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 ـة یــــدعى ـة فـــي الفضـــاءات التبولوجیــ ــات المغلقــ ـة نـــوع جدیـــد مـــن المجموعــ ــد قمنـــا فـــي هـــذا البحـــث بتقــــدیم ودراســ لقـ -  -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. SCI. VOL.24 (2) 2011  - Semi-Regular Closed Sets N. F. Mohammed Department of Mathematics, College of Education, -Ibn-Al-Haitham, University of Baghdad Received in: 23, June, 2010 Accepted in: 27, September, 2010 Abstract In this paper, a new class of sets, namely - semi-regular closed sets is introduced and studied for topological spaces. This class properly contains the class of semi--closed sets and is property contained in the class of pre-semi-closed sets. Also, we introduce and study sr- continuity and sr-irresoleteness. We showed that sr-continuity falls strictly in between semi-- continuity and pre-semi-continuity. Key words: - semi-regular closed set, - semi-regular continuous, - semi-regular irresolute. Introduction Najasted [1] and Levine [2] introduced -open sets and generalized closed sets, Kummar introduced -generalized regular closed set and pre-semi closed set, see [3] and [4]. Alot of work was done in the field of generalized closed sets. In this paper we employ 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 open set. It is shown that the class of -semi- regular closed sets p roperly contains the class of semi--closed sets and is properly contained in the class of p re-semi-closed sets. We also introduce and study two classes of maps, namely, -semi-regular continuity and -semi-regular irresoluteness, -semi-regular continuity falls strictly in between semi--continuity and pre-semi-continuity. 1- Preliminaries Throughout this paper (X,) and (Y,') represent non-empty topological spaces. For a subset A of a space (X,), cl(A) and int(A) represent the closure of A and the interior of A respectively. 1.1 Definition: A subset A of a space (X,) is called (1) an -open set [1], [5] if A  int(cl(int(A))) and -closed if cl(int(cl(A)))  A. (2) a semi--open 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 open 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 open in (X,). The complement of a g-closed set is called a g-open set. (6) an -generalized closed set (briefly g-closed) [13] if  cl(A)  U whenever A  U and U is open in (X,). (7) a generalized -closed set (briefly g-closed) [14] if cl(A)  U whenever A  U and U is -open in (X,). (8) a generalized *-closed set (briefly g*-closed) [14] if cl(A)  int(U) whenever A  U and U is -open in (X,). IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 open in (X,). (10) a generalized **-closed set (briefly g**-closed) [14] if cl(A)  int(cl(U)) whenever A  U and U is -open in (X,). (11) a regular generalized closed set (briefly rg-closed) [15] if cl(A)  U whenever A  U and U is regular open in (X,). (12) an -generalized regular closed set (briefly gr-closed) [3] if cl(A)  U whenever A  U and U is regular open in (X,). (13) a generalized semi-preclosed set (briefly gsp-closed ) [16] if spcl(A)  U whenever A  U and U is open in (X,). (14) a pre-semi-closed set [4] if spcl(A)  U whenever A  U and U is g-open in (X,). The semi--closure (resp. -closure, semi-pre-closure) of A in (X,) is the intersection of all semi--closed (resp. -closure, semi-pre-closure) sets of (X,) that contain A and is denoted by Scl(A) (resp . cl(A), spcl(A)). 1.2 Proposition: (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 pre-semi-closed set ia gsp-closed set [4]. (6) Every semi--closed set is semi-pre-closed set (the proof follows directly from the definitions). 1.3 Remark: [6] Let X be a topological space, 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 Definition: 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) pre-semi-continuous [4] if f – 1(V) is a pre-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 open set in (X,) for every regular open 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 Proposition: (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 pre-semi-continuous map is gsp-continuous map [4]. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 (4) Every gr-irresolut map is gr-continuous map [3]. (5) Every continuous and open map is semi--irresolute map [20]. 2-  -Semi-Regular Closed Sets In this section we introduce the class of -semi-regular closed sets and study some of it's basic properties. 2.1 Definition: 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 open in (X,). SRC(X) denotes the collection of all sr-closed subset of (X,) 2.2.Proposition: Every semi--closed set is an sr-closed set. Proof: Let A be a semi--closed set, let U be a regular open set of (X,) such that A  U. Since Scl(A) = A for any semi--closed set (by part 1 of remark 1.3), then Scl(A)  U. Therefore A is also an sr-closed set. The following example shows that the converse of the above proposition 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 open 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 the class of sr-closed set p roperly contains the class of semi--closed sets. 2.4 Proposition: Every gr-closed set is an sr-closed set. Proof: Let A be an gr-closed set, let U be a regular open set of (X,) such that A  U. Since A is gr-closed set and Scl(A)  cl(A) (by part (2) of remark 1.3), then Scl(A)  U. Therefore A is also an sr-closed set. The following example shows that the sr-closed set need not to 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 open 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 proposition 1.2 part (2) the proof is over. The following example shows that the reveres implications in the above corollary are not true in general. 2.7 Example: Let X,  and A be as in example 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, part (4) of proposition 1.2 is applicable. The following example shows that an sr-closed set needs not to be a g*-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 -open set in ( � ,U), but cl(A) = [a,b]  (a,b). 2.10 Proposition: 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 open 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) (part 2 of remark 1.3). Thus Scl(A)  U, therefore A is an sr- closed set. The converse of above proposition is not always true as the following example shows. 2.11 Example: Let X and  be as in example 2.3, let A={c} and U={c} is regular open 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 Proposition: Every g-closed set is an sr-closed set Proof: Let A be an g-closed set, let U be a regular open set of (X,) such that A  U. Since A is g-closed and every regular open set is an open 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 proposition is not true as it can be seen from the following example. 2.13 Example: In example 2.3 cl(A) = X  {a,c}. Thus A is not g-closed set, but it is sr-closed set. 2.14 Proposition: Let A be an sr-closed set of (X,). Then Scl(A)-A does contain any non-empty regular closed set. Proof: Let F be any regular closed set of (X,) such that F  Scl(A) – A. Then F  X – A implies that A  X – F. Since A is sr-closed and X – F is a regular open 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-empty regular closed set. 2.15 Proposition: Every sr-closed set is a pre-semi-closed set. Proof: Let A be an sr-closed set of (X,), let U be a regular open 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- pre-closed set (by part 6 of proposition 1.2), then spcl(A)  Scl(A) and every regular open set is g-open set. Thus A is pre-semi-closed set. Thus the class of sr-closed set properly contained in the class of p re-semi-closed sets. 2.16 Corollary: Every sr-closed set is gsp-closed set. Proof: Follows the above proposition and part (5) of p roposition 1.2. 2.17 Corollary: Every gr-closed set is pre-semi-closed set. Proof: Follows from the fact every gr-closed set is sr-closed and proposition 2.15. 2.18 Proposition: If A is regular open and sr-closed set then A is semi--closed set. Proof: It is clear. 2.19 Proposition: Let A be an sr-closed subset of (X,). If B  X such that A  B  Scl(A), then B is sr-closed set. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 Proof: Let U be a regular open 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 types of weakly closed sets that were studied in this section. 3-  -Semi Regular Continuous Maps and  -Semi-Regular-Irresolute Maps 3.1 Definition: 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 Proposition: Every semi--continuous map is sr-continuous. Proof: Follows from proposition 2.2. We show that the class of sr-continuous maps properly contains the class of gr- continuous maps. 3.3 Proposition: 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 proposition 2.4 f – 1(V) is an sr-closed set of (X,). Thus f is an sr-continuous map. The implications in proposition 3.3 is not reversible. Follows from the following example. 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 roposition 1.5 and proposition 3.3. The converse of the above corollary is not true in general as we see in the following example. 3.6 Example: Let X, Y,  and the definition of f as in example 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 part (2) of p roposition 1.5 and proposition 3.3. The converse of the above corollary is not true in general as we see in the following example. 3.8 Example: See example 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 roposition 1.5 and proposition 3.3. The converse of the above corollary is not true in general as we see in the following example IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 Theorem: Let f:(X,)  (Y,') be an sr-continuous map. Then f is a pre-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 proposition (2.15) f – 1(V) is a pre-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 proposition and part (3) of p roposition 1.5. 3.13 Definition: 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 Proposition: 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 maps property continuous the class of sr-irresolute map. 3.15 Corollary: Every sr-irresolute map is a pre-semi-continuous. Proof: Follows from the above proposition and proposition 3.11. 3.16 Corollary: Every sr-irresolute is a gsp- continuous. Proof: Follows from proposition 3.14 and corollary 3.12. 3.17 Theorem: 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 open 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 open subset of X. Since f is semi--irresolute map, then f – 1 (Scl(A)) is semi--closed subset of X. This implies Scl(f – 1 (Scl(A)))= f – 1 (Scl(A)) (by part (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, open and regular irresolute map is sr-irresolute. Proof: It is clear by part (5) of p roposition 1.5 and the above theorem. 3.19 Definition: 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 example shows: IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 prove the following theorem. 3.22 Theorem: 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 open set of (Y,') such that f(A)  U. Since f is regular irresolute, then f – 1(U) is a regular open set of (X,). Since A  f – 1(U) and A is an sr-closed, then Scl(A)  f – 1(U). This implies 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. Therefore 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 pre-semi-closed set of (Y,') for every sr-closed set of (X,). Proof: It is clear. Fig. (2) explains the relationships among the different types of weakly continuous function. References 1. Njasted, Olav, (1965), On Some Classes of Nearly Open Sets, Pacific J. Math., 15 (3): 961-970. 2. Levine, N.( 1970), Generalized Closed sets in Topology Rend. Cire. Math. 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 Math., 44(2): 165-181. 5. Popa, Valeriu and Noiri, Taleashi,( 2000), Some Properties of -irresolute Multifunctions, Arab J.M ath. Sc., 6(2):17-26. 6. Al-Tabatabai, Nadia M.Ali, (2004), On New Types of Weakly Open Sets -Open and Semi--Open Sets, M.Sc. Thesis, University of Baghdad. 7. Nasir, Ahmed, Ibrahem, (2005), Some Kind of Strongly Compact and Pair-Wise Compact Spaces, M.Sc. Thesis, University of Baghdad. 8. Andrijevic, D. (1986), Semi-Pere Sets, Math. Vesnik, 38(1): 24-32. 9. Al-Maliki, Najlaa Jabbar, (2005), Some Kinds of Weakly Connected and Pairwis Connected Space, M.Sc. Thesis, University of Baghdad. 10. Adams, Colin and Franzosa, Robert, (2008), Introduction to Topology Pure and Applied, Upper Saddle River, NJO4458. 11. Popa, Valeriu and Noiri, ToKashi, (2001), On The Definitions of Some Generalized Forms of Continuty Under Minimal Conditions, Mem. Fac. Sci., Kochi Univ.(Math.), 22:9-18. 12. Nagata, J., (2002), On Preclosed Sets and Their Generalizations, Houston Journal of Math., 28(4). 13. Maki, H.; Devi, R. and Balachandran, K. (1994), Associated Topologies of Generalized -Closed Sets and -Generalized Closed Sets, Mem.Fac. Sci. Kohi., Univ. Ser. A. Math., 15:51-63. 14. Maki, H.; Devi, R. and Balachdran, K. 91993), Generalized -Closed Sets in Topology, Bull. Fukuoka Univ. Ed. Part III, 42:13-21. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (2) 2011 15. Sundaram, P.; Maki, H. and Balachandran, K. (1991), Semi-Generalized Continuous map and Semi-T1/2 Space, Bull. Fukuoka Univ. Fd. Part III, 40:33-40. 16. Dontchev, J. (1995), On Generalized Semi-Preopen Sets, Mem. Fac.Sci.Kochi.Univ.Ser. A.M ath., 16:35-48. 17. Balachandran, K.; Sundaram, P. and Maki, H. (1991), On Generalized Continuous Maps in Toplogical Spaces, Mem.Fac.Sci.Kochi. Univ. Ser.A.Math., 1: 5-13. 18. Gnanambal, Y. (1997) On Generalized Preregular Closed Sets in Topological Spaces, Indian J. Pure Appl. Math., 28(3):351-360. 19. Palanreppan, N. and K. C. Roo, (1993), Regular Generalized Closed Sets, Kyung Pook Math. J., 33(2): 211-219. 20. Mohammed, Nadia, Faiq, (2010), On Semi--Connected Subspaces, Baghdad Science Journal (Physics and Mathematics), 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  pre-semi-closed regular open gsp-closed Fig. (1) the relations among the different types of weakly closed sets continuous open semi--irresolute regular irresolutesr-irresolute g-continuous  g-continuous  gr-continuous  sr-continuous g-continuous  gr-irresolute  pre-semi-continuous gsp-continuous Fig. (2) the relationships among the different types of weakly continuous function.