Internat. J. Math. & Math. Sci.
VOL. ii NO. 4 (1988) 713-720
713
CONTINUOUS FUNCTIONS
SUBWEAKLY
DAVID A. ROSE
Department of Mathematics
Francis Marion College
Florence, South Carolina 29501
(Received June 8, 1987 and in revised form August 25, 1987)
ABSTRACT.
In a recent paper by T. Nolrl [I], a function
f:X
e-contlnuous if
the
e- topolooy.
e
X
Y is weakly continuous where
Smilarly, we define subweak
e
f:X
Y is said to be weakly
is the space X endowed with
s-continuity and almost
e-continuity and show that almost e-contlnuity coincides with the a]Dost continuity
of T. Husain [2] and H. Blumberg [3].
This implies a functional trldecomposition of
continuity using almost continuity and subweak
KEYWORDS AND PHRASES.
e- continuity.
Weakly continuous, subweakly contiauous, almost continuous,
semi-continuous, e-continuous
weakly
locally weak* continuous, A-function,
e-contlnuous, subweakly e-contlnuous, w*.c.,
e-functlon
semi-open, almost open, e-open
trl-decomposition of continuity.
1980 AMS SUBJECT CLASSIFICATION CODE.
54(]10.
INTRODUCTION.
In 1961, Norman Levlne showed that a function
Y between arbitrary
f:X
topological spaces X and Y is continuous if and only if it is both weakly continuous
and w*.c. [4], where weak continuity and w*.c. are independent strict generalizations
of continuity.
1940 [5]).
(This weak continuity was called weak
8-contlnulty by S. Fomln in
Levlne’s decomposition of continuity was strengthened in 1978 [6] by
replacing w*.c. with a strictly weaker condition, local weak* continuity.
important generalization of continuity is the almost continuity of T.
Another
Hsaln [2] and
H. Blumberg [3].
For closed graph functions between complete metric spaces almost
continuity implies continuity [7] [8]. Recently, M. Ganster and I.L. Reilly [9] have
found a strict generalization of continuity which for almost continuous functions
between arbitrary topological spaces implies continuity.
In this paper, a
t rldecomposltlon of continuity is found using almost continuity and based on the
improved version of Levlne’s decomposition of continuity.
results of T. Nolri [I] on weakly
Along the way, some recent
e-contlnuous functions are extended, a new
characterization of seml-open functions is found, and some results of A.S. Mashhour
et.al [12] are improved.
D.A. ROSE
714
Z.
PRELIMINARIES.
Unless specified, no structure beyond a topology O(X is assumed for a space X.
Pelative to this topology/, the interior and closure of a subset A
For the boundary of A we write Bd A.
A and C[ A respectively.
X are denoted Int
The collection of
almost open subsets of the space X, denoted AO(X), consists of those subsets A
such that A clnt CI A.
X
The collection of semi-open subsets of X, written SO(X),
CI Int A [I0].
consists of those subsets A satisfying A
A subset A
X is called
a-open subsets of X are precisely the sets of the
The
N where U
O(X) is open and N c X is nowhere dense, i.e. Int Cl N
form U
collection of a-open subsets of X, written sO(X), is a topology for X called the
a-topology for X, and it contains O(X) Ill]. The underlying set for the space X,
Int C1 Int A, and the
a-open if A
=.
A
subset
X
O(X )
aO(X)
X
a
AO(X)
DEFINITION I.
n SO(X)
f
-I
(b)
semi-continuous if
f
(c)
a-continuous if
f: X
Y is
f:X
AO(X) for each
(V)
SO(X) for each
(V)
V
O(Y) [2];
O(Y) [I0];
V
Y is continuous [121 [13].
Clearly, a function f is a-continuous
continuous and semi-continuous [14]
DEFINITION 2.
a-topology for X) will be
It is also useful to note that
[14] till.
A function
almost continuous if
O(X a). For a
sO(X)
so that
the interior and closure (relative to the
a-lnt A and a-Cl A respectively
denoted
(a)
_
s-topology for X, will be written
endued with the
A function
f:X
f-1(V)
c
if and only if it is both almost
[15].
Y is
f-l(c1
Int
V) for each
V e O(Y) [5] [4];
(a)
weakly continuous if
(b)
subweakly continuous if there is an open basis B for O(Y) such that
(c)
(d)
weakly
w*.c. if
(e)
locally weak* continuous if there is an open basis B for O(Y) such that
f
f-l(cl
f-l(v)
CI
-I
f
-I
V e B [16];
V) for each
f:X
a-continuous if
a
Y is weakly continuous [I];
V e O(Y) [4];
(Bd V) is closed for each
(Bd V) is closed for each
V e B [6].
V. Popa [17] showed that every almost continuous and seml-continuous function is
So
weakly continuous.
a-continuity implies weak continuity which implies
individually both subweak continuity and weak
a-contlnuity
None of these
implications individually is reversible and subweak continuity and weak a-contlnulty
are independent [16]
[I].
It is known [4] ([6]) that a function f is continuous if
and only if it is both weakly continuous and w*.c. (locally weak* continuous).
DEFINITION 3.
A function
(a)
subweakly
(b)
almost
(c)
semi-a-continuous if f: X
f:X
a-continuous if
a
(Xa)
X
[11].
f:X
a
Y is subweakly continuous;
Y is almost continuous;
Y is semi-continuous.
O. Njastad [11] showed that
semi-continuity
a
a
Y is
f:X
a-contlnuous if
SO(X a)
SO(X) so that seml-a-continuity is simply
aO(X)
O(X a) so that
It is also true that aO(X a)
Therefore,
f:X
a
Y is
is continuous which occurs if and only if
section we will show that
AO(X a)
a-continuous if and only if
f:X
Y
is
a-continuous.
AO(X) and consequently almost
f:(Xa) a
In the next
a-contlnuity
Y
SUBWEAKLY a -CONTINUOUS FUNCTIONS
We conclude this section with two more definitions.
coincides with almost continuity.
(a)
(b)
(c)
715
DEFINITION 4. A function f:X
Y is
semi-open if f(U) e SO(Y) for each U e O(X) [I0];
almost open if f(U)
a-open
if
e AO(Y) for each U
ya
f:X
O(X);
[12].
is open
Almost openness in the sense of A. Wilansky [18] is equivalent to almost openness
here [16].
A function
DEFINITION 5.
(a)
A-function if
(b)
a-function if
and
a-functlons
Y is an
f:X
-I
(B) e AO(X) whenever
Y is continuous.
f:X
f
B e AO(Y);
In the literature, A-functions have been called pre-irresolute functions [19],
a-lrresolute
were introduced as
functions by S.N. Maheshwari and
S.S. Thakur [23].
ALMOST -CONTINUITY IS ALMOST CONTINUITY.
3.
It is known [16] that functions which are both almost continuous and subweakly
continuous are weakly continuous.
and subweakly
Thus, functions which are both almost
are weakly
-continuous
a-continuous
-continuous
It will be shown that such
functions are actually weakly continuous by proving the title result of this
We begin with a lemma giving a new characterization of semi-open functions.
section.
A function
LEMMA 1.
f
-1
(Int C1 B)
PROOF.
f-l(y_
c Cl
f
-1
Y is semi-open if and only if
f:X
(B) for each subset B
(sufficiency)
For any subset B
Int CI B)
X-
f-l(Int
f-i (Y
B))
c
Thus, f(Int
subset A c
X:
y
f-l(B)
CI Int(Y
Y -f(A), we have Y
f-l(y
f(Int
X- CI
Int CI B
by setting B
Thus, f(Int A)
CI B)
Y.
y,
B))
Int(X
f-l(B))
y.
B) for all B
B
f(A) and A
c
f-1
Int
f-I(y-B).
Now for any
(y
B).
CI Int f(A).
Therefore, if A e O(X), f(A) e SO(X) showing that f is semi-open.
(necessity) For any subset B c y,
(by semi-openness) C1 Int f(Int f-l(y
Therefore,
f-l(B)c
X- C1
X-
f(X
f-l(Int
CI
f-l(B))
f(Int
f-l(y
B))
B))c C1 Int(Y
Y- Int C1 B.
B)
C1 B) and f-l(Int C1 B)
C1 f-l(B)
y.
for every subset B
The next theorem was obtained by D.S. Jankovic [24, Proposition 3.4].
THEOREM I.
If
Y is an almost continuous and semi-open function, then f is
f:X
an A-function.
PROOF.
For B e A0(Y),
B
Int
CI B, so that
f-l(Int
f-l(B) c f-l(Int C1
f-l(B).
B)c (by
almost continuity) Int C1
(by Lemma I) Int Cl
C1 B)
-i
Therefore, f (B) e AO(X) showing that f is an A-function.
COROLLARY I.
PROOF.
If
For any space X,
f:X
X
a
continuous and semi-open.
AO(X a)
AO(X).
is the identity function then f and
f-I
are each both almost
D.A. ROSE
716
As an immediate consequence of Corollary
we have the title result of this
section.
A function
COROLLARY 2.
if and only if it is
Y is almost a-continuous
f:X
almost continuous.
A TRIDECOMPOSITION OF CONTINUITY.
4.
In the literature, a decomposition (or more accurately, a bidecomposition) of
continuity consists of two independent function properties each strictly weaker than
continuity which jointly imply continuity [4]
[20].
Such a decomposition is improved
or strengthened by weakening one or both of the function properties while maintaining
continuity jointly [6] [9].
In this section we find a tridecomposition of continuity
into three independent function properties each strictly weaker than continuity.
By
three independent properties we mean that not one of them is implied jointly by the
Our trldecomposition is based on the improved version of
other two properties.
Levine’s decomposition [6], and the fact that subweak continuity and almost continuity
jointly imply weak continuity [16].
The following theorem is an almost immediate
consequence of this fact.
THEOREM 2.
If
f:X
Y is almost continuous and subweakly
a-continuous
then it
is weakly continuous.
PROOF.
By the remarks above and corollary 2 of theorem I, for a function
Y satisfying the hypotheses of this theorem,
f:X
subweakly continuous.
Thus,
f:X
a
continuous and weakly a-continuous
f:X
a
Y is almost continuous and
Y is weakly continuous.
So,
f:X
Y is almost
and T. Noiri showed [I] that such functions are
weakly continuous.
below shows that subweak
Example
a-continuity
Noiri
a-continuity
is strictly weaker than weak
so that theorem 2 is a strict improvement of the supporting result of
[I, theorem 4.9].
The next theorem gives three properties each strictly weaker than continuity
which jointly are equivalent to continuity.
A discussion of existing examples will
show that these three properties are independent and thus comprise a tridecomposition
of continuity.
THEOREM 3.
A function
continuous, subweakly
PROOF.
f:X
Y is continuous if and only if it is almost
a-continuous
and locally weak* continuous.
The necessity is obvious and the sufficiency follows from theorem 2 and
the improved version of Levine’s decomposition of continuity [6].
To see that none of the weaker than continuity properties of theorem 3 is implied
by the other two jointly, recall first that a function was found [6, Example 3] which
was almost continuous, weakly continuous, and hence also weakly
therefore subweakly
a-continuous,
-continuous
but not locally weak* continuous.
and
Next, the
identity function from a set with at least two elements having the indiscrete topology
onto the same set with the discrete topology
[16, Example I],
is almost continuous
since the domain is indiscrete, and w*.c. and hence also locally weak* continuous
since the range is discrete.
-continuous
But by theorem 3, this function cannot be subweakly
and therefore not weakly -continuous
since it is not continuous.
SUBWEAKLY
= -CONTINUOUS
FUNCTIONS
717
Finally, consider the identity function from a set with non-discrete T topology onto
the same set with the discrete topology [16, Example 4]. Because the domain is
e-contlnuous using the
function is subweakly continuous and hence subweakly
T|, the
Also, this function is w*.c. and thus
basis of singleton subsets of the range.
Again, by theorem 3, the
locally weak* continuous since the range is discrete.
function cannot be almost continuous since it is not continuous.
These examples not
only show the independence of the three weaker than continuity properties of theorem
3, but also of the triplets
{almost continuity, subweak continuity, w*.c.
and
Hence, each of
almost continuity, subweak continuity, local weak* continuity
these triplets is a tridecompositlon of continuity with the latter being strictly
stronger that the preceding one since local weak* continuity is strictly weaker than
w*.c. [6].
The following example shows that subweak
e-contlnulty is strictly weaker
than subweak continuity so that the trldecomposltion of continuity presented in
theorem 3 is strictly the strongest of these three trldecomposltions.
X
{1,2,...}
EXAMPLE I.
Let N
{0,I,2,...}
have the topology O(X)
have the coflnite topology O(N) and let
O(N) U {X}. I.L. Re illy and M.K.
Vamanamurthy have shown [21, Example 2] that X is not
if
{0,I,2
Y
T
X
but
e
T
is
Therefore,
I.
has the discrete topology O(Y), the identity function
Y
f:X
cannot be subweakly continuous since every open basis for O(Y) contains the singleton
subsets of Y and for some
T
However,
for O(Y) since
X
T
is
f-l({n})
n e Y, C1
Y
f:
e
f-l(c1{n})
# {n}
since X is not
is subweakly continuous using the basis of singleton sets
Therefore,
I.
Y
f:X
is subweakly
s-contlnuous.
Further, because Reilly and Vamanamurthy have shown [21, Proposition 1)] that a space
X is discrete if and only if
X
e
X
is discrete,
regular spaces are continuous [4, theorem 2],
5.
is not discrete and
f:X
e
Y is not
Therefore, since Y is regular, and weakly continuous functions into
continuous.
and hence
e
Y is not weakly
f:X
f:X
e+
Y
cannot be weakly continuous,
s-contlnuous.
s-CONTINUITY
PRODUCT THEOREMS FOR SUBWEAK
In this section it is shown that the class of subweakly
s-continuous functions
is closed under arbitrary products.
THEOREM 4.
If the function
subweakly
s-contlnuous
subweakly
s-continuous
PROOF.
If each
theorem 4] that
subweakly
f:
Qa :Ya
where
Y
Qaof:
Ya
Ya
f:X
a
Ya
O of:X
is such that each
is the
a
th
"a
Y
is
a
projection, then f is
is subweakly continuous, then it is known
is subweakly continuous and therefore,
f:X
Y
[22
a
is
s-continuous.
COROLLARY 3.
Y is a subweakly e-continuous function and if
If each f :X
a
a
{f (x)}, then f is subweakly
is the function defined by f(x)
a
a
s-continuous
th
PROOF. If
is the a
projection, then
f"
f:X
Y
Qa:Ya Ya
LEMMA 2.
If
f:X
continuous function then
fa Qa
Y is a continuous function and
gof:X
Z
g:Y
is subweakly continuous.
Z
is a subweakly
D.A. ROSE
718
Let B be an open basis for O(Z) for which g is subweakly continuous
PROOF.
(gof)-l(v)
B, CI
-I -I
f (g (Cl V))
V
each
(gof)
COROLLARY 4.
f: X
a
If
Is an
Y
f:X
a
gof: X
by lemma 2,
Y
l(g-l(v))
CI
f-l(c
g
-I
(V))
f
-1( C1
g
-I
For
(V))
(Cl V).
function, then
a-continuous
PROOF
If
-I
CI f
a
gof:X
a-functlon and
g:Y
Z is subweakly
a-continuous.
is continuous and
g:
ya
Z
Z
is a subweakly
is subweakly continuous, then
Z is subweakly continuous so that
gof:X
Z is subweakly
a-continuous
THEOREM 5.
Hf :HX
a
Xa
a
Y is a subweakly
a
a
a-continuous.
f :X
If each
a
is subweakly
a-continuous
function, then
Y and P :X
X be the
Hf
and let Q :EY
PROOF. Let X
HX and f
a
a
a
a
a
a
a
th
a
projections It was shown by A.S. Mashhour et.al. [12, Theorem 3.2] that
functions which are both
Qa
is an
Is subweakly
a-contlnuous
Thus, f:X
a-continuous.
P of
a
Thus, each
a-continuous and almost open are -functions
a-function being continuous and open.
by corollary 5, each
Y
a
Therefore, if each
faOQa
is subweakly a-continuous
f :X
a
a
Y
a
is subweakly
by theorem 4 since each
f Q
a a"
ACKNOWLEDGEMENT
The author would like to thank the referee for his helpful suggestions.
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