Kyungpook Mathematical Journal Volume 32, Number 1, June, 1992 ON I-OPEN SETS AND I-CONTINUOUS FUNCTIONS M.E. Abd EI-Monsef, E.F. Lashien and A.A. Nasef In 1990, D. Jankovic’ and T.R. Hamlett have introduced the notion of I-open sets in topological spaces. The aim of this paper is to introduce more new properties of I-openness. Also, we introduce and study new topological notions via ideals, namely, I-c\osed sets, I-continuous func­ tions, I -open (c1osed) functions. Relationships between these c\asses and other relevant c1 asses are investigated 1. Introduction Throughout the present paper, (X, T) and (Y, (7) (or simply X and Y) denote topological spac얹 on wl삐1 no separation axioms are assumed unl얹 s explicitly stated. Let A be a subset of (X, T). The c\ osure of A and the interior of A are denoted by Cl(A) and int(A) , respectively. Recall that A is said to be regular closed if Cl(intA) = A. A is said to be 0 • open (0.0.) [8] (resp. semi-open (5.0.) [5] , preopen (P. O.) [6 ], β-open (β 0.) [1]) if A c ir벼 CI(int(A))) (resp. A C Cl(int(A)), A c int(Cl(A)), A C CI(int(CI(A))). The complement of an o-open (resp. serni-open, preopen, β-open) set is called o-c\osed (resp. semi-closed, pre-closed, β­ c1osed). T he family of all o-open (resp. sem야pen ， preopen, ß-open) sets of (X, T) is denoted by TQ (resp. SO(X, T) , PO(X, T) , ( 더O(X ， T)) Q lt is shown in [8] that T'" is a topology on X and T C T • An ideal on a nonempty set X is a collection 1 of subsets of X which atkmditivlty)is closed [lO]uIldel Wethe denot opeI e by (X , T, I) a topological space (X, T) and an R.e ceived J anuary 4, 1991 Key Words and Phrases. Ideal, regular open set , a-open set , semi-open set, preopen set, β-open set , 1←。 pen set, / - c1 osed set , pre-continuous function , M-pre-continuous function , I-open function , I-closed function , I-continuous function 1980 AMS Subject Classification Codes Primary: 54C10;Secondary; 54D25, 54D30 21 22 M.E . Abd EI-Monsef. E.F. Lashien and A.A. Nasef ideal 1 on X. Given a space (X,T ,1) and a subset A 드 X , we denote by A*(I) = {x E X : U n A rf. 1 for every (open) neighborhood U of x} , wri t ten simply as A* when there is no chance for confusion; CI*(A) = A U A‘ defines a Kuratowski closure operator [10] for a topology T*(I) (also de­ noted T* when there is no chance for confusion) finer than T. The topol ogy T* has as a basis 더 (I， T) = {U\ E : U E T , E E I} [9] . Recall that A c (X, T , 1) is called *-dense-in-itself [2] (resp. T ‘ -closed [3] , *-perfect [2]) iff A c A* (resp. A* c A, A = A*) A function f : (X, T) • (Y, (7) is said to be pre-continuous [6] (resp M-pre-continuous [7]) if for each V E σ (resp. V E PO(Y)), f-l(V) E PO(X). f is called preopen [6] (resp. preclosed [6]) if the image of each open (resp.closed) set in X is preopen (resp. preclosed). 2. On I-open and I-closed sets Definition 2.1 [4]. Given a space (X, T , 1) and A 드 X , A is said to be 1-open if ...1 드 int(A*) We denote by IO(X,T) = {...1 드 X:A 드 int(A‘)} or simply write 10 forIO(X,T) 、애 en there is no chance for conf뼈 on Remark 2. 1. It is clear that, 1-openness and openness are independent concepts (Examples 2.1 , 2.2) Example 2.1. Let X = {a , b, c, d} with a topologyr = {X, ø, {c} , {a , b} , {a , b, c}} and 1 = {</>, {a}} . Then {b, c, d} E IO( X , T) but {b, c, d} rf. T Example 2.2. Let X be as in Example 2‘1, T = {X, ø, {d} , {a , c} , {a , c, d} } and 1 = {ø , {c} ,{d} ,{c ,d}}. It is clear that {a ,c,d} E T , but {a ,c,d} rf. IO(X,T) Remark 2.2. One can deduce that: I-open set =수 preopen set, and the converse is not true, in general, as shown by the following example. Example 2.3. Let X , T and 1 be as in Example (2.2). Then, we can easily deduce that {d} E PO(X,T) , but {d} rf. IO(X,T). Remark 2.3. The intersection of two I-open sets need not be 1-open 잃 IS illustrated by the following example. Example 2.4. Let X = {a ,b,c,d} , T = {X,ø, {a ,b}, {a ,b,c}} and 1 = {</>}‘ Then {a , c}, {b, c, d} E IO(X, T) , but {a , c} n {b , c, d} rf. IO(X, T). Theorem 2.1. For a space (X, T , 1) and ...1 c X , ψe have: (i) 1f 1 = {ø }, then ...1*(1) = CI(A) , and hence each of I-open set and On I-open sets and /-continuous functions 23 preope η set are coi η cide. (ii) /1/ = P(X), then A*(1) = </> and hence A is /-open iJJ A = </>. Theorem 2.2. For a η y 1 -open set A 01 a space (X, T , 1), we have A* = (int(A*))*. Definition 2.2. A subset F 드 (X, T , 1) is called /-closed if its complement is /-open. Remark 2.4. The concept of /-closeness makes a very important deviation from the closeness for the topology in ordinary sense. Theorem 2.3. For A ζ (X, T ,I) ψ e haν e ((int(A))*)C 카 int((AC)*) in ge η eral (Example 2.5) 뼈eπ AC denotes the compleme η t 01 A. Example 2.5. If X = {a ,b,c,d}, T = {X,</>, {a} , {a ,b} , {a ,b,c}} and / = {</>, {a}}. Then it is clear that if A = {a , b} , then : (( int( A)) γ = {a }, but int((Ac)*) = </> Theorem 2.4. /1 A 드 (X,T ,I) is /-c/osed, then A ::J (intA))‘ Proof Follows from the definition of /-closed sets and Theorem 2.3(c) [3] Theorem 2.5. Let A 드 (X, T , 1) and (X\ (int(A))*) = int((X\ A)‘) The η A is / -c/osed iJJ A 그 (int(A))‘ Proof Obvious Theorem 2.6. Let (X, T , 1) be a space and A, B ç X. Then: (i) /1 {Uo : a E ,",} 드 IO(X,T) , the η U{uo: a E ,",} E IO(X,r) μ]. (ii) /J A E IO(X,T) and B E T, then A n B E IO(X,T) μl 끼(i페111 (iω비v끼) IIAEIO(X,T) andBESO(X,T) , thenAnBESO(A). (v) /J A E IO(X,T) aη d B E T, the η AnB 드 int(B n (B n A)*). Proof (i) Since {UO : a E ,",} 드 IO(X, T) , then Uo 드 int(U~) ， for every a E ,"" thus, UU" 드 U(in tU~)) 드 int(UU~) 드 int(UUo )'‘ for every a E ,",. (ii) A n B 드 1η t(A*) n B = int(A* n B) (since B E T) , from Theorem 2.3 (g) [3], we have : A n B ç int(A n B)*. (iii) Obvious, since A *(1) is closed and A* 드 Cl(A). (iv) Follows from Theorem (2.3) (c) [3]. (v) Follows directly [rom Theorem 2.3 (g) [3] Corollary 2. 1. (i) The union 01 /-closed set and c/osed set is /-closed. 24 M.E. Abd 티 Monsef. E.F. Lashien and A.A . Nasef (ii) Th e union of 1 -c/osed set and an a-closed se t is preclosed Theorem 2.7. If A 드 (X , T ,I) is 1 ~op e n and semi-closed, then A = int(A-). Proof Follows (rom Theorem 2.3(c) [3] . Theorem 2.8. Let A E IO(X ) and B E IO(Y ), th en A x B E IO(X x Y ) ν A- x B- = (A x B )" , ψh eπ X x Y is the product space Proof AxB 드 int(A") x int(B" ) = int(A‘ x B-), from hypothesis, = int(A x B)". Therefore, A x B E IO(X x y) Theorem 2.9. If A c W C CI(A) and A E IO( χ T) ， then W is ß-open. Proof Follows directly from Theorem 2.3 (c) [3J. Theorem 2.10. If(X,T,I) is a space and W E IO(X,T) , then CI(V) n W c (V n W)* , for every V E SO(X). Proof Let V E SO(X), then: CI(V) = CI(int(V)), since W E IO(X), then CI(V) n W C CI(int(V)) n int(W") C CI(int(V) n W*) c CI(V n W)" , by using Theorem 2.3 (c) [3] = (vnw)". Theorem 2.11. If(X,T ,I) is a space, A E T and B E IO(X,T), then there exists an open subset G of X such that AnG = 4>, implies AnB = 4> Proof Since B E IO (X , T) , then B 드 int(B-), by talcing G = int(B") to be an open set such that B c G, but A n G = 4> , then G ç X\A implies that CI(G) 드 (X \ A).