Internat. J. Math. & Math. Sci.
VOL. 12 NO.
(1989) 9-14
A NOTE ON WEAKLY 0-CONTINUOUS FUNCTIONS
M. Mrevl6
Department of Mathematics
and I.L. Rellly
Faculty of Sciences
University of Belgrade
Department of Mathematics
University of California, Davis
Davis, California 95616
Yugoslavia
U.S.A.
(Received June 8, 1987 and in revised form February 9, 1988)
ABSTRACT.
Recently a new class of functions between topological spaces, called weakly
In this paper we show how an
e-continuous functions, has been introduced and studied.
appropriate change of topology on the domain of a weakly 8-continuous function reduces
This paper examines some of the consequences of
it to a weakly continuous function.
this result.
KEY WORDS AND PHRASES.
Weak continuity, weakly e-continuous, change of topology.
1980 MATHEMATICS SUBJECT CLASSIFICATION.
54C10, 54AI0.
PRELIMINARIES.
X and
Throughout this paper
Y
denote topological spaces on which no separation
axioms are assumed unless explicitly stated.
be regular open (resp. regular closed) if
clS
and
denote the closure of
intS
Henceforth we denote
(resp. S N cl(U) # )
and is denoted by
{resp. O-closed)
if
for each open set
c16(S
cl {S)
A subset is called
{resp.
O-closed.
topology
T6
TO)
S
is said to
[resp. cl(intS)
and the interior of
S,
where
S)
respectively.
A point
e(S)
S
U containing x.
is called the
(resp. cl 8(S))
S
{resp.
x E X
is
Clo{S
The set of all
6-clostae (resp. e-closure) of
A subset
S
is said to be
on
6-cZoeel
S).
f-open (resp. 0-open) if its complement is
The family of all
{resp.
X
of a space
8-cluster) point of S if S N a(U) #
f-cluster (resp. e-cluster) points of
S
int(clS)
S
by the Bourbaki notation
int(clS)
called a f-cluster (resp.
S
A subset
f-closed
f-open (resp. 0-open) subsets of
{X,[)
forms a
X
Since the intersection of two regular open subsets of
the family of all regular open subsets of
(X,T
(X,T)
is regular open,
forms a base for a topology
T s on X,
10
MREVI AND
M.
called the semi-regularizatn of
TO
Ts
Ts
c
T
c
[MRV].
T
and only if
(X,T)
T6
and that
The relationship between
In particular, we note that
(X,T)
T and
is semi-regular if
Ts
A function
X
f
X
x
neighbourhood
Is
It turns out that
for any topological space
is considered in
each point
I.
I.L. REILLY
Y
is called
weakly continuous (resp. O-continuou8) if for
and each open neighbourhood
U of
such that
x
[L]
were considered by Levine
f(U)
c clV
[F]
and Fomin
Y of
f(x)
there is an open
(resp. f(clU) c clV)
These concepts
respectively.
CHANGE OF TOPOLOGY
DEFINITION I.
A function
if for each point
neighbourhood
U
X
x
of
x
Cammaroto and Noiri
(,T)
f
(Y,U)
is called weakly O-continuous
and each open neighbourhood
such that
V of f(x)
there is an open
f((U)) c elY.
[CN] point
out with their Examples 3.2 and 3.3 that the
class of weakly 0-continuous functions lies strictly between the class of 0-continuous
functions and the class of weakly continuous functions.
In a recent paper
[GMRV],
Gauld, Vamanamurthy and the present authors observed that there are many properties
closely related to the notion of continuity which in fact coincide with continuity if
the topology on either the domain or the range or both is changed.
They called such
a property a continuity property of functions.
PROPOSITION i.
Weak 0-continuity is not a continuity property.
We refer to Lemma 2 of [GMRV]
It is shown in the proof of Theorem 2
PROOF.
of
[GMRV]
while
h
that
f
and
are 0-continuous, and hence they are weakly 0-continuous,
g
is not weakly continuous and hence not weakly 0-continuous.
follows from Lemma 2 of
So the result
[GMRV]
We now show that a change of topology is germane to the discussion of weak
0-continuity.
We observe that if the domain of a weakly 8-continuous function is
retopologized appropriately, the function reduces to a weakly continuous function.
Baker
[B2]
has also made this observation.
place of weakly 0-continuous.
continuity property
[GMRV,
He used the term weakly 6-continuous in
The point should be made that weak continuity is not a
Theorem
2].
Recently, Noiri
IN2]
has studied another
class of functions, the weakly a-continuous functions, which also reduce to weakly
continuous functions when the domain is retopologized appropriately
The proof of Theorem 1 follows directly from Definition 1
IN2, Lemma 2.2].
NOTE ON WEAKLY e-CONTINUOUS FUNCTIONS
A function
THEOREM i.
(Y,U)
(X,Ts)
f
if
(X,T)
f
(Y,U)
is weakly @-continuous if and only
is weakly continuous.
[CN]
Now most of the results of Cammaroto and Noiri
can be reinterpreted in the
For example, their Theorem 3.4 is an immediate consequence of
light of Theorem 1
Theorem I,
11
is a restatement of Theorem 1 of Levine
their Theorem 3.5 (f)
[EJ].
while Theorem 3.S(e) follows from Theorem l(e) of Espelie and Joseph
[EJ]
parts of Theorem 1 of
[L],
Other
enable us to provide some further alternative
characterizations of weak @-continuity.
For a function
PROPOSITION 2.
(X,[)
f
Y
the following are equivalent:
is weakly @-continuous.
(a)
f
(b)
For each
(c)
For each open set
(d)
For each regularly closed
Tscl(f-lint (cl@B))) c f-i (cleB)
B c y
V
Tscl(f-l((V))) c f-l(clV)
subset B of y
scl(f-l(intB))
Y,
in
f-I (B)
c
On the other hand, we can go in the opposite direction and use Theorem 3.7 of
[CN]
to obtain the following result for weak continuity.
PROPOSITION 3.
f-l(F)
f-l(V)
(a)
(b)
If
X
f
is weakly continuous then
Y
subset
is closed for each @-closed
V of Y.
is open for each @-open subset
The Example 3.9 of
[CN]
F of Y,
can be modified to show that the converse of
Proposition 3 is false.
We observe that weak continuity is preserved under restriction of domain and under
This observation together with Theorem 1 can be used to obtain the results
products.
[CN, 4].
of
eemma
3 of
For example,
[MRV]
DEFINITION 2.
(ii)
-open
(iii)super open if
DEFINITION 3.
C
of
f: (X,T)
A function
cIraost open if f: (X,Ts)
f:
[MRV]
4 of
{Y,U)
(Y,Us)
f: (X,T)
A subset
S
while
[CN]
is called
is open,
(Y,Us)
(X,Ts)
{Y,U)
[CN],
is open,
of
is open.
is called N-closed if for any open cover
(X,T)
there is a finite subcollection
S
gives Theorem 4.1 of
provides a proof of Theorem 4.2 and 4.3 of
(i)
if
eemma
P
of
C
such that
S c U {s(V)
Applying this change of topology technique to Lemma 2 of Rose
[R] gives
P}.
V
the
following result.
PROPOSITION 4.
(a)
If
g
f
Let
f
X
Y
and
is weakly @-continuous and
is weakly continuous.
Y
g
f
Z.
is an almost open surjection, then
g
MRSEVI6
M.
12
(b)
If
is weakly 0-continuous and
f
g
AND I.L. REILLY
is an d-open surjection, then
f
g
is
weakly 0-continuous.
(c)
If
is weakly continuous and
f
g
weakly 0-continuous.
PROPOSITION 5.
If f
X
C
of
for each N-closed subset
PROOF.
f
(X,Ts)
then
(X,T)
[BI],
Baker
+
C
X, f[C) is d-closed in Y.
(Y,U) is weakly continuous and U s c U,
(X,Ts)
is compact in
(Y,Us)
is closed in
f(C)
[BI],
PROPOSITION 6.
th’en
If
Let
g
X
Y
is
and Corollary 2
Y
is weakly 0-continuous and
f-I (C)
Then the set
X.
is 6-closed in
Y be a Hausdorff space,
{x
X
f
X
is Hausdorff,
be d-continuous and
Y
g(x)}
f(x)
is a d-closed
X.
subset of
PROOF.
We have
(X,T)
f
Theorem 7.8 of
y, T s
(X
(X,Ts)
g
(X,Ts)
f
x
[CN]
(Y,Us)
(Y,Us)
[BI].
f
is weakly continuous.
and thus d-closed in
Us)
is
is a corollary of Theorem 10 of
is weakly 0-continuous" then
(Y,U)
is weakly 0-continuous, so that
is closed in
is continuous,
is Hausdorff, and apply Theorem 6 of Baker
(Y,Us)
Finally we observe that
For, if
(Y,Us)
(X,Ts)
f
weakly continuous and
G(f)
C
If
and hence d-closed in
of Y,
C
Y be weakly 8-continuous.
[NI].
we have
we obtain the following result.
X
f
for each N-closed subset
PROPOSITION 7.
is
so by Corollary 3 to Theorem 2 of
Using a similar argument to that in the proof of Proposition 5,
to Theorem 2 of Baker
g
is Hausdorff, then
Y
is weakly continuous and hence subweakly continuous.
(Y,Us)
N-closed in
Y is weakly 0-continuous and
(X,Ts)
f
Since
is a super open sur.jection, then
f
(X
y,
Hence
T
REFERENCES
[BI]
BAKER, C.W.
....;
[B2]
Properties of subweakly continuous functions,
Math. J.
1984), 39-43.
BAKER, C.W.
Characterizations of some near-continuous functions and near-open
functions, Internat. J. Math.
[CN]
Yokoha_a
CAMMAROTO, F.
3_.8(1986),
and
33-44.
NOIRI, T.
&
Math. Sci.
9_(1986), 715-720.
On weakly 0-continuous functions, Mat. Vesnik
NOTE ON WEAKLY 8-CONTINUOUS FUNCTIONS
ESPELIE, M.S.
and
JOSEPH, J.E.
13
Two weak forms of continuity, Canad. Math.
Bull. 25 (1982), 59-63.
FOMIN, S.
Extensions of topological
GAULD, D.B.,
MREVI,
spaces,
M., REILLY, I.L.
and
Ann. of Math. 44(1943), 471-480.
VAMANAMURTHY, M.K.
Continuity
properties of functions, Colloquia Math. Soc. Janos Bolyai 41(1983), 311-322.
EL]
LEVINE, N.
A decomposition of continuity in topological spaces, Amer. Math.
Monthly 68(1961), 44-46.
[MRV]
MREVI,
M., REILLY, I.L.
and VAMANAMURTHY, M.K.
On semiregularization
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IN1]
NOIRI, T.
Between continuity and weak continuity, Boil.
U.M.I. 9(1974),
647-654.
IN2]
NOIRI, T.
[]
ROSE, D.A.
Math. Sci.
Weakly e-continuous functions, Internat. J. Math.
&
Math. Sci.
Weak continuity and strongly closed sets, Internat. J. Math.
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&