PROPERTIES OF SOME WEAK FORMS OF CONTINUITY

As weak forms of continuity in topological spaces, weak continuity [I], quasi continuity [2], semi continuity [3] and almost continuity in the sense of Husain [4] are well-known. Recently, the following four weak forms of continuity have been introduced: weak quasi continuity [5], faint continuity [6], subweak continuity [7] and almost weak continuity [8]. These four weak forms of continuity are all weaker than weak continuity. In this paper we show that these four forms of continuity are respectively independent and investigate many fundamental properties of these four weak forms of continuity by comparing those of weak continuity, semi continuity and almost continuity.

and is denoted by sInt(S).A subset S is said to be 8-open [6] if for each x e S there exists an open set U such that x e U C CI(U) C S. DEFINITION 2.1.A function f X Y is said to be semi continuous [3] (resp. -i e-continuous [13]) if for every open set V of Y, f (V) is a semi-open set (resp.an e-set) of X.
A function f X Y is said to be quasi continuous at x e X [2] if for each open set V containing f(x) and each open set U containing x, there exists an open set G of X such that # # G U and f(G) V.If f is quasi continuous at every x e X, then it is called quasi continuous.In [9, Theorem i.i], it is shown that a function is semi continuous if and only if it is quasi continuous.
DEFINITION 2.2.A function f X Y is said to be weakly continuous [i] if for each x e X and each open set V containing f(x), there exists an open set U containing x such that f(U) C CI(V).
DEFINITION 2.3.A function f X Y is said to be almost continuous [4] if -i for each x e X and each open set V containing f(x), Cl(f (V)) is a neighborhood of x.
In [13, Theorem 3.2], it is shown that a function is e-continuous if and only if it is almost continuous and semi continuous.In [14] (resp. [i0]), almost continuous functions are called precontinuous (resp.nearly continuous).DEFINITION 2.4.A function f X Y is said to be weakly quasi continuous [5] at x e X if for each open set V containing f(x) and each open set U containing x, there exists an open set G of X such that G C U and f(G) C CI(V).If f is weakly quasi continuous at every x e X, then it is called weakly quasi continuous (briefly w.q.c.).
Both weak continuity and semi continuity imply weak quasi continuity but the converses are not true by Examples 5.2 and 5.10 (below).DEFINITION 2.5.A function f X Y is said to be faintly continuous -i (briefly f.c.) [6] if for every 8-open set V of Y, f (V) is open in X.
It is shown in [6] that every weakly continuous function is faintly continuous PROPERTIES OF SOME WEAK FORMS OF CONTINUITY 99 but not conversely.
DEFINITION 2.6.A function f X Y is said to be subweakly continuous (briefly s.w.c.) [7] if there exists an open basis Z for the topology of Y such -i -i that Cl(f (V)) C f (CI(V)) for each V e Z.
It is shown in [7] that every weakly continuous function is subweakly continuous but not conversely.
A function f X Y is weakly continuous if and only if for every open set V of Y, f (V) C Int(f (CI(V)) continuous if and only if f (V) C Int(Cl(f (V))) for every open set V of Y [7,   Theorem 4].Therefore, almost weak continuity is implied by both weak continuity and almost continuity.
From some remarks and definitions previously stated, we obtain the following diagram.In Section 5, it will be shown that the four weak forms of continuity which are all weaker than weak continuity are respectively independent.w.q.c.
In this section, we obtain some characterizations of a.w.c, functions and some relations between almost weak continuity and almost continuity (or weak continuity).
THEOREM 3.1.For a function f X Y the following are equivalent: (a) f is a.w.c. - Therefore, we obtain x e f-l(v)C Int(Cl(f-l(cl(V)))) and hence CI(f-I(cI(V))) is a neighborhood of x.
-i (c) (a): Let V be any open set of Y and x e f (V).Then f(x) e V and CI(f-I(cI(V))) is a neighborhood of x.Therefore, x e Int(Cl(f-l(cl(V)))) and we T. NOIRI Jankovit [8] remarked that a.w.c, functions into regular spaces are almost continuous.
It will be shown in Example 5.8 (below) that an almost continuous function into a discrete space is not necessarily weakly continuous.Therefore, it is not true in general that if Y is a regular space and f X Y is a.w.c, then f is weakly continuous.
Rose [7] defined a function f X Y to be a/most open if for every open set U of X, f(U) C Int(Cl(f(U))) and showed that a function f , and almost open, then it is almost continuous.
PROOF.Let x e X and V an open set containing f(x).By Theorem ii of [7] we have x e f-l(v is a neighborhood of x and hence f is almost continuous. COROLLARY 3.3 (Rose [7]).It will be shown in Examples 5.2 and 5.8 that semi continuity and almost weak continuity are independent of each other.Therefore, semi continuity does not imply weak continuity.However, we have THEOREM 3.4.If a function f X Y is a.w.c, and semi continuous, then it is weakly continuous.
PROOF.Let V be an open set of Y. Since f is semi continuous, we have f-l(v) g SO(X) and hence CI(f-I(v)) Cl(Int(f-l(v))) [15,Lemma 2].On the other -i -i hand, since f is aoW.C., by Theorem 3.1 we have Cl(Int(f (V))) C f (CI(V)) and hence CI(f-I(v)) f-I(cI(V)).It follows from Theorem 7 of [7] that f is weakly continuous.
In this section, we obtain some characterizations of w.q.c, functions.
THEOREM 4.1.A function f X Y is w.q.c, if and only if for each x g X and each open set V containing f(x), there exists U g SO(X) containing x such that f(U) C CI(V).PROOF.Necessity.Suppose that f is w.q.c.Let x g X and V an open set containing f(x).Let A be the family of all open neighborhoods of x in X.
Then for each N e A there exists an open set G N of X such that # G NC N and C CI(V).Put G {GNI N e A}, then G is open in X and x e CI(G).Let f(G N) U G {x}, then we have x e U e SO(X) and f(U) C CI(V).
Sufficiency.Let x e X, U be an open set containing x and V an open set containing f(x).There exists an A e SO(X) containing x such that f(A) C CI(V).
Put G Int(A/'U).Then, by Lemmas 1 and 4 of [15], G is a nonempty open set of X such that G C U and f(G) C CI(V).This shows that f is w.q.c.THEOREM 4.2.A function f X Y is w.q.c, if and only if for every F e RC(Y) -i f (F) e SO(X).PROOF.Necessity.Suppose that f is w.q.c.Let F e RC(Y).By Theorem 2 of [5], we have f-l(F) f-l(gl(Int(F)))C Cl(Int(f-l(el(Int(F))))) C Cl(Int(f-l(F))). -i Therefore, we obtain f (F) e SO(X).
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY I01 -i -i -i f (CI(V)) e SO(X) and hence f (CI(V)) C Cl(Int(f (CI(V)))).It follows from Theorem 2 of [5] that f is w.q.c.
THEOREM 4.3.For a function f X Y the following are equivalent: (a) f is w.q.c.
-i -i for every open set V of Y.
PROOF.(a) (b): Let B be a subset of Y. Assume that x f-I(cI(B)).
Then f(x) CI(B) and there exists an open set V containing f(x) such that V( B ; hence CI(V)' Int(Cl(B)) .By Theorem 4.1, there exists U e SO(X) containing x such that f(U) C/ CI(V).Therefore, we have U/'f-l(Int(Cl(B))) and hence x sCl(f-l(Int(Cl(B)))). Thus, we obtain For an open set V of Y, CI(V) e RC(Y) and by (c) we have Therefore, we obtain x f-l(v) and hence f-l(v) C slnt(f-l(cl(V))).
(e) (a): Let x e X and V be an open set containing f(x).We have -i -i x f (V) C sInt(f (CI(V))) e SO(X).
-i Put U sInt(f (CI(V))).Then, we obtain x e U e SO(X) and f(U) C CI(V).It follows from Theorem 4.1 that f is w.q.c.

EXAMPLES.
In this section, we shall show that semi continuity, almost continuity and weak continuity are respectively independent.Moreover, it will be shown that each two of quasi weak continuity, faint continuity, almost weak continuity and subweak continuity weakly continuous and Y is regular then f is continuous.Theorem ii of [6] shows that "weakly continuous" in the above result can be replaced by "f.c.".However, we shall observe that "weakly continuous" in the above result can not be replaced by "semi continuous", "almost continuous", "s.w.c.", "w.q.c." or "a.w.c.".
REMARK 5.1.There exists a semi continuous function into a regular space which is neither f.c., s.w.c, nor a.w.c.Therefore, semi continuity implies neither weak continuity nor almost continuity.
REMARK 5.5.There exists a s.w.c, function into a discrete space which is neither w.q.c., f.c.nor a.w.c.Therefore, a s.w.c, function is not necessarily weakly continuous even if the range is a regular space.
EXAMPLE 5.6.Let X be the set of all real numbers, the countable complement topology for X and o the discrete topology for X.Let f (X, T) (X, o) be the identity function.Then f is s.w.c, since the set {{x}l x e X} is an open basis for o and (X, ) is T I.However, f is neither w.q.c., f.c.nor a.w.c.
REMARK 5.7.There exists an almost continuous function into a regular space which is neither w.q.c., f.c.nor s.w.c.Therefore, almost continuity implies neither weak continuity nor semi continuity.
EXAMPLE 5.8.Let X be the real numbers with the indiscrete topology, Y the real numbers with the discrete topology and f X Y the identity function.Then f is almost continuous and hence a.w.c.However, f is neither w.q.c., f.c.nor S.W.C.
REMARK 5.9.There exists a weakly continuous function which is neither semi continuous nor almost continuous.

PROPERTIES OF SEVEN WEAK FORMS OF CONTINUITY.
In this section, we investigate the behavior of seven weak forms of continuity under the operations like compositions, restrictions, graph functions, and generalized products.And also we study if connectedness and hyperconnectedness are preserved under such functions.Many results stated below concerning semi continuity, weak continuity and almost continuity have been already known.Many properties of faint continuity and subweak continuity are also known in [6], [17] and [18].The known results will be denoted only by numbers with the bracket ).In contrast to this, new results will be denoted by THEOREM, LEMMA, EXAMPLE etc. 6.1.COMPOSITIONS.
The following are shown in [3, Example ii] and [18, Example 2].(6.1.1)The composition of two semi continuous (resp.weakly continuous, s.w.c.) functions is not necessarily semi continuous (resp.weakly continuous, s.w.c.).THEOREM 6.1.2.The composition of two almost continuous functions is not necessarily almost continuous.
In the sequel we investigate the behaviour of compositions in case one of two functions is continuous.PROOF.The proof is obvious and is thus omitted.
The next results follow from the facts stated in [18, p. 810 and Lemma i].(6.1.5)If f X Y is weakly continuous (resp.s.w.c., f.c.) and g Y Z is continuous, then go f is weakly continuous (resp.s.w.c., f.c.).THEOREM 6.1.6.If f X Y is w.q.c.(resp.a.w.c.) and g Y Z is continuous, then go f is w.q.c.(resp.a.w.c.).PROOF.First, by using Theorem 4.1 we show that go f is w.q.c.Let x e X -i and W an open set containing g(f(x)).Then g (W) is an open set containing -i f(x) and there exists U e SO(X) containing x such that f(U) C Cl(g (W)).
Since g is continuous, we obtain (g f)(U)C g(Cl(g-l(w)))C CI(W).Next, we show This shows that go f is a.w.c.THEOREM 6.1.7.The composition go f of a continuous function f X Y and a semi continuous function g Y Z is not necessarily w.q.c.
PROOF.Let X Y Z be the set of real numbers.Let be the usual topology, the indiscrete topology and 8 the discrete topology.Let f (X, T) (Y, ) and g (Y, ) (Z, 8) be the identity functions.Then f is continuous and g is almost continuous by Example 5.8.However, g, f is not a.w.c.
The following is shown in Lemma i of [18].(6.1.9)If f X Y is continuous and g Y Z is weakly continuous, then g f is weakly continuous.THEOREM 6.1.10.If f X Y is continuous and g Y Z is s.w.c.(resp. f.c.), then g f X Z is s.w.c.(resp.f.c.).
PROOF.Suppose that f is continuous and g is s.w.c.There exists an open -I -i basis E of Z such that Cl(g (W)) g (CI(W)) for every W E. Since f is continuous, we have Cl((go f)-l(w)) C f-l(cl(g-l(w))) C (g= f)-I(cI(W)).Therefore, go f is s.w.c.Suppose that f is continuous and g is f.c.For every 8-open is open in Y and hence (g f) (W) is open in X. Hence g, f is f.c.
6.2 RESTRICTIONS.THEOREM 6.2.1.The restriction of a semi continuous function to a regular closed subset is not necessarily w.q.c, and hence it need not be semi continuous.
PROOF.In Example 5.2, f (X, T) (X, ) is semi continuous and A {a, c} e 104 T. NOIRI RC(X, ).The restriction flA A (X, o) is not w.q.c, and hence it is not semi continuous.
The following is shown in Example 3 of [19].
(6.2.2) The restriction of an almost continuous function to any subset is not necessarily almost continuous.
THEOREM 6.2.3.If f X Y is weakly continuous and A is a subset of X, then the restriction flA A Y is weakly continuous.
PROOF.Let V be an open set of Y. Since f is weakly continuous, by Theorem -l -1 4 of [20] we have Cl(f (V))Ci f (CI(V)).Therefore, we obtain CIA((flA where CIA(B) denotes the closure of B in the subspace A. It follows from [7,  Theorem 7] that flA is weakly continuous.
THEOREM 6.2.5.The restriction of an a.w.c, function to a subset is not necessarily a.w.c.
PROOF.In Example 3 of [19], f R R is almost continuous and hence a.w.c.
However, the restriction flM M R is not a.w.c, at x 0.
In The following are immediate consequences of Theorem 6.2.3 and (6.2.4).(6.2.7)The restriction of a weakly continuous (resp.s.w.c., f.c.) function to an open set is weakly continuous (resp.s.w.c., f.C.)o THEOREM 6.2.8.If f X Y is w.q.c, and A is open in X, then the restriction flA A Y is w.q.c.
PROOF.Let x e A and V be an open set of Y containing f(x).Since f is w.q.c., by Theorem 4.1 there exists U e SO(X) containing x such that f(U) C CI(V).
Since A is open in X, by Lemma i of [15] x e A('U e SO(A) and (flA)(A(%U) CI(V).It follows from Theorem 4.1 that flA is w.q.c.THEOREM 6.2.9.If f X Y is a.w.c, and A is open in X, then the restriction flA A Y is a.w.c.

GRAPH FUNCTIONS.
Let f X Y be a function.A function g X X Y, defined by g(x) (x, f(x)) for every x e X, is called the graph function of f.The following are shown in [21, Theorem 2], [22, Theorem 2] and [20, Theorem i].(6.3.1)The graph function g of a function f is semi continuous (resp.
almost continuous, weakly continuous) if and only if f is semi continuous (resp. almost continuous, weakly continuous).

PROPERTIES OF SOME WEAK FORMS OF CONTINUITY 105
The following is shown in Theorem 7 of [17].(6.3.2) If a function is s.w.c., then the graph function is s.w.c.
The following is shown in Theorem 13 of [6].
(6.3.3)A function is f.c.if the graph function is f.c.THEOREM 6.3.4.The graph function g X X Y is w.q.c, if and only if f X Y is w.q.c.
PROOF.Necessity.Suppose that g is w.q.c.Let x e X and V an open set containing f(x).Then X V is an open set containing g(x) and by Theorem 4.1 there exists U e SO(X) containing x such that g(U) C CI(X V).Therefore, we obtain f(U) C CI(V) and hence f is w.q.c, by Theorem 4.1.
Sufficiency.Suppose that f is w.q.c.Let x e X and W be an open set containing g(x).There exist open sets UIC X and V C Y such that g(x) (x, f(x)) e U I V W. Since f is w.q.c., by Theorem 4.1 there exists U 2 e SO(X) containing x such that f(U 2) C CI(V).Put U U I(' U 2, then x e U e SO(X) [15,   Lemma i] and g(U) C CI(W).It follows from Theorem 4.1 that g is w.q.c.THEOREM 6.3.5.-i PROOF.Necessity.Suppose that g is a.w.c.In general, we have g (X B) -i f (B) for every subset B of Y. Let V be an open set of Y.By Theorem 3.1, Sufficiency.Suppose that f is a.w.c.Let x e X and W be an open set of X Y containing g(x).There exists a basic open set U V such that g(x) e -i U V C W. Since f is a.w.c., by Theorem 3.1 Cl(f (CI(V))) is a neighborhood of x and U(CI(f-I(cI(V))) C CI(U/' f-I(cI(V))).On the other hand, we have -i -i -i U ( f-I(cI(V)) g (U CI(V)) C g (CI(W)).Therefore, Cl(g (CI(W))) is a neighborhood of x and hence g is a.w.c, by Theorem 3.1.
Let {XI e V} and {YI e V} be any two families of topological spaces with the same index set V. The product space of {Xel e e V} (resp.{Y e e V}) is simply denoted by HX (resp.HY ).Let f X Y be a function for each e e V. Let f HX HY be the product function defined as follows: f({x }) {f(xe)} for every {x} e HX=.The natural projection of HXe (resp.HYe) onto X 8 X 8 (resp.q8 HY Ya).almost continuous, weakly continuous) for each e e V.
The following two results are shown in Theorems 3 and 5 of [18].funtion.If ge f X Z is w.q.c., then g is w.q.c.
-i PROOF.Let F e RC(Z).Since go f is w.q.c., (ge f) (F) SO(X) by Theorem 4.2.Since f is an open sontinuous surjection, by Theorem 9 of [3] we obtain -I -i f((go f) (F)) g (F) e SO(Y).It follows from Theorem 4.2 that g is w.q.c.

106
T. NOIRI THEOREM 6.4.5.The function f X HY is w.q.c, if and only if f X Y is w.q.c, for each s e V.
PROOF.Necessity.Suppose that f is w.q.c.Let e V. Since q HYs Y8 is continuous, by Theorem 6.1.6f6 P6 qB f is w.q.c.Moreover, P6 is an open continuous surjection and by Lemma 6.4.4 f8 is w.q.c.

Sufficiency. Let x
{x s} e HX s and W be an open set containing f(x).
There exists a basic open set V s such that f(x) e HVsC W, where for a finite number of V, say, s I, e2' s V is open in Y and otherwise V Y n s s " J e SO(Xs) containing x s such that f (U s) C Since f is w q c there exists U s then x e U e SO(HX) [15, Theorem 2] and Therefore, it follows from Theorem 4.1 that f is w.q.c.LEMMA 6.4.6.PROOF.Let W be an open set of Z. Since g f is a.w.c., we have Since f is an open surjection, we obtain g (W) j Int(Cl(g (CI(W)))).This shows that g is a.w.c.THEOREM 6.4.7.The function f HX HY is a.w.c, if and only if f X Y is a.w.c, for each a e V. PROOF.Necessity.Suppose that f is a.w.c.Let 6 e V. Since f is a.w.c. and q6 Ys Y6 is continuous, by Theorem 6.1.6f8 P8 q8 f is a.w.c, and hence f8 is a.w.c, by Lemma 6.4.6.Sufficiency.Let

S.
Theorem 3.1 CI(f-I(cI(Ve ))) is a neighborhood of x and J n CI(f-I(cI(Va ))) n X C CI(f-I(cI(W))).-I Therefore, Cl(f (CI(W))) is a neighborhood of x and f is a.w.c, by Theorem 3.1.
It is well-known that a function f X HY is continuous if and only if qB f.: X YB is continuous for each B e V. We investigate if weak forms of continuity have this property.
The following are shown in [15, Theorem 6] and [3, Example i0].(6.4.8)If a function f X HY is semi continuous, then q. f X Y is semi continuous for each 8 e V.However, the converse is not true.q6 THEOREM 6.4.9.A function f X IFf is almost continuous if and only if X YB is almost continuous for each B e V.
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY 107 PROOF.Necessity.Since q8 is continuous, this is an immediate consequence of Theorem 6.1.4.

Sufficiency. Let x e X and W an open set containing f(x) in HY.
There is open in Y exists a basic open set V such that f(x) e VC W, where V J for i, 2 n and otherwise Ve Y-I Since qB(f(x)) e V8 and q8o f is almost continuous for each B e V, Cl((q f) (Ve)) is a neighborhood of x for n J for i, 2, n and $ Cl((q.=)-I(v )) is a neighborhood of X.
j=l Assume that z

CI(f-I(Nv)).
There exists an open set U containing z such that U f-I(Hv) .Therefore, U f (qk f)-l(Vk for some k (i k n).This shows that z Cl((q f)_l(Vk ---n] -i ek and hence we obtain z Cl((qe.of) (V.)). -i Consequently, Cl(f (W)) is a neighborhood of x and hence f is almost continuous.
The following three results are shown in Theorems 2, 4 and 6 of [18].(6.4.10)A function f X NY is weakly continuous if and only if qs f X Y8 is weakly continuous for each 8 e V.
(6.4.11)A function f X HYe is s.w.c, if qD f X Y8 is s.w.c, for each 8 e V. (6.4.12)If a function f X HYe is f.c., then q8 each 8 e V. f X Y is f.c. for THEOREM 6.4.13.If a function f X HYa is s.w.c., then q=m f X Y8 is s.w.c, for each 8 e V.
PROOF.Since q8 is continuous, this follows immediately from (6.1.5).THEOREM 6.4.14.If a function f X HY is w.q.c., then q f X Y8 is w.q.c, for each 8 e V.However, the converse is not true in general.
PROOF.Since q8 is continuous, by Theorem 6.1.6q8 f is w.q.c.In Example i0 of [3], f.X-X. is semi continuous for i i, 2. However, a function X2, defined as follows: f(x) (fl(x), f2(x)) for every x e X, is not w.q.c.THEOREM 6.4.15.A function f X Ya is a.w.c, if and only if qB=f X YB is a.w.c, for each V.
PROOF.The necessity follows from Theorem 6.1.6.By using Theorem 3.1, we can prove the sufficiency similarly to the proof of Sufficiency of Theorem 6.4.9.
For a function f X Y, the subset {(x, f(x))l x X} of the product space X y is called the graph of f and is denoted by G(f).It is well known that if f X Y is continuous and Y is Hausdorff then G(f) is closed in X y.We shall investigate the behaviour of G(f) in case the assumption "continuous" on f is replaced by one of seven weak forms of continuity.THEOREM 6.5.1.If f X Y is semi continuous and Y is Hausdorff, then G(f) is semi-closed in X y but it is not necessarily closed.
PROOF.By Theorem 3 of [21], G(f) is semi-closed in X y.In Example 8 of [3], f X X* is semi continuous and X* is Hausdorff.However, G(f) is not closed in X X* because (1/2, O) e Cl(G(f)) -G(f).

108
T. NOIRI COROLLARY 6.5.2.A w.q.c, function into a Hausdorff space need not have a closed graph.
THEOREM 6.5.3.An almost continuous function into a Hausdorff space need not have a closed graph.PROOF.In Example I of [19], f R R is almost continuous and R is Hausdorff.However, G(f) is not closed since (p, -p) e CI(G(f)) G(f) for a positive integer p. COROLLARY 6.5.4.An a.w.c, function into a Hausdorff space need not have a closed graph.
The following is shown in [23, Theorem i0].(6.5.5)If f X Y is weakly continuous and Y is Hausdorff, then G(f) is closed.
The above result was improved by Baker [17] as follows: (6.5.6)If f X Y is s.w.c, and Y is Hausdorff, then G(f) is closed.
In this section we investigate if connected spaces and hyperconnected spaces are preserved under seven weak forms of continuity.A space X is said to be hyperconnected if every nonempty open set of X is dense in X.The following are shown in Example 2.4 and Remark 3.2 of [24] and [22, Example 3].
(6.6.1)Neither semi continuous surjections nor almost continuous surjections preserve connected spaces in general.
The following is shown in [20, Theorem 3].
PROOF.Let X be real numbers with the finite complement topology, Y real numbers with the discrete topology and f X Y the identity function.Then f is a s.w.c, surjection and X is connected.However, Y is not connected.
PROOF.This is an immediate consequence of (6.6.1).
PROOF.In Example 5.8, f X Y is an almost continuous surjection and X is hyperconnected.However, Y is not hyperconnected.THEOREM 6.6.8.Weakly continuous surjectlons need not preserve hyperconnected spaces.
PROOF.This follows immediately from Theorem 6.6.8.
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY 109 6.7.SURJECTIONS WHICH IMPLY SET-CONNECTED FUNCTIONS.
DEFINITION 6.7.1.Let A and B be subsets of a space X.A space X is said to be connected between A and B if there exists no clopen set F such that A C F and F/'AB .A function f X Y is said to be set-connected [27] provided that f(X) is connected between f(A) and f(B) with respect to the relative topology if X is connected between A and B.
The following lemma is very useful in the sequel.
LEMMA 6.7.2 (Kwak [27]).A surjection f X Y is set-connected if and only -i if f (F) is a clopen set of X for every clopen set F of Y.
PROOF.In Example 5.2, f is a semi continuous surjection but it is not set-connected since f-l({a}) is not closed in (X, T).THEOREM 6.7.4.An almost continuous surjcetion need not be set-connected.
PROOF.In Example 5.8, f is an almost continuous surjection but it is not set-connected.
PROOF.This is an immediate consequence of Theorems 6.7.3 and 6.7.4.
PROOF.In Example 5.6, f (X, T) (X, ) is a s.w.c, surjection but it is not set-connected since f-l({x}) is not open in (X, Y) for a clopen set {x} of (x, o).
In this section we sum up several questions concerning subweak continuity and faint continuity.
QUESTION i. Are the following statements for s.w.c, functions true i) A function is s.w.c, if the graph function is s.w.c. 2 QUESTION 2. Are the following statements for f.c.functions true i) The composition of f.c.functions is f.c.

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: the sequel we investigate the case of restrictions to open sets.The following are shown in [15, Theorem 3] and [19, Theorem 4].(6.2.6)The restriction of a semi continuous (resp.almost continuous) function to an open set is semi continuous (resp.almost continuous).
(6.4.2) If f X Y is s.w.c, for each e e V, then f HX HY is s.w.c.(6.4.3)If f HX HY is f.c., then f X Y is f.c. for each e V. LEMMA 6.4.4.Let f X Y be an open continuous surjection and g Y Z a

First
Round of Reviews March 1, 2009 Every weakly continuous and almost open function is almost continuous.
An a.w.c, and almost open function is not necessarily weakly continuous since the function in Example 5.8 (below) is almost continuous and almost open but not weakly continuous.