2011) 2( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد شبه المنتظمة المغلقة -  -المجموعات نادیة فائق محمد جامعة بغداد،ابن الهیثم -كلیة التربیة،قسم الریاضیات 2010،حزیران، 23: استلم البحث في 2010،ایلول ، 27: قبل البحث في الخالصة لقـــد قمنـــا فـــي هـــذا البحـــث بتقــــدیم ودراســـة نـــوع جدیـــد مـــن المجموعــــات المغلقـــة فـــي الفضـــاءات التبولوجیـــة یــــدعى -  -قـة لان هذا النوع من المجموعـات المغلقـة تحـوي مجموعـات شـبه مغ اذشبه المنتظمة المغلقة، -  -بالمجموعات مـن الـدوال المسـتمرة والمتـرددة تـدعى دالـة اجدیـد ادمنا ودرسـنا نوعـوكمـا قـ. وتكـون محتـواه فـي المجموعـات قبـل شـبه المغلقـة كمـا وجـدنا ان االسـتمراریة مـن . شـبه المنتظمـة المتـرددة -  -شبه المنتظمة المستمرة ودالة من الـنمط -  -من النمط .شبهالمراریة من النمط قبل واالست -  -به ششبه المنتظمة تكون واقعة تماماً بین االستمراریة من النمط -  -النمط ، شـبه المنتظمـة المسـتمرة --الدالة مـن الـنمط ، شبه المنتظمة المغلقة --المجموعة من النمط : الكلمات المفتاحیة .شبه المنتظمة المترددة --الدالة من النمط IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011  - Semi-Regular Closed Sets N. F. Mohamme d Departme nt of Mathematics, College of Education, -Ibn-Al-Haitham, Unive rsity of Baghdad Received in: 23, June , 2010 Accepte d in: 27, Septe mber, 2010 Abstract In this p aper, a new class of sets, namely - semi-regular closed sets is introduced and st udied for top ological sp aces. This class p rop erly contains the class of semi--closed sets and is p rop erty contained in the class of p re-semi-closed sets. Also, we introduce and st udy sr- continuity and sr-irresoleteness. We showed that sr-continuity falls st rictly in between semi-- continuity and pre-semi-continuity . Key words: - semi-regular closed set, - semi-regular continuous, - semi-regular irresolute. Introduction Najast ed [1] and Levine [2] introduced -op en sets and generalized closed sets, Kummar introduced -generalized regular closed set and p re-semi closed set, see [3] and [4]. Alot of work was done in the field of generalized closed sets. In this p aper we emp loy a new technique to obtain a new class of sets, called -semi-regular closed sets. This class is obtained by semi--closed set and regular op en set. It is shown that the class of -semi- regular closed sets p rop erly contains the class of semi--closed sets and is p rop erly contained in the class of p re-semi-closed sets. We also introduce and study two classes of map s, namely, -semi-regular continuity and -semi-regular irresoluteness, -semi-regular continuity falls st rictly in between semi--continuity and pre-semi-continuity . 1- Preliminaries Throughout this p aper (X,) and (Y,') represent non-emp ty top ological sp aces. For a subset A of a sp ace (X,), cl(A) and int(A) represent the closure of A and the interior of A resp ectively. 1.1 De fini tion: A subset A of a sp ace (X,) is called (1) an -op en set [1], [5] if A  int(cl(int(A))) and -closed if cl(int(cl(A)))  A. (2) a semi--op en set [6], [7] if A  cl(int(cl(int(A)))) and semi--closed if int(cl(int(cl(A))))  A. (3) a semi-preopen set [8], [9] if A  cl(int(cl(A))) and semi-preclosed if int(cl(int(A)))  A. (4) a regular op en set [10], [11] if A = int(cl(A)) and regular closed if A = cl(int(A)). (5) a generalized closed set (briefly g-closed) [2], [12] if cl(A)  U whenever A  U and U is op en in (X,). The complement of a g-closed set is called a g-op en set. (6) an -generalized closed set (briefly g-closed) [13] if  cl(A)  U whenever A  U and U is op en in (X,). (7) a generalized -closed set (briefly g-closed) [14] if cl(A)  U whenever A  U and U is -op en in (X,). (8) a generalized *-closed set (briefly g*-closed) [14] if cl(A)  int(U) whenever A  U and U is -op en in (X,). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 (9) an **-generalized closed set (briefly **g-closed) [14] if  cl(A)  int(cl(U)) whenever A  U and U is op en in (X,). (10) a generalized **-closed set (briefly g**-closed) [14] if cl(A)  int(cl(U)) whenever A  U and U is -op en in (X,). (11) a regular generalized closed set (briefly rg-closed) [15] if cl(A)  U whenever A  U and U is regular op en in (X,). (12) an -generalized regular closed set (briefly gr-closed) [3] if cl(A)  U whenever A  U and U is regular op en in (X,). (13) a generalized semi-p reclosed set (briefly gsp -closed ) [16] if sp cl(A)  U whenever A  U and U is op en in (X,). (14) a pre-semi-closed set [4] if sp cl(A)  U whenever A  U and U is g-op en in (X,). The semi--closure (resp . -closure, semi-p re-closure) of A in (X,) is the intersection of all semi--closed (resp . -closure, semi-p re-closure) sets of (X,) that contain A and is denoted by Scl(A) (resp . cl(A), sp cl(A)). 1.2 Proposi tion: (1) Every -closed set is semi--closed set, not conversely, [6]. (2) Every closed set is -closed set, so it is semi--closed set, not conversely, [6]. (3) Every closed (resp . -closed, g-closed, g-closed) set is an gr-closed set, [3]. (4) Every g*-closed (resp . **g-closed, g**-closed set is an gr-closed set, [3]. (5) Every p re-semi-closed set ia gsp -closed set [4]. (6) Every semi--closed set is semi-p re-closed set (the proof follows directly from the definitions). 1.3 Remark: [6] Let X be a top ological sp ace, A and B be two subsets of X, then (1) A is semi--closed set if and only if A = Scl(A). (2) A  Scl(A)  cl(A)  cl(A). (3) Scl(A)  Scl(B), whenever A  B. 1.4 De fini tion: A function f:(X,)  (Y,') is said to be: (1) semi--continuous [6], [7] if f – 1 (V) is a semi--closed set in (X,) for every closed set V of (Y,'). (2) g-continuous [17] if f – 1 (V) is a g-closed set in (X,) for every closed set V of (Y,'). (3) g-continuous [18] if f – 1 (V) is an g-closed set in (X,) for every closed set V of (Y,'). (4) g -continuous [14] if f – 1 (V) is a g-closed set in (X,) for every closed set V of (Y,'). (5) gr-continuous [3] if f – 1 (V) is an gr-closed set in (X,) for every closed set V of (Y,'). (6) p re-semi-continuous [4] if f – 1 (V) is a p re-semi-closed set in (X,) for every closed set V of (Y,'). (7) gsp -continuous [16] if f – 1 (V) is a gsp -closed set in (X,) for every closed set V of (Y,'). (8) semi--irresolute [6] if f – 1 (V) is a semi--closed set in (X,) for every semi--closed set V of (Y,'). (9) gr-irresolute [3] if f – 1 (V) is an gr-closed set in (X,) for every gr-closed set V of (Y,'). (10) regular irresolute [19] if f – 1 (V) is a regular op en set in (X,) for every regular op en set V of (Y,'). (11) semi-*-closed [6] if f (U) is a semi--closed set in (Y,') for every semi--closed set U in (X,). 1.5 Proposi tion: (1) Every g-continuous map is gr-continuous map [3]. (2) Every g-continuous (resp .g-continuous) map is an gr-continuous map [3]. (3) Every p re-semi-continuous map is gsp -continuous map [4]. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 (4) Every gr-irresolut map is gr-continuous map [3]. (5) Every continuous and op en map is semi--irresolute map [20]. 2- -S emi-Regul ar Close d S ets In this section we introduce the class of -semi-regular closed sets and st udy some of it's basic p rop erties. 2.1 De fini tion: A subset A of (X,) is called -semi-regular closed set (briefly sr-closed) if Scl(A)U whenever A  U and U is regular op en in (X,). SRC(X) denotes the collection of all sr-closed subset of (X,) 2.2.Proposi tion: Every semi--closed set is an sr-closed set. Proof: Let A be a semi--closed set, let U be a regular op en set of (X,) such that A  U. Since Scl(A) = A for any semi--closed set (by p art 1 of remark 1.3), then Scl(A)  U. Therefore A is also an sr-closed set. The following examp le shows that the converse of the above p rop osition is not true in general. 2.3 Example: Let X={a,b,c} and ={X,,{a},{c},{a,c}}. Let A={a,c}, X is the only regular op en set containing A. It is clear A is an sr-closed set. But A is not semi--closed set since Scl({a,c}) = X  {a,c}. Thus t he class of sr-closed set p rop erly contains the class of semi--closed sets. 2.4 Proposi tion: Every gr-closed set is an sr-closed set. Proof: Let A be an gr-closed set, let U be a regular op en set of (X,) such that A  U. Since A is gr-closed set and Scl(A)  cl(A) (by p art (2) of remark 1.3), then Scl(A)  U. Therefore A is also an sr-closed set. The following examp le shows t hat t he sr-closed set need not t o be an gr-closed set. 2.5 Example: Let X={a,b,c} and ={X,,{a},{b},{a,b}}. Let A={b}, let {b} is the regular op en set containing A. Trivially A is an sr-closed set since Scl(A)={b}  {b}. But A is not gr- closed set since cl(A)={b,c}  {b}. 2.6 Corollary: Every closed (resp . -closed, g-closed, g-closed) set is an sr-closed set. Proof: Since every gr-closed set is an sr-closed set, then in vitue of p rop osition 1.2 part (2) the proof is over. The following examp le shows t hat t he reveres imp lications in the above corollary are not true in general. 2.7 Example: Let X,  and A be as in examp le 2.3. A is neither closed (since cl(A) = X  A) nor -closed (since cl(A) = X  A) and also it is neither g-closed (since A={a,c}  {a,c} whenever {a,c}  , but cl(A) = X  {a,c}) nor g-closed (since A={a,c}  {a,c} whenever {a,c}  O(X), but cl(A) = X  {a,c}). 2.8 Corollary: Every g*-closed (resp . **g-closed, g**-closed) set is an sr-closed set. Proof: Since every gr-closed set is an sr-closed set, p art (4) of p rop osition 1.2 is app licable. The following examp le shows t hat an sr-closed set needs not to be a g*-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 2.9 Example: Let X= � and  = U, let A=(a,b) is sr-closed set but not a g*-closed set, since (a,b)  (a,b) and (a,b) is -op en set in ( � ,U), but cl(A) = [a,b]  (a,b). 2.10 Proposi tion: Every r-g- closed is an sr-closed set. Proof: Let A be a regular generalized closed set of (X,). Let U be a regular op en set of (X,) such that A  U. Then cl(A)  U since A is r-g closed set. Since every closed set is semi-- closed set, then Scl  cl(A) (p art 2 of remark 1.3). Thus Scl(A)  U, t herefore A is an sr- closed set. The converse of above prop osition is not alway s t rue as t he following examp le shows. 2.11 Example: Let X and  be as in examp le 2.3, let A={c} and U={c} is regular op en set containing A. It is clear A is an sr-closed set since Scl(A) = {c}  {c}. But is not r-g closed set since cl(A) = {b,c}  {c}. 2.12 Proposi tion: Every g-closed set is an sr-closed set Proof: Let A be an g-closed set, let U be a regular op en set of (X,) such that A  U. Since A is g-closed and every regular op en set is an op en set, then cl(A)  U. But Scl(A)   cl(A) since every -closed set is semi--closed set. Therefore A is also an sr-closed set. The converse in the above p rop osition is not true as it can be seen from the following examp le. 2.13 Example: In examp le 2.3 cl(A) = X  {a,c}. Thus A is not g-closed set, but it is sr-closed set. 2.14 Proposi tion: Let A be an sr-closed set of (X,). Then Scl(A)-A does contain any non-emp ty regular closed set. Proof: Let F be any regular closed set of (X,) such that F  Scl(A) – A. Then F  X – A imp lies that A  X – F. Since A is sr-closed and X – F is a regular op en set of (X,), then Scl(A)  X – F, so F  X - Scl(A). Therefore F  Scl(A)  (X – Scl(A)) = . Hence Scl(A) – A does not contain any non-emp ty regular closed set. 2.15 Proposi tion: Every sr-closed set is a p re-semi-closed set. Proof: Let A be an sr-closed set of (X,), let U be a regular op en set of (X,) such that A  U. Then Scl(A)  U since A is sr-closed set. Since every semi--closed set is semi- p re-closed set (by p art 6 of p rop osition 1.2), then sp cl(A)  Scl(A) and every regular op en set is g-op en set. Thus A is p re-semi-closed set. Thus the class of sr-closed set p rop erly contained in the class of p re-semi-closed sets. 2.16 Corollary: Every sr-closed set is gsp -closed set. Proof: Follows t he above prop osition and p art (5) of p rop osition 1.2. 2.17 Corollary: Every gr-closed set is p re-semi-closed set. Proof: Follows from the fact every gr-closed set is sr-closed and p rop osition 2.15. 2.18 Proposi tion: If A is regular op en and sr-closed set t hen A is semi--closed set. Proof: It is clear. 2.19 Proposi tion: Let A be an sr-closed subset of (X,). If B  X such that A  B  Scl(A), t hen B is sr-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 Proof: Let U be a regular op en set of (X,) such that B  U. Then A  U, since A is sr- closed set, Scl(A)  U. Now, Scl(B)  Scl(Scl(A)) = Scl(A)  U. Therefore B is also an sr-closed set. Fig. (1) shows the relations among the different ty p es of weakly closed sets that were st udied in this section. 3- -S emi Regul ar Conti nuous Maps and -S emi-Regul ar-Irresol ute Maps 3.1 De fini tion: A function f:(X,)  (Y,') is called an -semi-regular continuous map (briefly sr- continuous if f – 1 (V) is an sr-closed set of (X,) for every closed set V of (Y,'). 3.2 Proposi tion: Every semi--continuous map is sr-continuous. Proof: Follows from p rop osition 2.2. We show that the class of sr-continuous map s p rop erly contains the class of gr- continuous maps. 3.3 Proposi tion: Let f:(X,)  (Y,') be an gr-continuous map. Then f is an sr-continuous map . Proof: Let V be a closed set of (Y,'). Since f is an gr-continuous map , then f – 1 (V) is an gr-closed set of (X,). By p rop osition 2.4 f – 1 (V) is an sr-closed set of (X,). Thus f is an sr-continuous map . The imp lications in p rop osition 3.3 is not reversible. Follows from the following examp le. 3.4 Example: Let X = {a,b,c} = Y,  = {X,,{a},{b},{a,b}} and ' = {Y,,{a,c}}. Define f:(X,)  (Y,') by f(a) = c, f(b) = b and f(c) = a, {b} is a closed set of (Y,') but f – 1 ({b}) = {b} is not gr-closed set of (X,). So f is not gr-continuous map . However f is an sr- continuous map. 3.5 Corollary: Every g-continuous map is sr-continuous. Proof: Follow from part (1) of p rop osition 1.5 and p rop osition 3.3. The converse of the above corollary is not true in general as we see in the following examp le. 3.6 Example: Let X, Y,  and the definition of f as in examp le 3.4, let ' = {Y,,{a},{b,c}}. f is not g- continuous map since {b,c} is a closed set of (Y,') but f – 1 ({b,c}) = {a,b} is not g-closed set of (X,). However f is an sr-continuous map . 3.7 Corollary: Every g-continuous (resp . g-continuous) is an sr-continuous. Proof: Follows from p art (2) of p rop osition 1.5 and p rop osition 3.3. The converse of the above corollary is not true in general as we see in the following examp le. 3.8 Example: See examp le 3.4 f is sr-continuous map but not g-continuous map. 3.9 Corollary: Every gr-irresolute map is an sr-continuous. Proof: Necessity follows from part (4) of p rop osition 1.5 and p rop osition 3.3. The converse of the above corollary is not true in general as we see in the following examp le IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 3.10 Example: Let X = {a,b,c} = Y,  = {X,,{a},{b},{a,b}} and '=I. Define f:X  Y by f(a) = c, f(b) = b and f(c) = a, {b} is an sr-closed set of (Y,') but f – 1 ({b}) = {b} is not gr-closed set of (X,). So f is not gr-irresolute map. However f is an sr-continuous map . 3.11 The orem: Let f:(X,)  (Y,') be an sr-continuous map . Then f is a p re-semi-continuous map . Proof : Let V be a closed set of (Y,'). Since f is sr-continuous map , then f – 1 (V) is an sr-closed set of (X,). By p rop osition (2.15) f – 1 (V) is a p re-semi-closed set of (X,). Thus f is a pre-semi-continuous map . 3.12 Corollary: Every sr-continuous map is gsp -continuous. Proof: Follows from the above prop osition and p art (3) of p rop osition 1.5. 3.13 De fini tion: A function f:(X,)  (Y,') is called an -semi-regular irresolute (briefly sr- irresolute) if f – 1 (V) is an sr-closed set of (X,) for every sr- closed set of (Y,'). 3.14 Proposi tion: Let f:(X,)  (Y,') be an sr-irresolute map . Then f is an sr-continuous map . Proof: Let V be a closed set of (Y,'). By corollary 2.6 V is an sr-closed set of (Y,'). Since f is an sr-irresolute map , f – 1 (V) is an sr-closed set of (X,). Therefore f is an sr-continuous map . Thus the class of sr-continuous map s p rop erty continuous the class of sr-irresolute map . 3.15 Corollary: Every sr-irresolute map is a pre-semi-continuous. Proof: Follows from the above prop osition and p rop osition 3.11. 3.16 Corollary: Every sr-irresolute is a gsp - continuous. Proof: Follows from p rop osition 3.14 and corollary 3.12. 3.17 The orem: Let f:(X,)  (Y,') be a regular irresolute and semi--irresolute map . Then f is sr- irresolute map. Proof: Let A be an sr-closed set of (Y,'), then there exists a regular op en set U of Y such that Scl(A)  U whenever A  U. By taking the inverse image we get f – 1 (Scl(A))  f – 1 (U). Since f is regular irresolute map , then f – 1 (U) is regular op en subset of X. Since f is semi--irresolute map , then f – 1 (Scl(A)) is semi--closed subset of X. This imp lies Scl(f – 1 (Scl(A)))= f – 1 (Scl(A)) (by p art (1) of remark 1.3), then Sclf – 1 (A)  Scl( f – 1 (Scl(A)). Thus Sclf – 1 (A)  f – 1 (U). Therefore f – 1 (A) is sr-closed set in X. Therefore f is sr-irresolute map . 3.18 Corollary: Every continuous, op en and regular irresolute map is sr-irresolute. Proof: It is clear by p art (5) of p rop osition 1.5 and the above theorem. 3.19 De fini tion: Let f:(X,)  (Y,') be a function, then f is said to be: (1)-semi-regular closed (briefly sr-closed) if f(A) is an sr-closed set of (Y,') for every closed set A of (X,). (2) *-semi-regular closed (briefly *sr-closed) if f(A) is an sr-closed set of (Y,') for every sr-closed set A of (X,). 3.20 Remark: It is clear that every closed function is -semi-closed function, but the converse is not true in general as the following examp le shows: IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 3.21 Example: Let X={a,b,c,d},  = {X,,{a},{b},{a,b},{a,b,c}}. Define f:(X,)  (X,) by f(a) = a, f(b) = b, f(c) = f(d) = d we observe f is -semi-regular closed function which is not closed function since {a,c,d} is closed set in X, but f({a,c,d}) = {a,d} is not closed set in X. Hence f is -semi-regular closed function, which is not closed function. Finally, we p rove the following theorem. 3.22 The orem: Let f:(X,)  (Y,') be a regular irresolute and semi-*-closed map . Then f is *-semi-regular closed map . Proof: Let A be an sr-closed set of (X,), let U be a regular op en set of (Y,') such that f(A)  U. Since f is regular irresolute, then f – 1 (U) is a regular op en set of (X,). Since A  f – 1 (U) and A is an sr-closed, then Scl(A)  f – 1 (U). This imp lies f(Scl(A))  U. Since f is semi-*-closed map , then f(Scl(A)) = Scl (f(Scl(A))). So Scl(f(A))  Scl(f(Scl(A))) = f(Scl(A)))  U. T herefore f(A) is an sr-closed set of (Y,'). 3.23 Corollary: Let f:(X,)  (Y,') be a regular irresolute and semi-*-closed map . Then f(A) is a p re-semi-closed set of (Y,') for every sr-closed set of (X,). Proof: It is clear. Fig. (2) exp lains the relationship s among the different ty p es of weakly continuous function. Re ferences 1. Njasted, Olav, (1965), On Some Classes of Nearly Op en Sets, Pacific J. M ath., 15 (3): 961-970. 2. Levine, N.( 1970), Generalized Closed sets in Top ology Rend. Cire. M ath. Palermo, 19(2): 89-96. 3. Kumar, M . K. R. S. Veera, (2002), -Generalized Regular Closed Sets, Acta Ciencia Indica, XXVIIIM (2), 279. 4. Kummar, M . K. R.S. Veera, (2002), Pre-Semi-Closed Sets, Indian Journal of M ath., 44(2): 165-181. 5. Pop a, Valeriu and Noiri, Taleashi,( 2000), Some Prop erties of -irresolute M ultifunctions, Arab J.M ath. Sc., 6(2):17-26. 6. Al-Tabatabai, Nadia M .Ali, (2004), On New Ty p es of Weakly Op en Sets -Op en and Semi--Op en Sets, M .Sc. Thesis, University of Baghdad. 7. Nasir, Ahmed, Ibrahem, (2005), Some Kind of Strongly Compact and Pair-Wise Compact Sp aces, M .Sc. Thesis, University of Baghdad. 8. Andrijevic, D. (1986), Semi-Pere Sets, M ath. Vesnik, 38(1): 24-32. 9. Al-M aliki, Najlaa Jabbar, (2005), Some Kinds of Weakly Connected and Pairwis Connected Sp ace, M .Sc. Thesis, University of Baghdad. 10. Adams, Colin and Franzosa, Robert, (2008), Introduction to Top ology Pure and Ap p lied, Up p er Saddle River, NJO4458. 11. Pop a, Valeriu and Noiri, ToKashi, (2001), On The Definitions of Some Generalized Forms of Continuty Under M inimal Conditions, M em. Fac. Sci., Kochi Univ.(M ath.), 22:9-18. 12. Nagata, J., (2002), On Preclosed Sets and Their Generalizations, Houston Journal of M ath., 28(4). 13. M aki, H.; Devi, R. and Balachandran, K. (1994), Associated Top ologies of Generalized -Closed Sets and -Generalized Closed Sets, M em.Fac. Sci. Kohi., Univ. Ser. A. M ath., 15:51-63. 14. M aki, H.; Devi, R. and Balachdran, K. 91993), Generalized -Closed Sets in Top ology , Bull. Fukuoka Univ. Ed. Part III, 42:13-21. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (2) 2011 15. Sundaram, P.; M aki, H. and Balachandran, K. (1991), Semi-Generalized Continuous map and Semi-T1/2 Sp ace, Bull. Fukuoka Univ. Fd. Part III, 40:33-40. 16. Dont chev, J. (1995), On Generalized Semi-Preopen Sets, M em. Fac.Sci.Kochi.Univ.Ser. A.M ath., 16:35-48. 17. Balachandran, K.; Sundaram, P. and M aki, H. (1991), On Generalized Continuous M aps in Top logical Sp aces, M em.Fac.Sci.Kochi. Univ. Ser.A.M ath., 1: 5-13. 18. Gnanambal, Y. (1997) On Generalized Preregular Closed Sets in Top ological Sp aces, Indian J. Pure Ap p l. M ath., 28(3):351-360. 19. Palanrepp an, N. and K. C. Roo, (1993), Regular Generalized Closed Sets, Ky ung Pook M ath. J., 33(2): 211-219. 20. M ohammed, Nadia, Faiq, (2010), On Semi--Connected Subsp aces, Baghdad Science Journal (Phy sics and M athematics), 7(1), issn:1815-4808. closed  g-closed  rg-closed g-closed  gr-closed -closed g-closed **g-closed g*-closed g**-closed semi--closed  sr-closed  p re-semi-closed regular op en gsp -closed Fig. (1) the rel ations among the different types of weakly closed se ts continuous op en semi--irresolute regular irresolute  sr-irresolute g-continuous  g-continuous  gr-continuous  sr-continuous g-continuous  gr-irresolute  p re-semi-continuous gsp -continuous Fig. (2) the rel ationshi ps among the different types of weakly continuous function.