AMERICAN JOURNAL OF MATHEMATICAL SCIENCE AND APPLICATIONS 2(2) • July-December 2014 • ISSN : 2321-497X • pp. 115-118
*-NORMAL SPACES AND PRE- g -CLOSED FUNCTIONS IN TOPOLOGY Govindappa Navalagi 1
304, Ashirwad Apartment, Rajatgiri, Dharwad-580004, E-mail :
[email protected]
ABSTRACT: In 1983,M.E. Abd El-Monsef et al. have introduced and studied the concepts of �-open sets, �-closed sets and �-continuity in topology. In I986,D.Andrijevic had defined and studied the concepts of semipreopen sets and semipreclosed sets which are nothing but �-open sets and �-closed sets. In 1995, Julian Dontchev had defined and studied the concepts of generalized semipreopen (in brief, gsp-open) sets and generalized semipreclosed (in brief, gsp-closed i.e., g�-closed ) sets in topology as generalizations of semipreopen i.e., g�-open sets and semipreclosed sets. In 2001 and 2006, Caldas et al and Baker have studied the concepts of contra �-continuity, almost contra �continuity and preservation properties of, �-normal spaces, respectively. The aim of this paper is to define and study a new class of functions called pre-g�-closed functions in topological spaces and *-normal spaces. We, also obtain some more preservations of �-normal spaces. 2010 Mathematics Subject Classifications: 54A05,54B05,54C08 ; 54D10, Key words and phrases: �-open sets, �-closed sets, g�-closed sets, g�-open sets, preopen sets, semiopen sets, �-continuity, �-irresoluteness,g �-closed functions.
1. INTRODUCTION In 1983, M.E.Abd El-Monsef et al. have introduced and studied the concepts of �-open sets, �-closed sets and �-continuity in topology. In I986,D.Andrijevic had defined and studied the concepts of semipreopen sets and semipreclosed sets which are nothing but �-open sets and �-closed sets. In 1995, Julian Dontchev had defined and studied the concepts of generalized semipreopen (in brief, gsp-open) sets and generalized semipreclosed (in brief, gsp-closed i.e., g�-closed ) sets in topology as generalizations of semipreopen i.e., g�-open sets and semipreclosed sets. In 2001 and 2006, Caldas et al and Baker have studied the concepts of contra �-continuity, almost contra �-continuity and preservation properties of, �-normal spaces, respectively. The aim of this paper is to define and study a new class of functions called pre-g�-closed functions in topological spaces and �*-normal spaces. We, also obtain some more preservations of �normal spaces. 2. PRELIMINARIES Throughout the present paper, spaces always mean topological spaces on which no separation axioms are assumed unless explicitly stated. Let X be a space and A a subset of X. We denote the closure of A and the interior of A by Cl(A) and Int(A), respectively. The following definitions and results are useful in the sequel : Definition 2.1: A subset A of a space X is said to be : (i) semipreopen [3](= �-open [1]) if A Cl Int Cl(A).
116
American Journal of Mathematical Science and Applications 2(2), 2014
The complement of a semipreopen i.e. �-open set is called semipreclosed[3] i.e. �-closed set [1] in X. The family of all semipreopen = �-open sets of X is denoted by SPO(X) or �O(X) and that of semipreclosed = �-closed sets of X is denoted by SPF(X) or by �F(X). Definition 2.2 : The intersection of all semipreclosed = �-closed sets containing A is called the the semipreclosure[3] or the �-closure [2] of A and is denoted by spCl(A)[3] or by �Cl(A) [2]. Dually, the the semipreinterior or �-interior of A, denoted by spInt(A) or by �Int(A), is defined to be the union of all semipreopen =�-open sets contained in A. Definition 2.3 : A function f: X � Y is called : (i) ����-continuous [1], if the inverse image of each open set of Y is �-open set in X. (ii) ���-irresolute [7] if the inverse image of each �-open set of Y is �-open set in X. Definition 2.4 : A function f: X � Y is called : (i) ����-closed[1], if the image of each closed set of X is �-closed set in Y. (ii) pre-�-closed [ 5 ] if the image of each �-preclosed set of X is �-closed set in Y. Definition 2.5 [7]: A space X is said to be �-normal provided that every pair of non-empty disjoint closed sets can be separated by disjoint b-open sets. Definition 2.7 : A subset A of a space X is said to be : (i) gsp-closed(=g�-closed) [6] if spCl(A) � U whenever A� U and U is open in X. The complement of a gsp-closed set is called gsp-open set in X. Definition 2.8 : A function f: X �Y is called : (i)
g�-continuous[6], if the inverse image of each open set of Y is g�-open set in X.
(ii) g�-irresolute [6] if the inverse image of each g�-open set of Y is g�-open set in X. Definition 2.9 [5]: A function f : X � Y is said to be contra �-continuous if the inverse image of each open set of Y is �-closed set in X. Definition 2.10[4]: A function f : X � Y is said to be almost �-continuous if the inverse image of each regular open set of Y is �-open set of X. Definition 2.10[4 ]: A function f : X � Y is said to be contra almost �-continuous if the inverse image of each regular open set of Y is �-closed set of X. Definition 2.11 [6]: A space X is called a g�-T1/2 space if every g�-closed subset in X is �-closed set in it. 3. PRE-g -CLOSED FUNCTIONS Definition 3.1: A function f : X � Y is said to be pre-g�- closed if the image of each �-closed set of X is g�-closed set in Y. Note 3.2 : A function f : X � Y is said to be pre-g�-open if the image of each �-open set of X is g�open set in Y.
� *-Normal Spaces and Pre-g � -Closed Functions in Topology
117
Note 3.3 : From the above definitions, we state the following implications : (i)
Every closed function is �-closed function.
(ii) Every pre-�-closed function is �-closed function. (iii) Every pre-�-closed function is pre-g�-closed function. (iv) Every �-closed function is g�-closed function. (v) Every pre-g�-closed function is g�-closed function. Remark 3.4 : (i) closedness and pre-g�-closedness are independent each other. (ii) pre-g�-closedness and �-closedness are independent of each other. Theorem 3.5 : If f : X � Y is �-irresolute pre-g�-closed function and F is g�-closed set of X, then f(F) is g�-closed set in Y. Proof : Obvious. Theorem 3.6: A surjective function f: X � Y is pre-g�-closed if and only if for each subset B of and each �-open set U containing f-1(B), there exists a g�-open set V of Y such that B � V and f-1(V) � U. The routine proof of the Theorem is omitted. It is proved in [ ] that, if f : X � Y and g: Y � Z are pre-�-closed functions, then gof : X � Z is also pre-�-closed function. But, it is not so in case of pre-g�-closedness, which is given in the following. Theorem 3.7 : If f : X� Y is pre-�-closed function and g: Y�Z is pre-g�-closed function, then gof : X� Z is pre-g�-closed function. Easy proof of the theorem is omitted. We, define the following. Definition 3.8 : A function f: X Y is called always g�-closed if the image of each g�-closed set of X is g�-closed set in Y. Clearly, every always g�-closed function is pre-g�-closed function. In the similar way, we give some decompositions of pre-g�-closedness in the following. Theorem 3.9 : If f : X � Y and g: Y � Z are two functions. (i)
If f is �-closed and g is pre-g�-closed, then gof is g�-closed.
(ii) If f is g�-closed and g is always g�-closed, then gof is g�-closed. (iii) If f is pre-g�-closed and g is always g�-closed, then gof is pre-g�-closed. Proof : Obvious. Theorem 3.10 : Let f : X� Y and g: Y�Z be two functions such that gof : X � Z is pre-g�-closed. (i) If f is �-irresolute surjection, then g is pre-g�-closed. (ii) If g is g�-irresolute injection, then f is pre-g�-closed.
118
American Journal of Mathematical Science and Applications 2(2), 2014
Proof : Obvious. We, define the following. Definition 3.11: A space X is said to be �*-normal provided that every pair of non-empty disjoint �closed sets can be separated by disjoint b-open sets. Theorem 3.12: If f : X � Y is a pre-g�-closed, �-irresolute surjection and X is a �*-normal space, then Y is �*-normal. Proof : Obvious. Corollary 3.13: �-normality is preserved under pre-�-closed �-irresolute surjections. Proof is easy and hence omitted. REFERENECES [1] M.E.Abd El-Monsef, S.N.El-Deeb and R.A.Mahamoud, �-open sets and b-continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90. [2] M. E. Abd El-Monsef, R. A. Mahamoud and E.R. Lashin, �-closure and �-interior, J.Fac.Ed. Ain Shams Univ., 10(1986),235-245. [3] D. Andrijevic, Semipreopen sets, Mat. Vesnik, 38(1986), 24-32. [4] C. W. Baker, On contra almost -�-continuous functions and weakly �-continuous functions, Kochi. J. Math., 1 (2006), 1-8. [5] Miguel Caldas and Saeid Jafari, Some properties of contra �-continuous functions, Mem. Fac. Sci., Kochi Univ..(Math.), 22(2001), 19-28. [6] J. Dontchev, Generalization of semipreopen sets, Mem. Fac. Sci. Kochi Univ. Ser. A. Math., 16 (1995), 35-48. [7] R. A. Mahamoud and M. E. Abd El-Monsef, �-irresolute and �-topological invariant, Proc. Pakistan Acad. Sci., 27(1990), 285-296. [8] G. B. Navalagi, Semiprecontinuous functions and properties of generalized semipreclosed sets in topology, IJMMS, 29(2) (2002), 85-98. [9] Govindappa Navalagi and Mallavva Shankrikop,Semipre – regular and semipre-normal spaces, The Global Jour. of Appl. Math. and Math. Sci., Vol. 2, (1-2), (2009), 27-39.
