg-closed set in X for every rg-closed set V in Y . Generalized Regular Continuous Functions In this section we introduce the concept of generalized regular continuous functions in topological spaces and study the characterizations and basic properties of generalized regular continuous functions.Also,we study another types of generalized regular continuous functions and we study the relation among them . Definition: A function YX:f → from a topological space X into a topological space Y is called a generalized regular continuous (briefly gr-continuous) if )V(f 1− is gr-closed set in X for every closed set V in Y . Theorem . A function YX:f → from a topological space X into a topological space Y is gr-continuous iff )V(f 1− is gr-open set in X for every open set V in Y . Proof: It is Obvious . Theorem: Let YX:f → be a function from a topological space X into a topological space Y. If YX:f → is gr-continuous, then ))A(f(cl))A(grcl(f ⊆ for every subset A of X . Proof: Since ))A(f(cl)A(f ⊆ ⇒ )))A(f(cl(fA 1−⊆ . Since ))A(f(cl is a closed set in Y and f is gr-continuous ,then by (2.1) )))A(f(cl(f 1− is a gr-closed set in X containing A . Hence )))A(f(cl(f)A(grcl 1−⊆ . Therefore ))A(f(cl))A(grcl(f ⊆ . Theorem: Let YX:f → be a function from a topological space X into a topological space Y . Then the following statements are equivalent:- i)For each point x in X and each open set V in Y with V)x(f ∈ ,there is a gr-open set U in X such that Ux ∈ and V)U(f ⊆ . ii) For each subset A of X, ))A(f(cl))A(grcl(f ⊆ . iii)For each subset B of Y, ))B(cl(f))B(f(grcl 11 −− ⊆ . Proof: )ii()i( → . Suppose that (i) holds and let ))A(grcl(fy ∈ and let V be any open neighborhood of y . Since ))A(grcl(fy ∈ ⇒ )A(grclx ∈∃ s.t y)x(f = . Since V)x(f ∈ ,then by (i) ∃ a gr-open set U in X s.t Ux ∈ and V)U(f ⊆ . Since )A(grclx ∈ ,then by (1.4) φ≠AU  and hence φ≠V)A(f  . Therefore we have ))A(f(cl)x(fy ∈= . Hence ))A(f(cl))A(grcl(f ⊆ )i()ii( → Mathematics - 380 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 If (ii) holds and let Xx ∈ and V be any open set in Y containing f(x). Let )V(fA c1−= ⇒ Ax ∉ . Since cV))A(f(cl))A(grcl(f ⊆⊆ ⇒ A)V(f)A(grcl c1 =⊆ − . Since Ax ∉ ⇒ )A(grclx ∉ and by (1.4) there exists a gr-open set U containing x such that φ=AU  and hence V)A(f)U(f c ⊆⊆ . )iii()ii( → . Suppose that (ii) holds and let B be any subset of Y . Replacing A by )B(f 1− we get from (ii) )B(cl)))B(f(f(cl)))B(f(grcl(f 11 ⊆⊆ −− . Hence ))B(cl(f))B(f(grcl 11 −− ⊆ . )ii()iii( → . Suppose that (iii) holds, let B = f(A) where A is a subset of X . Then we get from (iii) )))A(f(cl(f))A(f(f(grcl)A(grcl 11 −− ⊆⊆ . Therefore ))A(f(cl))A(grcl(f ⊆ . Definition: A function YX:f → from a topological space X into a topological space Y is said to be perfectly generalized regular continuous (briefly perfectly gr-continuous) if )V(f 1− is gr-clopen (gr-open and gr-closed) set in X for every open set V in Y . Definition: A function YX:f → from a topological space X into a topological space Y is said to be contra generalized regular continuous (briefly contra gr-continuous) if )V(f 1− is gr- closed set in X for every open set V in Y . Theorem: Let YX:f → be a function. Then 1) If f is r-continuous ,then f is gr-continuous . 2) If f is gr-continuous ,then f is g-continuous . 3) If f is gr-continuous ,then f is rg-continuous . 4) If f is gr-continuous ,then f is gb-continuous . 5) If f is continuous ,then f is rg-continuous . 6) If f is perfectly gr-continuous ,then f is gr-continuous . 7) If f is perfectly gr-continuous ,then f is rg-continuous . 8) If f is perfectly gr-continuous ,then f is g-continuous . 9) If f is perfectly gr-continuous ,then f is gb-continuous . 10) If f is perfectly gr-continuous ,then f is contra gr-continuous . Proof: 1) Let F be a closed set in Y, Since f is r-continuous, then by (1.5) no.3, )F(f 1− is r-closed in X . Since every r-closed set is gr-closed ,then )F(f 1− is gr-closed in X . Hence f is gr-continuous . 3) Let F be a closed set in Y, Since f is gr-continuous, then by (2.1), )F(f 1− is gr-closed in X . Since every gr-closed set is rg-closed ,then )F(f 1− is rg-closed in X . Hence f is rg-continuous . 5) Let F be a closed set in Y, Since f is continuous, then )F(f 1− is closed in X . Since every closed set is rg-closed ,then )F(f 1− is rg-closed in X . Hence f is rg-continuous . 9) Let U be an open set in Y, Since f is perfectly gr-continuous, then by (2.5), )U(f 1− is gr- closed and gr-open in X . Since every gr-open set is gb-open ,then )U(f 1− is gb-open in X . Hence f is gb-continuous . Similarly, we can prove (2),(4),(6),(7),(8) and (10) . Mathematics - 381 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Remarks : 1)Continuous functions and gr-continuous functions are in general independent .Consider the following examples:- Examples i)Let }c,b,a{X = , }}b{},c,b{},b,a{,,X{ φ=τ and }}b,a{,,X{ φ=τ′ . Let ),X(),X(:f τ′→τ be a function defined by : a)a(f = , b)b(f = and c)c(f = . It is clear that f is continuous ,but f is not gr-continuous ,since{c}is closed in ),X( τ′ ,but }c{})c({f 1 =− is not gr-closed in ),X( τ . ii)Let }c,b,a{X = , },X{ φ=τ and }}a{,,X{ φ=τ′ . Let ),X(),X(:f τ′→τ be a function defined by : a)a(f = , b)b(f = and c)c(f = . It is clear that f is not continuous ,but f is gr-continuous ,since φ=φ− )(f 1 , X)X(f 1 =− and }c,b{})c,b({f 1 =− are gr-closed in ),X( τ . 2)The converse of ((2.7),no.1) in general is not true .Consider the following examples:- Let }d,c,b,a{X = , }q,p{Y = , }}d,c{,,X{ φ=τ and }}p{,,Y{ φ=τ′ . Let ),Y(),X(:f τ′→τ be a function defined by : q)d(f)b(f)a(f === and p)c(f = . f is gr-continuous ,since φ=φ− )(f 1 , X)Y(f 1 =− and }d,b,a{})q({f 1 =− are gr-closed in ),X( τ . But f is not r-continuous ,since{q}is closed in ),Y( τ′ ,but }d,b,a{})q({f 1 =− is not r-closed in ),X( τ . 3)The converse of ((2.7),no.2,3,4) in general is not true .In (1,(i)),f is g-continuous(resp. rg- continuous, gb-continuous) since f is continuous, but f is not gr-continuous . 4)The converse of ((2.7),no.5)in general is not true .In (1,(ii)), f is rg-continuous ,but f is not continuous . 5)The converse of ((2.7),no.6,7,8,9) in general is not true .In (2),f is gr-continuous (resp. rg- continuous, g-continuous, gb-continuous), but f is not perfectly gr-continuous ,since{p}is open in ),Y( τ′ ,but }c{})p({f 1 =− is gr-open, but not gr-closed in ),X( τ . 6)The converse of ((2.7),no.10) in general is not true .Consider the following examples:- Let }b,a{YX == and }}a{,,X{ φ=τ′=τ . Let ),X(),X(:f τ′→τ be a function defined by : b)a(f = and a)b(f = . f is contra gr-continuous ,since φ=φ− )(f 1 , X)Y(f 1 =− and }b{})a({f 1 =− are gr-closed in ),X( τ .But f is not perfectly gr-continuous ,since{a}is open in ),X( τ′ ,but }b{})a({f 1 =− is gr- closed, but not gr-open in ),X( τ . 7)Continuous functions and contra gr-continuous functions are in general independent . Consider the following examples:- i) It is clear that In (1,(i)),f is continuous ,but f is not contra gr-continuous . ii) It is clear that In (6),f is contra gr-continuous ,but f is not continuous . 8)Contra gr-continuous functions and r-continuous functions are in general independent . Consider the following examples:- i) It is clear that In (6),f is contra gr-continuous ,but f is not r-continuous . ii)Let }d,c,b,a{YX == , }}d,b,a{},c,b,a{},b,a{},b{},a{,,X{ φ=τ and }}a{,,Y{ φ=τ′ . Let ),Y(),X(:f τ′→τ be a function defined by : a)a(f = , b)b(f = , c)c(f = and d)d(f = . f is r-continuous ,since φ=φ− )(f 1 , X)Y(f 1 =− and }d,c,b{})d,c,b({f 1 =− are r-closed in ),X( τ . But f is not contra gr-continuous ,since{a}is open in ),Y( τ′ ,but }a{})a({f 1 =− is not gr-closed in ),X( τ . Mathematics - 382 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Thus we have the following diagram: Definition: A topological space ),X( τ is called a grT - space if every rg-closed set is gr-closed set . Examples: 1)In Remarks ((2.8) no.1(ii)), ),X( τ is a grT - space, since rg-closed set = {X, φ ,{a},{b},{c}, {a,b},{a,c},{b,c}} = gr-closed . 2)In Remarks ((1.2) no.1(ii)), ),X( τ is not a grT - space, since }c{ is rg-closed set ,but not gr- closed . Theorem: Let YX:f → be a function such that X is a grT - space, then:- i)Every continuous function is gr-continuous . ii)Every rg-continuous function is gr-continuous . iii)Every g-continuous function is gr-continuous . Proof: i)Let F be a closed set in Y, Since f is continuous, then )F(f 1− is closed in X. Since every closed set is rg-closed ,then )F(f 1− is rg-closed in X. Since X is a grT - space ,then )F(f 1− is gr-closed in X . Hence f is gr-continuous . Similarly, we can prove (ii) and (iii) . Generalized Regular Irresolute Functions Definition: A function YX:f → from a topological space X into a topological space Y is called a generalized regular irresolute (briefly gr-irresolute) if )V(f 1− is gr-closed set in X for every gr-closed set V in Y . Theorem: A function YX:f → from a topological space X into a topological space Y is gr- irresolute iff )V(f 1− is gr-open set in X for every gr-open set V in Y . Proof: It is Obvious . Definition: A function YX:f → from a topological space X into a topological space Y is said to be contra generalized regular irresolute (briefly contra gr-irresolute) if )V(f 1− is gr- closed set in X for every gr-open set V in Y . Remarks : 1) gr-irresolute functions and gr-continuous functions are in general independent .Consider the following examples:- Mathematics - 383 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 Examples: i)Let }c,b,a{X = , }}a{,,X{ φ=τ and },X{ φ=τ′ . Let ),X(),X(:f τ′→τ be a function defined by : a)a(f = , b)b(f = and c)c(f = . It is clear that f is gr-continuous ,but f is not gr-irresolute, since{a}is gr-closed in ),X( τ′ ,but }a{})a({f 1 =− is not gr-closed in ),X( τ . ii)Let }c,b,a{X = , }}a{,,X{ φ=τ and }}b{},c,b{},b,a{,,X{ φ=τ′ . Let ),X(),X(:f τ′→τ be a function defined by : a)a(f = , b)b(f = and c)c(f = . f is gr-irresolute ,since φ=φ− )(f 1 , X)X(f 1 =− and }c,a{})c,a({f 1 =− are gr-closed in ),X( τ . But f is not gr-continuous ,since{a}is closed in ),X( τ′ ,but }a{})a({f 1 =− is not gr-closed in ),X( τ . Theorem: Let YX:f → be a function such that X is a grT - space, then:- i)Every rg-irresolute function is gr-irresolute . ii)Every g-irresolute function is gr-irresolute . Proof: i)Let F be a gr-closed set in Y, then by (1.2) no.3, F is rg-closed in Y . Since f is rg-irresolute, then by (1.5) no.6, )F(f 1− is rg-closed in X . Since X is a grT - space ,then )F(f 1− is gr-closed in X . Hence f is gr-irresolute. ii)Let F be a gr-closed set in Y, then by (1.2) no.2, F is g-closed in Y . Since f is g-irresolute, then by (1.5) no.5, )F(f 1− is g-closed in X . Since every g-closed set is rg-closed ,then )F(f 1− is rg-closed in X . Since X is a grT - space ,then )F(f 1− is gr-closed in X . Hence f is gr-irresolute. Theorem: If YX:f → and ZY:g → are functions, then:- 1) If YX:f → and ZY:g → are both gr-irresolute functions, then ZX:fg → is gr-irresolute. 2) If YX:f → is contra gr-irresolute and ZY:g → is gr-irresolute, then ZX:fg → is contra gr-irresolute . 3) If YX:f → is gr-irresolute and ZY:g → is gr-continuous ,then ZX:fg → is gr-continuous . 4) If YX:f → is gr-continuous and ZY:g → is r-continuous ,then ZX:fg → is gr-continuous . 5) If YX:f → is gr-continuous and ZY:g → is continuous ,then ZX:fg → is gr-continuous . 6) If YX:f → is contra gr-irresolute and ZY:g → is gr-continuous ,then ZX:fg → is contra gr-continuous . Proof: 1)Let U be a gr-open set in Z, since g is gr-irresolute ,then )U(g 1− is gr-open in Y, since f is gr-irresolute ,then ))U(g(f 11 −− is gr-open in X . Since ))U(g(f)U()fg( 111 −−− = ,then )U()fg( 1− is a gr-open set in X . Thus fg  is gr-irresolute. 2)Let U be a gr-open set in Z, since g is gr-irresolute ,then )U(g 1− is gr-open in Y, since f is contra gr-irresolute ,then ))U(g(f 11 −− is gr-closed in X .Since ))U(g(f)U()fg( 111 −−− = ,then Mathematics - 384 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 )U()fg( 1− is a gr-closed set in X . Thus fg  is contra gr-irresolute. 3)Let F be a closed set in Z, since g is gr-continuous ,then )F(g 1− is gr-closed in Y ,since f is gr-irresolute ,then ))F(g(f 11 −− is gr-closed in X . Since ))F(g(f)F()fg( 111 −−− = ,then )F()fg( 1− is a gr-closed set in X . Thus fg  is gr-continuous . 4)Let F be a closed set in Z, since g is r-continuous ,then )F(g 1− is r-closed in Y ,since every r-closed set is closed, then )F(g 1− is closed in Y, since f is gr-continuous ,then ))F(g(f 11 −− is gr-closed in X . Since ))F(g(f)F()fg( 111 −−− = ,then )F()fg( 1− is a gr-closed set in X . Thus fg  is gr-continuous 5)Let F be a closed set in Z, since g is continuous ,then )F(g 1− is closed inY, since f is gr- continuous, then ))F(g(f 11 −− is gr-closed in X . Since ))F(g(f)F()fg( 111 −−− = ,then )F()fg( 1− is a gr-closed set in X . Thus fg  is gr-continuous . 6)Let U be an open set in Z, since g is gr-continuous ,then )U(g 1− is gr-open in Y, since f is contra gr-irresolute ,then ))U(g(f 11 −− is gr-closed in X . Since ))U(g(f)U()fg( 111 −−− = , then )U()fg( 1− is a gr-closed set in X . Thus fg  is contra gr-continuous. References 1. Arya,S. P. and Gupta,R. (1974) On strongly continuous functions , Kyungpook Math. J., 14 :131-143. 2. Palaniappan ,N. and Rao, K.C.(1993) Regular Generalized Closed Sets , Kyungpook Math. J., 33 : (2) 211-219 . 3. Bhattacharya,S. (2011) On Generalized Regular Closed Sets ,Int . J. Contemp. Math. Sciences, 6 : (3) 145-152 . 4. Levine , N.(1970) Generalized Closed Sets In Topology , Rend. Circ. Math . Palermo , 19: (2) 89-96 . 5. Stone,M.(1937) Applications of the theory of Boolean rings to general topology ,Trans. Amer. Math. Soc., 41 : 375-381 . 6. Andrijevic,D.(1996) On b-open sets , Mat. Vesnik, 48: (1-2) 59-64 . 7. Ganster,M. and Steiner,M.,2007, On b τ -closed sets , Applied General Topology, 8 :(2) 243-247. 8. Balachandran, K.; Sundaram, P. and Maki, H.(1991) On generalized continuous maps in topological spaces , Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 12 : 5-13 . 9. Al-omari,A. and Noorani,M.S.,2009 , On Generalized b-closed sets , Bull. Malays. Math. Sci. Soc., 2: (32) 19-30. Mathematics - 385 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 3 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 3 Vol. 25 Year 2012 في الفضاءات التبولوجية حول الدوال المستمرة المنتظمة المعممة صبيحة إبراهيم محمود الجامعة المستنصرية ، كلية العلوم ، قسم الرياضيات 2012ايار 21قبل البحث في: 2012كانون الثاني 30استلم البحث في: الخالصة generalized regular)أسميناها بالدوال المستمرة المنتظمة المعممة في هذا البحث قدمنا نوعا جديدا من الدوال continuous functions هذه الدوال اضعف من الدوال المستمرة المنتظمة)(regular continuous functions كذلك درسنا بعض . )(regular generalized continuous functionsمن الدوال المستمرة المعممة المنتظمة وأقوى ذلك درسنا أنواعا أخرى من الدوال المستمرة فضًال عنوالخواص األساسية للدوال المستمرة المنتظمة المعممة. أتفالمكا المعممة ومن ثم درسنا العالقة بينها . المنتظمة ة .دوال المستمرة المعممة المنتظمالدوال المستمرة المنتظمة المعممة ، الدوال المستمرة المنتظمة ، ال الكلمات المفتاحية: S. I. Mahmood