On I-Open Sets and I-Continuous Functions

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). Hence B 드 (X\A) and this completes the proof.

Theorem 2.12. If (X ,T ,If) is a 되 -spa ce and A E IO(X), then A 드 d d int(A ) , ψ h e π A denotes the 따de ri ve d set of A and If denotes the ideal of finit e subsels. Proof Follows directly from the definition of I -open set and the fact that A*(If) = Ad in a T1-space [3] . On I-open sets and I-continuous functions 25

Theorem 2_13. Let {X" : a E l'-} be a family of spaces, X == n X。 ’‘ be the prodπ ct space and A == II A" x II xβ， a non empty subset of X , 0=1 o :J:. β where n is a positive integer and A" c X". Then, A" E IO(X,,) for each (1 ~ a ~ n) iJJ A E IO(X).

Proof (Nece5sity): Suppose A" E IO(X,,) for each (1 ~ (l' ~ n). Sin ce a A == II A" x II X ß 드 int(A')‘ Then A E IO(X). 0=1 0#β (Sufficiency): AS5 ume that A E 1 O( X). So A 드 int(A') == n:= ,A: x n. ， β Xβ. Since A fo rþ and A E I O(X) then int(A') fo rþ and hence

int(A:) 폼 rþ , for each (1 < (l' ~ η) . Therefore, A" 드 int( A:) and 50 , A" E IO(X) for each (1 ~ (l' ~ n).

Theorem 2.14. For a subset A 드 (X,r ,1) ψe have: (i) If A is r' -c/osed and A E IO(X), th e π ， int(A) == int(A*) (ii) A is r' - c/osed íJJ A is open and 1 -c/osed. (iii) If A 생 *-perfecl, then A == int(A'), for every A E IO(X,r) (iv) Jf A is regular closed and I-open, then A'(Jn) == int(A'Un)) ψhe1'e In is the ideal of nowhere dense sets. (Tn == {A c X : int(CαI (A꺼)) == rþ ηn} ) .

p좌roof 띠(ii) ,’ (ii ) and (ii폐l ( 비ivv) Follows from the defini tion of J-open and the fact that A is regular closed iff A == A* (Jn) [3]

3. I-continuous functions

Definition 3.1. A function f (X, r , 1) • (κ 0") is 5aid to be 1- continuous iffor every V E O", f-'(V) E IO(X,r). From the above definition one may notice that

1-continuity =추 precontinui ty [6]

and the converse is not true as 5hown by the foll ow ing example.

Example 3.1. Let X == Y == {a ,b,c,d} , r is the in띠1κ띠미cdliscre the discrete topology and 1 == { rþ , {c}} on X . T hen the idcntity function f : (X , r ,1) • ()이 0") is precontinuou5 but not I-continuous, because {c} E 0" , but f - ' ({c}) == {c} f/. IO(X). T he following two examples show that the concept of continuity and I -continuity are independent 26 M.E . Abd EI -Monsef, E.F. Lashien and A.A. Nasef

Example 3 .2. Let X = Y = {a , b, c} , T = {X, 4>, {a} , {c} , {a,b}, {a , c}} , 1 = {4>, {b }, {c }, {b, c}} on X and u = {Y, 4>, {a }, {c }, {a , c}}. Then the identity fun ction f : (X, T , 1) • (Y, u) is continuous but not I-continuous because {c} E u, but f-l({C}) = {c} rf. JO(X)

Example 3.3. Let X = Y = {a ,b,c}, T = U = {X, rþ , {a}} and 1 = {4>, {b}} on X. Then f: (X,T ,I) • (κ u) which is defined by: f(a) = a = f(b) and f(c) = c is I-continuous but not continuous.

Theorem 3.1. For a funclion f : (X,T ,I) • (Y, u) the follouη ng are equivalent: (i) f is 1 -contin.LOUS . (ii) For each x E X and each V ε u containing f( x), there exists W E IO(X) containing x such that f(W) C V. (iii ) For each x E X and each V E u containing f(x) , (J -l(V)t is a neighborhood of x.

Proof (i) =} (ii ) : Since V E u containing f (x) , then by (i), f-l(V ) E IO(X), by putting W = f -l(V) which containing x. Therefore f(W) C v (ii ) =추 (iii): Since V E u containing f( x) , then by (ii), there exisls W E IO(X) containing x such that f(W ) C V. So, x E W 드 int(W‘) 드 int(J - l(V))' ç U- 1 (V))* . Hence U- 1 (V))‘ is a neighborhood of x (iii ) =추 (i) : Obvious

Theorem 3.2. For f : (X , T , 1) • (Y, u) the following are equivalenl: (i) 1 is 1 -contin uous. (ii ) The inverse image of each closed set in Y is 1 -closed. (i ii ) (int(J - l (M)))‘ c f-l(M‘), for each *-dense-in-ilself subset M C Y (i v) 1((int(U))*) C (J (U))', for each U c X , and for each *-perfect subset 01 Y

Proof (i) =} (ii ): Let F C Y be c1osed, then Y \ F is open, by (i), 1- 1 (y\ F) = X\f-l(F) is I-open . Thus, 1-1 (F) is I- c1osed (ii) =송 (iii): Let M C Y , since M" is c1osed, then by (ü) f-l(M') is I- c1 osed. Thus, by using Theorem (2.4) f- l(M") ::l (int (J -l(M’)))", since M is *-dense-in-itself, then f - l(M") 그 (int(J-l (M")))" 그 (int(J - l(M)))". (iii ) =} (iv): Let U C X and W = I (U) , then by (iii ), f-l(W') 그 (int(J -l(W)))* ::l (int(U))*. Hence, f ((int(U))') C W' = (J(U))" . (iv) =} (i): Let V E u, W = Y \ V and U = f-l(W), then I(U) c W and by (iv) , 1((int(U))*) C (J(U))' C W" (by using T heorem 2.3(a) [3]) On I-open sets and I-continuous functions 27

= W (because W is *-perfect). Thus, f -l(W) 그 (int(U))* = (int(J-l(W)))*, and therefore, f-1(W ) = f-l (y\ V) is I-closed. Hence, f-l(V) is I -open in X and f is I-continuous

Theorem 3.3. The function f : (X, T, I) • (Y, a) is 1 -continuous 퍼 the graph function 9 : X • X x Y is 1 -continuous Proof (Necess ity): Let f be I -continuous. Now let x E X and let V be any open set in X x Y containing g(x) = (x, f(x) ). Then there exists a basic open set U x W such that g(x) E U x W C V. Since f is 1 continuous, there exists I -open set U!, in X sucht that x E U1 C X and f (U1) C W. Since U1 n U is I -open set in X and U1 n U c U , then g(U1 n U) c U x W C V showing that 9 is I-continuous. (Sufficiency): Let 9 : X • X x Y be I-continuous and let V be open containing f (x) . Then X x V is open in X x Y and the I-continuity of 9 1r때 li es there exists I -open set W such that g(W) C X x V. But this implies f (W) C V. Therefore, f is I-continuous.

Theorem 3.4. Let f : (X, T, I) • (Y, a) be an 1 -contínuous and U E T. Then the Testriction f lU is an 1 -continuous. Proof Let V E a. Then f - 1(V) ç int(J -1(V))* and so , U n f-l(V) 드 1 1 U n intU- (V))*. Thus (JlUt (V) 드 U n int(J-1(V))“ since U E T , then 1 1 (J lut (V) = int[U n (r (V))*] , by using Theorem 2.3 (g) [3] C i떠 [U n r 1 (V) ]‘ int[(J IUt1(V)]*. Therefore, flU is 1 - continuous

Theorem 3.5. Let f : (X, T , I) • (Y, a, J) be a function and {U", : Q E 6 } be an open cover of X. If the restriction fu nciion f lU '" 양 1 -continuous, for each Q E 6 , then f is 1 -continuous. Proof Similar to Theorem 3.4. The folJ owing results are immediate and the obvious proofs are omit ted.

Theorem 3.6. Let f : (X, T , I) • (Y, a ) be 1 -continuous and open func tion, then the inverse image of each 1 -open set in Y ís preopen ín X.

1 Theorem 3.7. Let f: (X,T ,I) • (Y,a) be I-continuous and f- (V*) C 28 M.E. Abd EI-Monsef. E.F. Lashien and A.A. Nasef

1 U- (V) t , Jor each V C Y. Then the inVel‘'s e image oJ each 1 -open set is I-open. Remark 3.1. The composition of two I -continuous functions need not be I-continuous, in general, as shown by the following example. Example 3 .4. Let X = Z = {a ,b,c} , Y = {a ,b,c,d} with topologies r = {X,Iþ, {a} }, 17 = {Y,Iþ, {a ,c}} and v = {Z,.p, {c}, {b ,c}} and let 1 = {.p, {c}} on X and J = {Iþ, {a}} on Y and let the identity function J : (X, T ,I) • (Y, σ) and 9 : (Y, 17, J) • (Z, v) defined as: g(a) = a,g(b) = g(d) = b and g(c) = c. It is clear that both J and 9 are 1- continuous. However, the composition function 9 0 J is not I-continuous because {c} E v, but (g 0 J)-I({C}) = {c} rt IO(X).

Theorem 3.8. The Jollowing hold Jor the Junctions: J: (X, T , 1) • (Y, (7) and 9 : (Y, 17, J) • (Z, μ) (i) 9 0 J is J-continuous, iJ J is J-continuous and 9 is continuous (ii) go J is preco η tinuous ν J is M-P-coη tinuous and 9 is 1 -continuous (iii) JJ J is surjectio η ， J-l(B*) c [J -l(B)]* Jor each B c Y and both J and 9 are J-continuous, then 9 0 J is also J-continuous Proof (i) This is obvious. (ii) Follows from the fact that each I-open set is preopen set. (iii) Is c1 ear by using Theorem 3.7.

4. I-open and I-closed functions

D efinition 4.1. A function J: (X, T) • (Y, 17, J) is called I-open (resp I -closed) if for each U E T (resp. U is closed), J(U) E IO(Y) (resp. J(U) is J-closed). Remark 4.1. (i) l-open (I-closed) function => preopen (preclosed) function and the converse is not true in general (Example 4.1) (ii) Each of l -open function and open function are independent (Ex amples 4.2, 4.3) .

Example 4. 1. Let X = Y = {a ,b,c} with two topologies T = {X,Iþ {a} , {a ,b} , {a ,c}} , 17 = {Y,Iþ, {a} ,{a ,b}} and J = {Iþ, {a} , {b} ,{a ,b}} on Y ‘ Then the identity function J : (X,T) • (Y, 17, J) is preopen but not J open, because, {a} E T , but J({a}) = {a} rt IO(Y).

Example 4.2. lf X = {a ,b,c,d} = Y , T = {X,Iþ, {a ,b} ,{a ,b,d}} , 17 = {Y,.p, {a ,b}, {a,b,c}} and J = {Iþ, {c} , {d }, {c,d}} on Y. Then the On 1-open sets and l-contin uous functions 29 identity function f: (X,T) • (Y, 17 , J) is 1-open function but not open function.

Exar매 le 4.3. If X = Y = {ι b ， c} , T = {X, 1>, {a}} , 17 = {Y, 1>, {a} , {a, b}} and J = {1>, {a}} , then the identity function f : (X, T) • (Y, 17, J) is open but not 1-open because {a} E T , but f({a}) = {a} (j 10(Y).

Theorem 4.1. Let f: (X,T) • (Y, σ ， J) be a function. Then the follo ψ ing are equivalent: (i) f is 1 -open function (ii) For each x E X and each neighborhood U of x , there exists an 1-open set W C Y containing f(x) such that W c f(U) Proof Immediate.

Theorem 4.2. Let f: (X,T) • (Y, 17, J) be an 1 -ope η (resp . 1 -closed) funclioη ， if W C Y and F c X is a closed (resp. ope 띠 set containmg f-1 (W), then there exists a η 1-closed(resp. 1-open) set H c Y containiηg W such that f-1(H) C F. Proof This is obvious.

Theorem 4.3. If f : (X,T) • (Y, 17 , J) is 1 -open, the η f-1(int(B)t C (f- 1(B))* such that f-l(B) is *-dense-in-itselj, for every B C Y Proof Obvious by using Theorem (4.2)

Theorem 4 .4. For any bijeclive function f (X, T) • (κ 17 , J) the following are equivalent: (i) f-1 : (Y, σ ， J) • (X, T) is 1-contiη uous (ii) f is 1-open. (iii) f is 1 -c/osed.

Theorem 4.5. 1f f : (X,T) • (Y, σ ， J) is 1 -open and for each A c X , f(A‘) c [J (A)]" , then the image of each 1-ope η 8et is J -open

Theorem 4.6. Let f : (X, T , 1) • (Y, 17, J) and 9 : (Y, 17 , J) • (Z, ι J() be two funclions, where 1, J and κ are ideals on X , Y and Z respeclively Then: (i) 9 0 f is J - ope η ， if f is open and 9 is J -open. (ii) f is 1 - ope η if 9 0 f is ope η and 9 is 1 -contin uous iη:j ective. (“패11l each V c Y , then 9 0 f is 1-open. Proof Obvious. 30 M.E. Abd EI-Monsef, E.F. Lashien and A.A. Nasef

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TANTA UNJV ERS1TY, EGYPT