413
Internat. J. Math. & Math. Sci.
VOL. 12 NO. 2 (1989) 413-415
A NOTE ON WEAKLY QUASI CONTINUOUS FUNCTIONS
TAKASHI NOIRI
Department of Mathematics
Yatsushiro College of Technology
866 Japan
Yatsushiro, Kumamoto,
(Received March i0, 1987 and in revised form September 15,1987)
-1
Y.
f
X
F
of
is weakly quasi
V
f (CI(V))C Cl(Int(f (CI(V)))) for every open set
By utilizing this result, the present author [2] showed that a function
continuous if and only if
of
Y
X
f
-1
In [i], it was shown that a function
ABSTRACT.
is weakly quasi continuous if and only if for every regular closed set
-i
Y, f (F) is semi-open in X. In this note, the author shows that these
Y
results are false and corrects the proofs of Theorem 6.1.7 and Lemma 6.4.4 of [2].
KEY WORDS AND PHRASES.
weakly quasi continuous.
1980 AMS SUBJECT CLASSIFICATION CODE.
54CI0.
The purpose of this note is to point out that Theorem 2 of [i] and Theorem 4.2
of [2] are false, and to correct the proofs of Theorem 6.1.7 and Lemma 6.4.4 of [2].
Let
S
be a subset of a topological space.
Int(S), respectively.
denoted by
CI(S)
[3] (resp.
regular-closed) if
f
Y
X
A function
of
Y (resp.
Y
X
f
[3] (resp.
(resp. f(V))
is said to be semi-open
Cl(Int(S))).
S
A function
[4]) if for every
semi-open
is semi-open in
X (resp.
Y).
[5] if for each x e X and
-I
f(x), Cl(f (V)) is a neighborhood of x.
f
Y is said to be weakly quasi continuous [i]
X
is said to be almost-continuous
each open neighborhood
DEFINITION.
f-l(v)
X),
S
A subset
S C Cl(Int(S)) (resp.
is said to be semi-continuous
V
open set
and
The closure and the interior of S are
V
of
A function
x e X, each open set
U
containing
f(x), there exists an open set
G
of
(briefly w.q.c.) if for each
X
x
and each open
such that
# G CU
set
V
and
f(G) C CI(V).
Popa and Stan [i] obtained the following characterization of w.q.c, functions.
Y is w.q.c, if and only if
X
THEOREM A ([i, Theorem 2]). A function f
containing
f-I(cI(V))C Cl(Int(f-l(cl(V))))
V
for every open set
of
Y.
In [2], among others, the author established the following three statements.
Y is w.q.c, if and only if
X
THEOREM B ([2, Theorem 4.2]). A function f
for every regular closed set
F
of
THEOREM C ([2, Theorem 6.1.7]).
function
w.q.c.
f
X
Y
-i
Y, f (F)
is semi-open in
X
go f
The composition
and a semi-continuous function
g
Y
Z
X.
Z of a continuous
is not necessarily
414
T. NOIRI
LEMMA D ([2, Lemma 6.4.4]).
and
Y
g
Z
If
a function.
Let
Y
X
f
Z
X
go f
be an open continuous surjection
is w.q.c., then
B was utilized in the proofs of Theorem C and Lemma D.
is w.q.c.
g
The author utilized Theorem A in order to prove Theorem B.
Moreover, Theorem
it follows from
However,
Example 2 (below) that the necessity of Theorem A is false and hence so is Theorem
B.
Thus, it is necessary to revise the proofs of Theorem C and Lemma D.
For this
purpose, we have the following modification of Theorem A.
V
of
x e f
U
set
U C
-1
(V).
X
of
-i
For each open set
V
# U
such that
(CI(V))))
G
Cl(Int(f-l(cl(V))))
Y
and
f(U)
CI(V).
# U
x, there exists an open
Therefore, it follows that
Since
X, G
f(x).
containing
G
X
any open set of
CI(V).
f(U)
and
.
Int(f-l(cl(V))),
#
Put
G
x e
containing x
-i
x e f (V) C
By hypothesis, we have
G(Int(f-l(cl(V)))
and hence
U, then we obtain
containing
Y
be any open set of
V
X
be any point of
x
any open set of
Let
is w.q.c.
of
# U C G
-i
-i
f (V) C Cl(Int(f (CI(V)))).
and hence
Let
G
Int(f-l(cl(V))).
U C
and
Sufficiency.
and
is w.q.c, if and only if for every open set
Y
Necessity.
f-I(cI(V))
Cl(Int(f
X
f
-1
(V) C Cl(Int(f (CI(V)))).
Suppose that f
Y, f
PROOF.
and
A function
THEOREM i.
-1
Int(f-l(cl(V)))
f
This shows that
is w.q.c.
The following example =iows that the necessities of Theorems A and B are both
false.
{, X, {a}, {b, c}} and o {, X, {a},
{a, b, c},
EXAMPLE 2. Let X
{b}, {a, b}}. Let f
(X, o) be the identity function. Then f is w.
(X, )
{a}
{a, c} is a regular closed set
q.c. by Theorem i. Let V
o, then CI(V)
{a}. Thus,
c}
and
of (X, o). However,
{a,
Cl(Int(f-l(cl(V))))
f-I(cI(V)) Cl(Int(f-l(cl(V)))).
f-I(cI(V))
f-I(cI(V))
(X, ) and
(I) It follows immediately from Theorem 1 that the sufficiencies of
is not semi-open in
REMARK 3.
Theorems A and B are both true.
(2) In the proof of [2, Theorem 6.1.7], the set
{b, c, d} and Cl(Int((g
(Z, ), (g
g f
by Theorem 1
{b, c, d}
V
f)-I(cI(V))))
f)-l(v)
is clopen in
{b}.
Therefore,
is not w.q.c, and hence it is not semi-continuous.
Next, we give the correct proof of Lemma D in the improved form.
Let
THEOREM 4.
g
Y
Z
Let
W
g f
If
a function.
PROOF.
be a semi-open almost continuous surjection and
Y
X
f
X
be any open set of
-i
-i
Z
Since
is w.q.c.
g
is w.q.c., then
Z.
is w.q.c., by Theorem 1
g f
(W) Cl(Int((g f) (CI(W)))). Since f is almost continuous,
for every subset A of X, f(Cl(Int(A)))Cl(f(Int(A))) [6, Theorem 6]. Moreover,
since f is semi-open, f(Int(A)) .Cl(Int(f(A))) [4, Theorem 9] and hence
(g f)
we have
f(Cl(Int(A)))
Cl(Int(f(A)))
-i
-i
g (W) Cl(Int(g (CI(W)))).
THEOREM 5.
g
Y
Z
subset
(g
B
f)-l(w)
f
Y
X
A
g
Necessity.
-i
of
W
Let
of
X.
Therefore, we obtain
It follows from Theorem 1 that
g
is w.q.c.
A function
be an open continuous surjection.
i .q.c. if and only if the composition
PROOF.
Theorem 1
Let
for every subset
be any open set of
g f
Z.
X
Since
Z
is w.q.c.
g
is w.q.c., by
-i
(W) Cl(Int(g (CI(W)))). Since f is open continuous, for every
-i
-i
Cl(Int(f (B))). Therefore, we obtain
Y f (Cl(Int(B)))
Cl(Int((g
Sufficiency.
f)-I(cI(W))))
and hence by Theorem 1
g f
is w.q.c.
Since an open continuous function is semi-open almost continuous,
this is an immediate consequence of Theorem 4.
NOTE ON WEAKLY QUASI CONTINUOUS
415
REFERENCES
i.
POPA, V. and STAN, C. On a decomposition of quasi-continuity in topological
spaces (Romanian), Stud. Cerc. Mat. 25 (1973), 41-43.
2.
NOIRI, T.
Properties of some weak forms of continuity, Internat. J. Math. Math.
Sci. i0
3.
LEVINE, N.
(1987) (to appear).
Semi-open sets and semi-continuity in topological spaces, Amer.
Math. Monthly
4.
5.
6.
70
(1963), 36-41.
BISWAS, N. On some mappings in topological spaces, Bull. Calcutta Math. Soc.
61 (1969), 127-135.
HUSAIN, T. Almost continuous mappings, Prace Mat. i__0 (1966), 1-7.
ROSE, D. A. Weak continuity and almost continuity, Internat. J. Math. Math.
Sci. 7 (1984), 311-318.