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0Journal of the Egyptian Mathematical Society (2012) 20, 7-13
Egyptian Mathematical Society Journal of the Egyptian Mathematical Society
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ORIGINAL ARTICLE
Pairwise weakly and pairwise strongly irresolute functions
F.H. Khedr a *, H.S. Al-Saadi b
a Department of Mathematics, Faculty of Sciences, University of Assiut, Assiut 715161, Egypt b Department of Mathematics, Faculty of Education for Girls, P.O. Box 4281, Makkah, Saudi Arabia
Available online 2 March 2012
KEYWORDS
//-Semi-open set;
//-Semi-continuous;
//-Irresolute;
//-Semi-closure;
//-Semi T2-space;
//-Semi-compact
Abstract In this paper we consider a new weak and strong forms of irresolute functions in bitopo-logical spaces, namely, //-quasi-irresolute functions and strongly irresolute functions. Several characterizations and basic properties of these functions are given. We investigate the relationships among some weak forms of continuity and other generalizations of continuous mappings in bito-pological spaces.
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1. Introduction
The study of bitopological spaces was first initiated by Kelly [3] and thereafter a large number of papers have been done to generalized the topological concepts to bitopological space. Irresolute mappings in bitopological spaces was defined by Mukherjee [11]. In 1991 Khedr [4] introduced and investigate a class of mappings in bitopological spaces called pairwise h-irresolute mappings. Khedr [7] defined the concept of quasi-irresolute mappings in these spaces and studied some of its
* Corresponding author.
E-mail addresses: Khedrfathi@gmail.com (F.H. Khedr), hasa112@ hotmail.com (H.S. Al-Saadi).
1110-256X © 2011 Egyptian Mathematical Society. Production and hosting by Elsevier B.V. All rights reserved.
Peer review under responsibility of Egyptian Mathematical Society. doi:10.1016/j.joems.2011.12.007
properties. The concepts of strongly irresolute mappings in bitopological spaces was defined by Khedr in [6] and he showed that quasi-irresoluteness and semi-continuity are independent of each other.
The aim of this paper is to introduce basic properties of quasi-irresolute and strongly irresolute functions in bitopolo-gical spaces. We study these functions and some of results on s-closed spaces and semi-compact spaces in bitopological spaces. Also, we investigate the relationships among some weak forms of continuity, irresoluteness, quasi-irresoluteness and strong irresoluteness.
Throughout this paper (X, t1; s2), (Y, r1; r2) and (Z, v1; v2) (or briefly X, Y and Z) denote bitopological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of X, we shall denote the closure of A and the interior of A with respect to s, (or ri) by i-cl(A) and i-int(A) respectively for i = 1,2. Also i,j =1,2 and i „ j.
A subset A is said to be /-semi-open [1], if there exists a s-open set U of X such that U c A c j-cl(U), or equivalently if A c j-cl(i-int(A)). The complement of an /-semi-open set is said to be /-semi-closed. An /-semi-interior [1] of A, denoted by ij-sint(A), is the union of all ij-semi-open sets contained in A. The intersection of all /-semi-closed sets containing A is called the /-semi-closure [1] of A and denoted by ij-scl(A).
A subset A of X is said to be i/'-semi-regular [7] if it is both /i-semi-open and i/-semi-closed in X. The family of all i/-semi-open (resp. i/'-semi-closed, i/'-semi-regular) sets of X is denoted by i/-SO(X) (resp. i/-SC(X), i/-SR(X)) and for x 2 X, the family of all i/'-semi-open sets containing x is denoted by i/-SO(X, x).
A point x 2 X is said to be i/-semi h-adherent point of A [2] if /i-scl(U) n A —/ for every i/'-semi-open set U containing x. The set of all i/-semi h-adherent points of A is called the i/-semi h-closure of A and denoted by i/-sclh(A). A subset A is called i/semi h-closed if i/-sclh(A) = A. The set fx 2 X\/'i-scl(U) C A, for some U is i/-semi-open} is called the i/-semi h-interior of A and is denoted by i/-sinth(A). A subset A is called i/-semi h-open if A = i/-sinth(A).
Now, we mention the flowing definitions and results:
Definition 1.1. A bitopological space (X, si, s2) is said to be:
(i) Pairwise semi-T0 [8] (briefly P-semi T0) if for each distinct points x, y 2 X, there exists either an i/'-semi-open set containing x but not y or a /'-semi-open set containing y but not x.
(ii) Pairwise semi-Ti [8] (briefly P-semi T^ if for every two distinct points x and y in X, there exists an i/'-semi-open set U containing x but not y and a /'-semi-open set V containing y but not x.
(iii) Pairwise semi-T2 [6] (briefly P-semi T2) if for every two distinct points x and y in X, there exists either U 2 i/-SO(X, x) and V 2 /ï-SO(Xy) such that U n V = /.
Definition 1.2. [7] A bitopological space (X, s1, t2) is said to be i/'-semi-regular (resp. i/-s-regular) if for each i/'-semi-closed (resp. s-closed) set F and each point x R F, there exists an i/'-semi-open set U and a /'-semi-open set V such that x 2 U, Fc V and U n V = /.
Lemma 1.3. [8] For every subset A o/ a space X, we have the /o//owing:
(i) X \ i/-scl(A) = i/-sint(X \ A).
(ii) X \ i/-sint(A) = i/-scl(X \ A).
Lemma 1.4. [7] Let A be a subset o/ a space X. Then we have:
(i) If U 2 i/-SO(X), then j'i-scl(U) 2 ./'i-SR(X).
(ii) If A 2 i/-SO(X), then ./'i-scl(A) = i/-sclh(A).
Lemma 1.5. [9] Let A be a subset of a space X. Then we have if A 2 i/-SR(X), then A is both i/-semi h-closed and/i-semi h-open.
Lemma 1.6. [8] / a/unction/: (X, s1, t2) ! (Y, r2) is an i/-pre-semi-closed, then for each subset S c Y and each U 2 i/-SO(X) containing (S), there exists V 2 i/-SO(Y) such that Sc Vand/^7(V) c U.
Lemma 1.7. [7] A bitopological space (X,ti,s2)i/-semi-regular (resp. i/-s-regular) if and only if/or each i/-semi-open (resp. T,-open) set G and each point x 2 G, there exists an i/-semi-open set U such that x 2 U, F c V and/'i-scl(U) c G.
Lemma 1.8. [7] A bitopo/ogica/ space (X, t1, t2) is i/-s-c/osed if andon/y //or every cover o/Xby ji-semi-regu/ar sets has afinite subcover.
Lemma 1.9. [9]
(i) Every ji-semi h-c/osed set is i/-h-sg-c/osed.
(ii) A bitopo/ogica/ space (X, t1, t2) is an P-semi T 1=2-space if and on/y if every i/-h-sg-c/osed set is i/-semi-c/osed.
Definition 1.10. [9] A function /: (X, t1, t2)!(Y, r1, r2) is called:
(i) i/'-h-semigeneralized continuous (briefly i/'-h-sg-continu-ous) if f _1(F) is i/'-h-sg-closed in X for every ji-semi-closed
V of Y.
(ii) i/'-h-semigeneralized irresolute (briefly i/'-h-sg-irresolute) if f _1(V) is i/'-h-sg-closed in X for every i/'-h-sg-closed set
V of Y.
(iii) i/'-h-sg-closed if for every ji-semi-closed set U of X,/(U) is an i/'-h-sg-closed in Y.
(iv) i/'-semi-generalized closed (briefly i/'-sg-closed) if for each t/-closed set F of X, /(F) is an i/'-sg-closed set in Y.
2. Characterization of pairwise quasi-irresolute functions.
Definition 2.1. [7] A function/: (X, t1, t2)! (Y, r1, r2) is said to be i/'-quasi-irresolute if for each x 2 X and each V 2 i/-SO(Y,/(x)), there exists U 2 i/-SO(X, x) such that /(U) Cji-scl(V). If/is 12-quasi-irresolute and 21-quasi-irresolute, then/ is called pairwise quasi-irresolute.
Definition 2.2. A function/: (X, t1, t2) ! (Y, r1, r2) is said to be i/'-irresolute [11] (resp. i/'-semi-continuous [1]) if/_1(V) is an i/'-semi-open set of X for every i/'-semi-open (resp. r,-open) set V of Y.
Theorem 2.3. The /o//owing statements are equiva/ent /or a /unction /: X ! Y:
(i) / is i/-quasi-irresolute
(ii) i/'-scl/^CB)) c/_1(i/-sclh(B)) for every subset B of Y.
(iii) /(i/-scl(A)) c i/-sclh/(A)) for every subset A of X.
(iv) /_1(F) 2 i/-SC(X) for every i/-semi h-closed set F in Y.
(v)/_1(V) 2 i/-SO(X) for every i/-semi h-open set V in Y.
Proof. (i) ) (ii): Let B c Y and x R/_1(i/-sclh(B)). Then /(x) R i/-sclh(B) and there exists V 2 i/-SO(Y,/(x)) such that ji-scl(v) n B = /. By (i), there exists U 2 i/-SO(X, x) such that /(U) c ji-scl(v). Hence /(U) n B = / and U n/_1(B) = /. Consequently, we obtain x R i/-scl (/_1(B)).
(ii) ) (iii): For any subset A of X, the inclusion i/-scl(A) c i/-scl(T1/(A))) holds. By (ii), we have i/-scl(r1(/(A))) c/^1(i/-sclh (/(A))) and hence/(i/-scl(A)) c i/-sclh(/(A)).
(iii)) (ii): For any subset B of Y, we have i/-sclh/(r1(B))) c i/-sclh(B). By (iii), we obtain/(i/-scl(T1(B))) c i/-sclh(/(/^1(B))) and hence i/--scl(/"1(B)) c/" 1(i/-sclh(B)).
(ii) ) (iv): Let F be an i/-semi h-closed set in Y. By (ii), we have j-sclf-1(F)) c/-1(i/-sclh(F)) = f-:(F). Therefore,/-1(F) is ij-semi-closed in X.
(iv) ) (v): If Vis i/-semi h-open in Y, then Y\Vis i/-semi h-closed. By (iv), f-1(Y\V) = X\/-1(V) is ¿/'-semi-closed in X. Thus f-\V) 2 ij-SO(X).
(v) ) (i): Let x 2 X and V 2 i/-SO(Y,/(x)). It follows from Lemmas 1.4 and 1.5, thatji-scl(v) is ji-semi h-closed and i/-semi h-open in Y. Set U — /-1(ji-scl(V)). By (v), we observe that U 2 ij-SO(X) andf(U) cji-scl (V). The proof is complete. □
The next theorem contains an unexpected result.
Theorem 2.4. The following statements are equivalent for a function f: X ! Y:
(i) f is pairwise quasi-irresolute
(ii) For each x 2 X and each V 2 i/-SO(Y,/(x)), there exists U 2 ij-SO(X) such that /(ij-scl(U)) cji-scl (V).
(iii) f -1(F) 2 ji-SR(X) for every F 2 ji-SR(Y).
(i) ) (ii): Let x 2 X and V 2 i/-SO( Y,f(x)). Then by Lemmas
1.4 and 1.5, ji-scl (V) is both i/-semi h-open and ji-semi h-closed. Put U — f -1 (ji-scl(u)). Then it follows from Theorem 2.3(v), that U 2 ji-SR(X). Thus we obtain U 2 ij-SO(X). U = ji-scl(U) and f(i/-scl((U)) cji-scl(v).
(ii) ) (i): Obvious.
(i) ) (iii): Let V 2ji-SR(Y). By Lemma 1.5, V is ji-semi h-closed and i/-semi h-open in Y. It follows from Theorem 2.3 that/-1(V) 2 ji-SR(X).
(iii) ) (i): Let x 2 X and V 2 ij-SO(Y,f(x)). By Lemma 1.4,
ji-scl(v) 2 ji-SR(Y,/(x)) and by hypothesis f-1(/i-scl(v)) 2ji-SR(X,x). Put U = /-1(i/-scl(v)), then U 2 i/-SO(X, x) and f(U) c ji-scl(v). This shows that f is i/-quasi-irresolute. □
The following Theorem offers several characterizations of i/-quasi-irresolute functions.
Theorem 2.5. The following statements are equivalent for a function /: X ! Y:
(i) f is pairwise quasi-irresolute
(ii) ij-sclh(f-1(B)) cf -1(ij-sclh(B)) for every subset B of Y.
(iii) f (ij-sclh(A)) c ij-sclh(f (A)) for every subset A of X.
(iv) f-l(F) is i/-semi h-closed in X for every i/-semi h-closed set F in Y.
(v)/-1(V) is i/-semi h-open in X, for every i/-semi h-open set V in Y.
Proof. By making use of Theorem 2.4, we can prove this Theorem in the similar way to the proof of Theorem 2.3. □
Theorem 2.6. Let/: (X, x1, x2) ! (Y, ri, r2) be y-h-sg-irresolute. If (X, Xj, s2) is pairwise semi T1 ¡2, then/ is/i-quasi-irresolute
Proof. Suppose that V is a /i-semi h-closed set in Y. By lemma 1.9(i), V is i/-h-sg-closed in Y. Since / is i/-h-sg-irresolute, f-1(V) is i/-h-sg-closed in X. By Lemma 1.9 (ii), f-1(v) is i/semi-closed. This shows that / is i/-quasi-irresolute, by Theorem 2.3. □
Theorem 2.7. If /: (X, s1, x2)!(Y, r1, r2) is an /-quasi-irresoulte and Y is /-semi-regular, then / is /-irresolute.
Proof. Let V 2 i/-SO(Y) and x 2 f- 1(V), there exists W 2 i/-SO(Y) such thatf(x) 2 Wc/i-scl(W) c V, since/is i/-quasi-irreso-lute, then there exists U 2 i/-SO(X,x) such that f(U) c/i-scl(W). Therefore, we have x Uc/-1(ij-scl(W)) cf-1(V) and hence /-1(V) 2 ij-SO(X). This shows that/is i/-irresolute. □
Theorem 2.8. Iff: (X, s1, s2) ! (Y, r1, r2) is an ij-quasi-irreso-lute and Y is ij-semi-regular, then / is ij-semi-continuous.
Proof. Similar to that of Theorem 2.7. □
Lemma 2.9. Let /: X ! Y and g : X ! X x Y the graph function of / where g(x) — (x, f(x)) for each x 2 X. If g is ij-quasi-irresolute, then / is ij-quasi-irresolute.
Proof. Let x 2 X and V 2 ij-SO(f(x)). Then X x V is an ij-semi-open set in X x Y containing g(x). Since g is i/-quasi-irresolute there exists U 2 ij-SO(X) such that g( U) ji-scl(X x V) X x ji-scl (V). Thus we obtain f(U) ji-scl (V). □
The converse of Lemma 2.9, is not true as the next example shows.
Example 2.10. Let X —fa, b, cg, s1 —{/, {a}, {b}, fa, bg, Xg and s2 — {/, {b}, {a, b}, X}. Define a function /: (X, s1, s2) ! (X, s1, x2) by setting /(a) — b, f(b) — a and f(c) — c. Then / is 12-irresolute and hence 12-quasi-irresolute but g is not 12-quasi-irresolute. It is apparent that 12-SO(X) — {/, {a}, {b}, {a, b}, {b, c}, {a, c}, X}. We prove that g is not 12-quasi-irresolute at c. Now, put V — {(a, a), (a, c), (c, c)}. Then V is 12-semi-open in Xx X, g(c) — (c,/(c)) — (c, c)2 V and V = 21-scl(V). Since 12-SO(c) — {{a, c}, {b, c}, X}, g(U) 21-scl(V) for every U 2 12-SO(c). Thus show that g not be 12-quasi-irresolute at c.
Lemma 2.11. A space X is pairwise semi-T2 if and only if for each pair distinct of points x,y of X, there exist U 2 ij-SO(X,x) and V 2 ij-SO(X,y) such that ij-scl(U) \ ij-scl(V) = /.
Proof. This follows immediately from Lemma 1.4. □
Theorem 2.12. If Y is pairwise semi-T2-space and /: (X, x1, x2) ! (Y, r1, r2) is a parwise quasi-irresolute injection, then X is pariwise semi-T2.
Proof. Let x1, x2 2 X and x1—x2. Then there exists V1 2 ij-SO(Y) containing/(xi) and V2 2 ij-SO(Y) containing /(x2) such that i/-scl(V1) n i/-scl(V2) = /, from Lemma 2.11. Since / is pairwise quasi-irresolute, there exists U1 2 ij-SO(X) containing x1 and U2 2 ij-SO(X) containing x2 such that f{Ui) C ij-scl(V1) and f(U2) C ij-scl(V2). Since / is injection, then U1 n U2 — /. Thus X is pairwise semi-T2. □
Theorem 2.13. / a /unction /: (X, t1, t2) ! (Y, r1, r2) is pair-wise quasi-irreso/ute and j-pre-semi-c/osed, then /or every i/-h-sg-c/osed set F o/ Y,/-1(F) is j-h-sg-c/osed set o/ X.
Proof. Suppose that F is an i/-h-sg-closed set of Y. Assume /-1(F) c U where U 2 i/-SO(X). Since / is j-pre-semi-closed and by lemma 1.6, there is an /-semi-open set V such that F c V and /-1(V) c U. Since F is j-h-sg-closed set and F c V, then i/-sclh(F) c V. Hence /-1(y'-sclh(F)) C/-1(V) C U. Since / is pairwise quasi-irresolute and by Theorem 2.5, i/-sclh(/ 1(F)) c U and hence/-1 (F) is j-h-sg-cloed set in X. □
Theorem 2.14. / a space X is pairwise semi-T1=2 and /: (X, t1, t2)! (Y, r1, r2) is sur/ective, i/-qмasi-irreso/мte and j-pre-semi-c/osed, then Y is pairwise semi T1=2.
Proof. Assume that A is an i/-h-sg-closed subset of Y. Then by Theorem 2.13, we have /_1(A) is an i/'-h-sg-closed subset of X. By Theorem 2.12, f 1(A) is i/-semi-closed and hence, A is /-semi-closed. It follows that Y is pairwise semi T1=2. □
Definition 2.15. Let X and Y be bitopological spaces. A subset S of X x Y is called /-strongly semi h-closed if for each (x, y) 2 (X x Y)\S, there exist U 2 ji-SO(X, x) and V 2 i/-SO(Y,y) such that [i/-scl(U) xji-scl(V) n S = /.
Definition 2.16. A function /: X! Y is said to be an i/'-strongly semi h-closed graph if its graph G(/ is i/'-strongly semi h-closed in Xx Ywhere G(/) = {(x,/(x)) : x 2 Xg.
Theorem 2.17. I/ a /unction /: (X, t1, t2) ! (Y, r1, r2) is a i/'-qмasi-irreso/мte and Y is pairwise semi T2, then G(/) is i/-strong/y semi h-c/osed.
Proof. Let (x,y) R G(/), then we have y—/(x). Since Y is pari-wise semi T2, by Lemma 2.11, there exist W 2 i/-SO(Y,/(x)) and V 2 i/-SO(Y,y) such that i/-scl(W) n /-scl(V) = /. Since / is /-quasi-irresolute, by Theorem 2.4, there exists U 2 ji-SO(X,x) such that /(/-scl(U)) c y'-scl(W). Therefore, we have /(/-scl(U)) n ji-scl(V) = /. This shows that G/) is i/-strongly semi h-closed. □
Theorem 2.18. I/ a /unction /: X ! Y has an i/-strong/y semi h-c/osed graph and g : X ! Y is an i/'-qмasi-irreso/мte /unction, then the set A = {(x1,x2)2 X x X:/(x1)) = g(x2)g is an i/-strong/y semi h-c/osed in X X X.
Proof. Let (x1,x2) R (Xx X)\A. Then we have/(x1)—g(x2) and hence (x1,g(x2)) 2 (X x Y)\G(/). Since G(/) is i/'-strongly semi h-closed, there exists U 2 i/-SO(x1) and W 2 i/-SO(g(x2)) such that /(/-scl(U)) nji-scl(W) = /. Since g is i/'-quasi-irresolute, then implies that there exists V 2 i/-SO(x2) such that g(i/-scl (V)) c /i-scl(W). Consequently, we obtain /(i/-scl (U)) n g(i/-scl (V)) = / and hence [/-scl(U) xji-scl( V) n A = /. Hence A is i/'-strongly semi h-closed in X x X. □
Corollary 2.19. If /: X ! Y is an i/'-quasi-irresolute function and Y is pairwise semi T2, then the set A = {(x1, x2) 2 Xx X\/(x1) = /(x2)g is i/'-strongly semi h-closed in Xx X.
Proof. This is an immediate consequence of Theorem 2.17 and 2.18. □
Definition 2.20. A space X is to be i/-semi-connected [10] if X cannot be expressed as the union of two disjoint non-empty subsets U and V such that U 2 i/-SO(X) and V 2 /i-SO(X).
Theorem 2.21. I//: X! Y is a P-qмasi-irreso/мte sur/ec^'on and X is P-semi-connected, then Y is P-semi-connected.
Proof. Suppose that Y is not P-semi-connected. Then Y is the union of two non-empty disjoint V1 2 i/-SO(Y) and V2 2 i/-SO(Y) such that V1 n V2 = / and V1 U V2 = Y. Let W1 = i/-scl( V1) and W2 = i/'-scl(V2). Then /-W1 2 i/-SR(Y) and W2 2 i/-SR(Y), by Lemma 1.4, such that W1 n W2 = / and W1 U W2 = Y. Therefore, we have f 1(W1)-/, f 1(W2)-/, r 1(W1)n/-1 (W2) = / and f 1(W1)U/-1(W2) =X. Moreover by Theorem 2.4, f 1(W1)2 i/-SR(X) and f 1(W2)2 i/-SR(X). This shows that X is not P-semi-connected and this is a contradiction. Therefore, Y is P-semi-connected. □
3. Characterization of //-strongly irresolute functions.
Definition 3.1. [6] A function/: X ! Y is said to be i/'-strongly irresolute if for each x 2 X and each V 2 i/-SO(Y/(x)), there exists U 2 i/-SO(X, x) such that/(i/-scl(U)) V. If/is 12-strongly irresolute and 21-strongly irresolute, then / is called pairwise strongly irresolute.
Theorem 3.2. The /o//owing statements are equiva/ent /or a /unction /: X ! Y.
(i) / is i/-strongly irresolute.
(ii) i/-sclh(/_1(b)) c f ~1(i/'-scl(B)) for every subset B of Y.
(iii) /(i/-sclh(A)) c i/-scl(/(A)) for every subset A of X.
(iv) /^(F) is i/'-semi-h-closed in X for every F 2 i/-SC(Y).
(v)/_1(V) is i/-semi h-open in X, for every V 2 i/-SO(Y).
Proof. (i) ) (ii): Let B c Y and x R/_1(i/'-scl(B)). Then /(x) R i/-scl(B) and there exists V 2 i/-SO(Y/(x)) such that V n B = /. By (i), there exists U 2 i/-SO(X,x) such that /(/-scl(U)) c V. Therefore, we have i/-scl(U) n/_1(B) =/ and hence x R i/-sclh(/^1(B)). □
(ii)) (iii): For any subset A of X, by (ii) we have i/-sclh(A) c ij - sclh(/ 1 (/(A))) c/-1(i/'-scl(f(A))) and hence f(i/-sclh(A)) c i/-scl(/(A)).
(iii)) (iv): For any F 2 i/-SC(Y), by (iii) we have /(i/-sclh(/-1(i))) C i/-scl(F) = F and hence i/-sclh (/-1(F)) c/-1(F). This shows that f 1(F) is i/-semi h-closed.
(iv) )(v): For any V 2 i/-SO(Y), Y\V 2 i/-SC(Y) and A/- 1(v) = /-1(Y \ V) is i/-semi h-closed. Therefore, /-1(V) is i/-semi h-open in X.
(v) ) (i): Let x 2 X and V2 i/-SO(Y/(x)). Then by (v), /-1(V) is i/-semi h-open in X. There exists U 2 i/-SO(X) such that x U Ci/-scl(U) c/-1(V). Therefore, we have /(i/-scl (U)) c V.
Theorem 3.3. I/ /: (X, x1, x2) —
solute, then/ is ij-h-sg-continuous.
(Y, a1, a2) is ij-strongly irre-
Proof. Let V be ji-semi-closed set of Y. Since / is ji-strongly irresolute, then by Theorem 3.2, /-1(V) is i/-semi h-closed. By lemma 1.9(i), f-1(V) is i/-h-sg-closed. Thus / is ij-h-sg-continuous. □
The converse of above theorem need not be true the following example show that.
Example 3.4. Let X — {a, b, c, d}, Y — {x, y, z}, x1 — {/, {c}, {b, c}, X}, X2 — {/, {c, d}, X}, r1 — {/, {z}, y} and r2 — {/, {y, z}, Y}. Define a function /: (X, x1, x2)— (Y, r1, r2) by setting /(a) — /(b) — /(d) — x and /(c) — z. Then / is 12-h-sg-continuous, since A — {a, b, d} — /-1({x}) is 12-h-sg-closed. But A is not 21-semi h-closed. Hence / is not 21-slrongly irresolute, since {x} is 21-semi-closed in Y.
Theorem 3.5. Let /: (X, x1, x2) — (Y, r1, r2) and g : (Y, r1, r2) — (Z, v1, v2) are two /unctions, then:
(i) If / is i/-strongly irresolute and g is i/-irresolute, then gof : X — Z is i/-strongly irresolute.
(ii) If / is i/-quasi-irresolute and g is i/-strongly irresolute, then g o/ is i/-strongly irresolute.
(i) Let V2 i/-SO(Z). Then g-1(V) 2 ij-SO(Y), since g is ij-irresolute. By Theorem 3.2, / -1(g-1(V)) = (gof)-1 (V) is i/-semi h-open in X. Thus gof is i/-strongly irresolute.
(ii) This follows immediately from Theorem 2.5 and 3.2. □
Theorem 3.6. An ij-irresolute /unction /: X — Y is ij-strongly irresolute i/ and only i/ X is ij-semi-regular.
Proof. Let/: X — X be the identity function. Then/is i/-irres-olute and i/-strongly irresolute by hypothesis. For any x 2 X and any F 2 i/-SC(X) not containing x, /(x) — x 2 X \ F 2 ij-SO(X) and there exists U 2 ij-SO(X) such that f(ij-scl(U)) c X\F. Therefore, we obtain x 2 U 2 i/-SO(X), F C X\ i/-scl(U) 2 ji-SO(X) and U n (X\ij-scl(U)) = /. This obvious that X is i/-semi-regular. □
Conversely, suppose that /: X — Y is i/-irresolute and X is i/-semi-regular. For any x 2 X and any V 2 ij-SO(f(x)), /-1(V) 2 ij-SO(X) and there exists U 2 ij-SO(X) such that x 2 U 2 ji-scl(U) c/-1(V), by Lemma 1.7. Therefore, we have /(i/-scl(U) c V. This shows that / is i/-strongly irresolute.
obtain /(i/-scl(U)) c V. Next, let x 2 X and U 2 i/-SO(x). Since g(x) 2 U x X 2 i/-SO(X x Y), there exists U0 2 ij-SO(X) such that g(ij-scl(U0) C U x Y. Therefore, we obtain x 2 U0 C ij-scl(U0) C U and hence X is i/-semi-regular. □
Remark 3.8. The converse to Theorem 3.7, is not true because in Example 2.10, / is 12-strongly irresolute and X is 12-semi-regular but g is not 12-strongly irresolute.
Theorem 3.9. If /: X — Y is a P-strongly irresolute injection and Y is P-semi T0, then X is P-semi T2.
Proof. Let x and y be any pair of distinct points of X. Since/is injective it follows that/(x)—/(y). Since Y is P-semi T0, there exists V 2 ij-SO(f(x)) not containing /(y) or W 2 ji-SO(f(y)) not containing /(x). If it holds that f(y) R V 2 ij-SO(f(x)) and since / is P-strongly irresolute then there exists U 2 ij-SO(X) such that /(/i-scl(U)) c V. Therefore, we obtain /(y) R /(/i-scl(U)) and hence y 2 X \ ji-scl(U) 2 ji-SO(X). If the other case holds, then we obtain the similar result. Therefore, X is P-semi T2. □
4. //-Semi-compact and ij-s-closed spaces.
Definition 4.1. Let A be a subset of a space X, then:
(i) A subset A is said to be i/-semi-compact relative to X (resp. i/-s-closed relative to X [7]) if for every cover {Va : a 2 V} of A by i/-semi-open sets of X, there exists a finite subset V0 of V such A C[{Va : a 2V0} (resp. A C U{ij-scl( Va) : a 2 V0}.
(ii) A space X is said to be i/-semi-compact [5] (resp. ij-s-closed [7]) if X is i/-semi-compact relative to X (resp. ij-s-closed relative to X).
(iii) A subset A is called i/-semi-compact if the subspace A is i/-semi-compact.
Theorem 4.2. Let /: X — Ybean ij-strongly irresolute function. If A is ij-s-closed relative to X, then /(A) is ij-semi-compact.
Proof. Let A be i/-s-closed relative to Xand {Va : a 2 V} any cover of /(A) by i/-semi-open sets of Y. For each x 2 A, there exists a(x) 2 $ such that/(x) 2 Va(x). Since/ is i/-strongly irresolute, there exists Ux 2ij-SO(X) such that/(ij-scl(Ux)) C Va(x). The family {Ux : x 2 A} form an i/-semi-open cover of A and there exists a finite number of points x1, x2,..., xn in A such that A C U{{/-sc1(Ux, ) : i — 1,2,..., n}. Therefore, we obtain /(A) C U{ Va(xi) : i — 1,2,..., n}. Thus /(A) is i/-semi-compact relative to Y. □
Theorem 3.7. Let /: X — Y be a /unction and g : X — X x Y the graph function of /. If g is ij-strongly irresolute, then / is ij-strongly irresolute and X is ij-semi-regular.
Proof. First, we show that/is i/-strongly irresolute. Let x 2 X and V 2 i/-SO(f(x)). Then Xx Vis an i/-semi-open set of Xx Y containing g(x). Since g is i/-strongly irresolute, there exists U 2 ij-SO(X) such that g(i/-scl(U)) c X x V. Therefore, we
Corollary 4.3. If X is i/-s-closed and /: X — Y is an i/-quasi-irresoulte (resp. i/-strongly irresolute) surjection, then Y is i/-s-closed (resp. i/-semi-compact).
Proof. The second case follows from Theorem 4.2. We shall shows the first. Let {Va : a 2 V}be an i/-semi-open cover of Y. By Lemma 1.4, the family {ij-scl(Va) : a 2 V} is a cover of Y by i/-semi-regular sets of Y. It follows from Theorem 2.4,
that the family {/ 1 (/ — sc/( Va)) : a 2 V} is a cover of X by /-semi-regular sets of X. Since X is i/-s-closed, then there exists a finite subset V0 of V such that X = U{/-1 (y'-scl( Va)) : a 2 V0} by Lemma 1.8. Since / is surjective, we have Y = U{i/-scl(Va) : a 2 V0}. This shows that Yis /-s-closed. □
A function /: X ! Y is said to be /-pre-semi-closed [8] if /(F) 2 j-SC(Y) for every F 2 /-SC(X).
Lemma 4.4. A sur/fiction /: X ! Y is j-pre-semi-c/osed i/ and on/y i//or each point y 2 Y and each U 2 i/-SO(X) containing /—1(y), there exists V2 i/-SO(Y) such thatZ-1(V) c U.
Proof. The first side follows from Lemma 1.6. On the other hand, let A be an i/-semi-open set of X. Suppose that y 2 Y\/(A) where X \ A is /-semi-closed set of X. By hypos-tasis, there exists an i/-semi-open set V c Y such that /—1(V)CX \ A. Thus A C /—1(Y \ V), this implies /(A) C Y\ V. Hence y 2 V C Y\/(A) and Y\/(A) is i/-semi-open set of Y. It follows that /(A) is /-semi-closed set in Y and hence / is an i/-pre-semi-closed. □
Theorem 4.5. Let /: X ! Y be an j-pre-semi-c/osed sur/fiction and/l(y) be j-s-c/osed re/ative to Y (resp. /-semi-compact re/-ative to Y) /or each y 2 Y. I/Kis /-semi-compact re/ative to Y, then / 1 (K) is j-s-c/osed re/ative to X (resp. /-semi-compact re/ative to Y).
Proof. Suppose that for each y 2 Y, /— 1(y) is /-s-closed relative to Y and K is i/-semi-compact relative to Y. Let {Ua : a 2 V} be a cover of /- 1(K) by /-semi-open sets of X. For each y 2 K, there exists a finite subset V(y) of V such that /—1(y) c U {i/'-scl (Ua) : a 2 V (y)}. By Lemma 1.4, i/'-scl (Ua) 2 i/-SO(X) for each a 2 V and hence U{i/ — sc/(Ua) : a 2 V (y)} 2 j-SO(X). By Lemma 4.4, there exists Vy 2 i/-SO(y) such that /" 1(Vy) C U{/ — sc/(Ua) : a 2 V (y)}. Since {Vy : y 2 K} is an i/-semi-open cover of K, for a finite number of points y1, y2,..., yn in K, we have K C U{ Vy : i = 1, 2,..., n} and hence /—1 (K) C U^L^^-1 (Vy) C UhUgVwOj-scl (U)). Therefore, /- 1(K) is /-s-closed relative to X. The proof of the case i/-semi-compact relative to Y is similar. □
Corollary 4.6. Let /: X ! Y be an /-pre-semi-closed surjec-tion and /-1(y) be /-s-closed relative to Y (resp. /-semi-com-pact relative to Y) for each y 2 Y. If Y is i/-semi-compact, then X is i/-s-closed (resp. i/-semi-compact).
Proof. This follows immediately from Theorem 4.5. □ 5. Comparisons.
Definition 5.1. A function/: (X, t1, t2) ! (Y, r1, r2) is said to be i/-semi-weakly continuous if for each x 2 X and each 07-open neighborhood V of /(x), there exists U 2 i/-SO(X) such that /(U) c j-cl(V).
Remark 5.2. i/-strongly irresolute implies i/-irresolute and i/irresolute implies i/-quasi-irresolute. However, i/-strongly irresolute and i-continuous are independent of each other as the following two examples show.
Example 5.3. Let X = Y = {a, b, c}, t1 = {/, {a}, {b}, {a, b}, X}, t2 = {/, {a}, {c}, {a, c}, {b, c}, X}, 01 = {/, {a}, {b, c}, Y} and r2 = {/, {b}, {b, c}, Y}. Then the identity function /: (X, t1, t2) ^ (Y, r1, r2) is 12-strongly irresolute but not 1-continuous.
Example 5.4. Let X = Y = {a, b, c}, t1 = {/, {b}, {a, b},{b, c}, X}, t2 = {/, {a}, {a, c}, X}, 01 = {/, {a}, {b, c}, Y} and r2 = {/, {a}, {c}, {a, c}, Y}. Then the function /: (X, t1, t2)! (Y, r1, r2) defined by /(a) = /(b) = b and /(c) = c. It is evident that/is 1-continuous. Since b 2/(U) and 12-SO(a) = {{a, b}, X}, then for each U 2 12-SO(a), we have /(U)a = 21-scl(a) for every U 2 12-SO(a). This shows that/ is not 12-quasi-irresolute.
Theorem 5.5. An i/-irreso/мteness imp/ies both i/-qмasi-irreso/мte and i/-semi-continмoмs.
Proof. Straightforward from the fact that every i-open set is i/semi-open and [[6], Remark 5.1]. □
The converse of Theorem 5.5 is not true as Example 5.4 and the following example show.
Example 5.6. Let X = {a, b, c}, t1 = {/, {a, b}, X}, t2 = {/, {a, c}, X}, r1 = {/, {b, c}, X} and r2 = {/, X}. Then the identity function /: (X, t1, t2) ! (X, r1, r2) is 12-quasi-irreso-lute. However, / is not 12-semi-continuous and hence not 12-irresolute.
Theorem 5.7. An i/-qмаsi-irreso/мte imp/ies i/-semi-weak continwity.
Proof. It follows from definition. □
The converse of the above theorem is not true, since in Example 5.4, / is 12-semi-weakly continuous but not 12-quasi-irresolute.
Remark 5.8. Every i/-semi-continuous function is i/-semi-weakly continuous but the converse is not true, the following example shows that.
Example 5.9. Let X ={a, b}, t1 ={/, {a}, X}, t2 = {/, X}, r1 l {/, {b}, X} and r2 = {/, {a}, {b}, X}. Then the identity function/: (X, t1, t2) ! (X, r1, r2) is 12-semi-weakly continuous but not 12-semi-continuous.
Remark 5.10. Every i/-strongly irresolute function is i/-irreso-lute. The converse need not be true, the following example shows that.
Example 5.11. Let X ={a, b, c}, t1 = {/, {a, b}, X}, s2 = {/, {b}, {a, b}, X}, 01 ={/, {b, c}, X} and r2 = {/, X}. Then the function /: (X, t1, t2)!(X, r1, r2) defined by /(a) = c, /(b) = b and /(c) = a. Then / is 12-irresolute but / is not 12-strongly irresolute, since {b, c} 2 12-SO(X) and 12-SO(X) = {/,{a, b},X} such that 21-scl({a, b}) = X. Thus /(X){b, c} and hence/is not 12-strongly irresolute.
By [[6], Remark 5.1] and for remarks in this section, we obtain the following diagram, where none of the implication is reversible.
ij-strongly irresolute
ij-irresolute
i-continuous -► ij-semi-continuous ij-quasi-irresolute
ij-semi-waekly continuous
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