A NOTE ON WEAKLY QUASI CONTINUOUS FUNCTIONS

The notion of weakly quasi continuous functions introduced by Popa and Stan ]. In this paper, the authors obtain the further properties of such functions and introduce weak* quasi continuity which is weaker than semi continuity [2] but independent ofweak quasi continuity.


INTRODUCTION
As weak forms of continuity in topological spaces, semi continuity, weak continuity [3], quasi continuity [4] and almost continuity in the sense of Husain [5] are well known.Neubrunnovi [6] showed that semi continuity is equivalent to quasi continuity.Also, Noiri [7] showed that semi continuity, weak continuity and almost continuity are respectively independent.In 1973, Popa and Stan [1] introduced weak quasi continuity which is implied by both weak a-continuity [8] and semi continuity.It is shown in [7] that weak quasi continuity is equivalent to weak semi continuity due to Arya and Bhamini [9].
Recently, Noiri in [7,8] investigated fundamental properties of weakly quasi continuous functions and compared the interrelation among weak quasi continuity, weak a-continuity, semi continuity and almost continuity.
The purpose of this paper is to obtain some characterizations of weakly quasi continuous functions and investigate the relationships between such functions and some separation axioms.We also introduce weak* quasi continuity which is weaker than semi continuity but independent of weak quasi continuity.2.

PRELIMINARIES
Throughout the present paper, spaces always mean topological spaces and f X --, Y denotes a single valued function of a space X into a space Y.Let X be a space and A a subset of X.We denote the closure of A and the interior of A by CI(A) and Int(A), respectively.A subset A is said to be semiopen [2] (resp.preopen [10], -open [11]) it" ACI(Int(A)) (resp.AInt(Cl()), C Int(Cl(Int()))).We denote the family of semiopen (resp.preopen, -open) sets of X by (resp.PO(X), o(X)).It is shown that (X)= SO(X)PO(X) [12].The complement of a semiopen set is said to be semiclosed The intersection of all semiclosed sets containing A is called the semi-closure [13] of A and is denoted by s-Cl(A).The semi-interior [13] of A, denoted by s-Int(A), is defined by the union of all semiopen sets contained in A. A subset A of X is said to be regular open (resp regular closed) [14] if A Int(Cl(A)) (resp A Cl(Int(A))).A point :r E X is in the 0-closure of A [15], denoted by Cl0(A), ira t CI(U) # 0 for each open set U containing x.A subset A is called 0-closed if Cl0 (A) A DEFINITION A. A function f X Y is said to be (a) semi continuous [2] (briefly, s c iff-l(V) E SO(X) for each open set V of Y; (b) almost continuous [5] if for each :r X and each open set V containing f(x), Cl(f-l(V)) is a neighborhood of z; (c) weakly continuous [3] (resp. 0-continuous [16]) if for each :r X and each open set V containing f(x), there exists an open set U containing x such that f(U)C CI(V) (resp f(CI(U)) C CI(V)); (d) weakly a-continuous [8] (briefly, w.a.c.) if for each z X and each open set V containing f(x), there exists a U a(X) containing x such that f(U) c CI(V). 3.
WEAKLY QUASI CONTINUOUS FUNCTIONS DEFINITION 3.1.A function f X --Y is said to be (a) weakly quasi continuous (briefly, w.q.c.) if for each x E X, each open set G containing x and each open set V containing f(z), there exists an open set U of X such that 0 # U C G and f(u) c o(v); (b) weakly semi-continuous [9] (briefly, w.s.c.) if for each X and each open set V containing f(x), there exists a U SO(X) containing x such that f(U) CI(V).
Noid showed in [7,Theorem 4. that a function f X Y is w.q.c, if and only if for each X and each open set V containing f(x), there exists a U SO(X) containing :r such that f(U) c CI(V).
Hence we know that w.q.c, and w.s.c, are equivalent concepts.
THEOREM 3.2.For a function f X Y, thefollowing are equivalent: (a) f is w.q.c.09 For each regular ctosea set B of Y, f-1 (B) SO(X).THEOREM 3.3.For a function f X ---.Y, thefollowing are equivalent: (a) f is w.q.c.
(c) For each subset A of X, f(s-Cl(A)) C CIo(f(A)).
(d) For each subset A of X, f(Int(Cl(A))) CIo(f(A)).The composition of two wq c functions may fail to be wq c [7] But Noiri showed m [7, Theorem 616] that under certain conditions the composition of two functions is w q c THEOREM 3.5.Let f X Y and 9 Y Z be functions. (a) Iff is w.q.c, and ts O-continuous, then o f is w.q.c. ('b) Iff IS s.c. and is weakly continuous, then o f is w.q.c.
PROOF.(a) Let :r E X and W be an open set of Z containing 9(f(z)) Since 9 is 0-continuous, there exists an open set V of Y containing f(:c) such that 9(CI(V)) c CI(W) Since f is w q c, there exists a U SO(X) containing :r such that f(U) C Cl(V) Hence g(f(U)) C 9(Cl(V)) c CI(W) (b) The proof is easy and hence omitted COROLLARY 3.6 (Noiri [7]) If f X Y ts w.q.c, and g Y Z ts continuous, then o f sw.q.c.LEMblA 3.7 (Noiri and Ahmad [18]) Let A and B be subsets of X.If A PO(X) and B SO(X), then A B SO(X).THEOREM 3.8.If f X Y ts w.q.c, and A E PO(X), then the restnctton flA A Y is w.q.c.PROOF.Let :tEA and V be an open set of Y containing f(z) Since f is wqc, there exists a U SO(X) containing :r such that f(U)c CI(V) Since A PO(X), by Lemma 3 7   :c A U SO(X) and (f[A)(A U) f(A U) c f(U) c CI(V) Hence f[A is w q c COROLLARY 3.9 (Noiri [7]) If f X Y is w.q.c, and A ts open In X, then the restncnon f]A A Y s w.q.c.COROLLARY 3.10 (Arya and Bhamini [9]) If f X Y is w.q.c, and A o(X), then the restrlcnon flA A Y s w.q.c.Sufficient condition for a function to be w q c, when it is given to be so in some subspace, is given in the following THEOREM 3.11.Let f X Y be a funcnon and {A,[i I} be a cover of X such that A SO(X) for each I. Iff[A, A Y is w.q.c, for each I, then f s w.q.c.
PROOF.Let V be a regular closed set of Y.
Then f-l(v) SO(X) because the union of semiopen sets is semiopen [2] Hence, by Theorem 3 2 f iswqc COROLLARY 3.12.Let f X Y be a function and {A[i E I} be a cover of X such that A a(X) foreach I. IfflA A Ytsw.q.c.foreach I, then f tsw.q.c.COROLLARY 3.13.Let f X Y be a function and {A[i I} be a cover of X such that A, is open in X for each I. Iff[A, A Y is w.q.c, for each I, then f is w.q.c.DEFINITION 3.14.Let A be a subset ofX A function f X A is called a w q c retracnon if f is w q c and flA is the identity function on A THEOREM 3.15.Let A be a subset of X and f X A be a w.q.c, retracnon.If X ts T2, then A ts semlclosed in X.
PROOF.Suppose that A is not semiclosed Then there exists a x X such that x s- CI(A) A Since .f is a w q c retraction, f(z) z By the T2 property of X, there exist disjoint open sets U and V such that zU and f(x)V which implies UACI(V)=O.Let WSO(X) containing z. Then U n W SO(X) and hence (U f3W)A # O because z s-CI(A).Let y (U f W) A. Since y A, we have f() U W N A C U and hence f(v) 6 CI(V) This implies that f(W) CI(V) because W. This is contrary to the fact that .f is w q c Hence A is semiclosed in X.
In [8], Noiri showed that if Y is T2, fl X Y is s c., f2 X Y is w a.c and f f2 on a dense subset of X, then .flf2 on X Similarly, we have THEOREM 3.16.Let Y be T and f X Y be almost continuous.If f2 X Y ts w.q.c. and tffl f on a dense subset D of X, then f f on X. PROOF.Similar to the proof of[8, Theorem 4 10] by using Lemma 3.7.THEOREM 3.17.Let Y be Urysohn and fl X Y be w.q.c.If f2 X Y is w.a.c, and if f f2 on a dense subset D of X, then f f2 on X. PROOF.Similar to the proof of [8, Theorem 4.10]. 4.

GRAPHS OF FUNCTIONS
The graph of a function f :X Y, denoted by G(f), is the subset {(x,f(z))lz X} of the product space X Y. Noiri [20] showed that if.f X Y is weakly continuous and Y is T2, then the graph G(f) is closed.Using "w.q.c." and "semiclosed" instead of "weakly continuous" and "closed" respectively, we obtain the following.THEOREM 4.1.If f X -Y is w.q.c, and Y is T2, then for each (z, y) G(f), there exist U SO(X) andopenset V in X such that z U, y V andf(U) Alnt(CI(V)) .
PROOF.Let (z, y) G(f).Then y f(z).Since Y is T2, there exist disjoint open sets V and W such that y V and f(z) W. This implies that Int(Cl(V)) A CI((W) .Since f is w.q.c., there exists U SO(X) containing z such that f(U) c CI(W).Hence f(U) fq Int(Cl(V)) .COROLLARY 4.2.Iff X Y is w.q.c, and Y is T, then the graph G(f is semiclosed PROOF.It follows from Theorem 4.1.THEOREM 4.3.Iff X -Y isaw.q.c, ands is O-closedsubset in X Y, then pl(S fG(f)) is semiclosed in X, where Pl is the projection of X Y onto X.
PROOF.Let x 8-C1(pl (S G(f))), where S is a 0-closed subset of X Y.
Let U and V be any open sets of X and Y containing z and f(x), respectively.Since f is w.q.c., by Theorem 3.2 This implies that (xo, f(zo)) S and f(zo) CI(V).Therefore, (U x CI(V)) AS C CI(U x V) S and consequently, (z, f(z)) CI0(S).Since S is 0-closed, (x, f(x)) S f G(f).Hence z Pl (S N G(f)).This shows that Pl (S G(f)) is semiclosed in X. COROLLARY 4.4.If f X Y has a O-closed graph G(f) and y X -Y is w.q.c., then {z XlY(z) v(z)} is semiclosed.
PROOF.Since {z xIf(z)=v(z)} =pl(G(f)G(g)) and G(f) is a 0-closed subset of X x Y, it follows from Theorem 4.3 that {x X]f(z) g(z)} is semiclosed.COROLLARY 4.5.If f X Y is O-continuous, y X --Y is w.q.c, and Y is Urysohn, then {z Xlf(z) g(z)} is semiclosed PROOF.It follows from Theorem 7 of[21 and Corollary 4.4.DEFINITION 4.6.Let f:X-Y be a function.The graph G(f) is said to be strongly semiclosed if for each (z, y) X Y-G(f), there exist U SO(X) and V SO(Y) such that x U, y V and (U x s-CI(V)) fq G(f) O. LEMMA 4.7.If f X Y has a strongly semtclosed graph G(f) tf and only t[ for each (x,l) E X Y G(f) there exist U E SO(X) and V SO(Y) such that z U, V and f(U) 3 s-El(V) 0 PROOF.It follows from Definition 4 6   THEOREM 4.8.If f X Y IS w.q.c, and Y ts Urysohn, then G(f) ts strongly ,emtclo.wdm XxY.PROOF.Since s-CI(U) C CI(U) for each subset U of X, it follows mmediately from Lemma 47 5.
WEAK* QUASI CONTINUITY DEFINITION 5.1.A function f X Y is weakly* quasi continuous (briefly, w* q c it" for each open set V of Y, f-(Fr(V)) is semiclosed in X, where Fr(V) denotes the frontier of V Every c function is w* q c but the converse is not true as the following Example 5 2 shows Moreover, Example 5 2 and 5 3 show that w q c and w* q c are independent of each other EXAMPLE 5.2.
Let X {a,b,c}, 7-= {qS, X,{a}} and a {cb, X,{a},{bc}} Let f (X, -) (X, a) be the identity function Then f is w* q c However, f is not c and hence not wqc EXAMPLE 5.3.Let X {a,b,c}, 7-= {,X, {a}} and a {q6, X, {b}} Let f (X, 7-) (Xa) be the identity function Then f is w q c but f is not w* q c The w q c functions are not generally c [7] The next two theorems give conditions under which w q c and c functions are equivalent A space X is said to be extremally disconnected if the closure of each open set is open in X THEOREM 5.4.Let f X Y be a function and X be extremally disconnected.7hen f IS &c.If and only if f is w.q.c, and w*.q.c.

PROOF. The necessity is clear
Sufficiency Let :r X and V be any open set containing f(:r) Since f is w q c, there exists a U SO(X) containing :r such that f(U) c CI(V) But since f is w* q c, f-l(Fr(V)) f-I(cI(V)-V) is semiclosed and hence by Proposition of [22] U-f-l(Fr(V)) SO(X) Further f(:c) Fr(V) implies :c f-a(Fr(V)) The proof will be complete if we show that f(z) f(U f-l(Fr(V))) C V Let V U f-l(Fr(V)) Then f(v) El(V) But f-l(Fr(V)) and so f(v) Fr(V) El(V) V which implies that f(u) V In Theorem 5.4, we cannot drop the assumption that X is extremally disconnected as Example 5 5   shows EXAMPLE 5.5.Let X {a,b,c,d}, 7-{4,,X, {b}, {c}, {b,c}, {a,b,c}, {b,c,d}} and a= {qh, X, {a}, {c}, {a,c}, {a,b,c}} Let f: (X, 7-) (X,a) be the identity function Then f is wqc andw*qc but notsc A space X is said to be tim-compact [14] if each point of X has a base of neighborhoods with compact frontiers THEOREM 5.6.If f X Y ts w.q.c, with the closed graph G(f) and Y is rim-compact, then f lss.c.PROOF.Let x c X and V be any open set containing f(x) Since Y is rim-compact, there exists an open set W of Ysuch that f(x) W c V and Fr(W) is compact Bec,ause f is w q c, there exists a U SO(X) containing :r such that F(U) c CI(W) Let V Fr(W) Since f(x) W which is disjoint from Fr(W), (x,V) -G(f) Then since G(f) is closed, there exist open sets U and V such that :r C Uv, V Vv and f(Uv) V v 0 The collection {Vv]v Fr(W)} is an open cover of Fr(W) Since Fr(W) is compact, there exist a finite number of points Vl,V2,...,W in Fr(W) such that Fr(W) C LIVw LetU0=U(iUw) ThenU0 ESO(X) and

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: (b) For each subset B ofY, s-Cl(f-l(Int(Cl(B)))) C f-(Cl(B)).(c) For each regular closedset F ofY, s-Cl(f-l(Int(F))) C f-l(F).(d) For each open set B ofY, s-CI(f-I(B)) C f-I(CI(B)).(e) For each open set B ofY, f-l(B) C s-Int(f-(Cl(B))).