97
Internat. J. Math. & Math. Sci.
(1987)97-111
Vol. I0 No.
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
TAKASHI NOIRI
Department of Mathematics
Yatsushiro College of Technology
Yatsushiro, Kumamoto, 866
Japan
(Received January 28, 1986)
ABSTRACT.
As weak forms of continuity in topological spaces, weak continuity [I],
quasi continuity [2], semi continuity
[4] are well-known.
[3] and almost continuity in the sense of
Husain
Recently, the following four weak forms of continuity have been
introduced: weak quasi continuity [5], faint continuity [6], subweak continuity
and almost weak continuity
than weak continuity.
[8].
[7]
These four weak forms of continuity are all weaker
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.
KEY WORDS AND PHRASES.
weakly continuous, semi continuous, almost continuous, weakly
quasi continuous, faintly continuous, subweakly continuous, almost weakly continuous.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
1.
54CI0.
INTRODUCTION.
The notion of continuity is one of the most important tools in Mathematics and
many different forms of generalizations of continuity have been introduced and
investigated.
Weak continuity [i], quasi continuity [2], semi continuity [3] and
almost continuity in the sense of Husain [4] are well-known.
quasi continuity is equivalent to semi continuity.
It is shown in [9] that
It will be shown that weak
continuity, semi continuity and almost continuity are respectively independent.
In
1973, Popa and Stan [5] introduced weak quasi continuity which is implied by both
weak continuity and quasi continuity.
Recently, faint continuity and subweak
continuity which are both implied by weak continuity have been introduced by Long and
Herrington
[6] and Rose [7], respectively.
Quite recently, Jankovid [8] introduced
almost weak continuity as a generalization of both weak continuity and almost
continuity.
In [i0], Piotrowski investigated and compared many properties of quasi
continuity, almost continuity and other related weak forms of continuity.
The main purpose of this paper is to show that these four weak forms of continuity
implied by weak continuity are respectively independent and to investigate many
fundamental properties of such weak forms of continuity by comparing with weak
continuity, semi continuity and almost continuity.
In Section 3, we obtain some
characterizations of almost weak continuity and some relations between almost weak
T. NOIRI
98
continuity and weak continuity (or almost continuity).
Section 4 deals with some
In Section 5, it is shown
characterizations of weakly quasi continuous functions.
that weak quasi continuity, faint continuity, subweak continuity and almost weak
In Section 6, we compare many fundamental
continuity are respectively independent.
properties of semi continuity, almost continuity, weak continuity, subweak continuity,
faint continuity, weak quasi continuity and almost weak continuity.
The last section
is devoted to open questions concerning subweak continuity and faint continuity.
2.
PRELIMINARIES.
Throughout this paper spaces always mean topological spaces on which no separation
f
X
into a topological space
Y.
By
axiom is assumed.
S
interior of
[3] (resp.
Cl(Int(S)),
regular closed) sets in
S
X
a space
S
sCI(S).
x e S
there exists an open set
DEFINITION 2.1.
U
A function
e-set) of
S
X
f
The
S
S [12] and is denoted by
is called the semi-interior
8-open [6] if for each
x e U C CI(U) C S.
is said to be semi continuous [3]
-i
Y, f (V) is a semi-open set
Y
V
is
The intersection of all semi
is said to be
such that
[13]) if for every open set
e-continuous
an
A subset
RC(X)).
SO(X) (resp.
is denoted by
is called the semi-closure of
sInt(S).
S
S C Cl(Int(S))
The family of all semi-open (resp.
The union of all semi-open sets contained in
[12] and is denoted by
A subset
e-set [ii]) if
complement of a semi-open set is called semi-closed.
closed sets containing
The closure and the
Int(S), respectively.
and
regular closed, an
X
of a topological space
be a subset of a space.
S C Int(Cl(Int(S)))).
S
f
we denote a function
CI(S)
are denoted by
said to be semi-open
(resp.
Y
Let
of
(resp.
(resp.
X.
A function
f
X
Y
V
containing
open set
G
of
at every
x e X, then it is called quasi continuous.
X
x e X [2] if for each
is said to be quasi continuous at
f(x)
open set
U
and each open set
# # G
such that
U
containing
f(G) V.
and
If
f
x, there exists an
is 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.
for each
containing
x e X
x
for each
V
x e X
is said to be weakly continuous [i] if
Y
X
f
and each open set
such that
DEFINITION 2.3.
of
A function
containing
A function
X
f
V
and each open set
is said to be almost continuous [4] if
-i
containing f(x), Cl(f (V)) is a neighborhood
Y
x.
In [13, Theorem 3.2], it is shown that a function is
functions are called precontinuous (resp.
DEFINITION 2.4.
x e X
A function
if for each open set
x, there exists an open set
G
X
f
V
is weakly quasi continuous at every
[i0]), almost continuous
nearly continuous).
Y
is said to be weakly quasi continuous
containing
such that
X
of
e-continuous if and only if
In [14] (resp.
it is almost continuous and semi continuous.
at
U
f(x), there exists an open set
f(U) C CI(V).
f(x)
and each open set
G C U
and
U
f(G) C CI(V).
If
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
(briefly f.c.) [6] if for every
f
[5]
containing
X
Y
8-open set
is said to be faintly continuous
-i
V of Y, f (V) is open in X.
It is shown in [6] that every weakly continuous function is faintly continuous
f
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
99
but not conversely.
A function
DEFINITION 2.6.
Y
X
f
is said to be subweakly continuous
Z
(briefly s.w.c.) [7] if there exists an open basis
Cl(f
that
-i
-i
(V)) C f (CI(V))
Y
of
for the topology
such
V e Z.
for each
It is shown in [7] that every weakly continuous function is subweakly continuous
but not conversely.
A function
DEFINITION 2.7.
(briefly a.w.c.) [8] if
A function
of
Y, f
f
is said to be a2most weak2y continuous
V
for every open set
A function
(CI(V))) [i, Theorem i].
(V) C Int(f
f
continuous if and only if
Theorem 4].
Y
X
f
-1
(V) C Int(Cl(f (CI(V))))
Y.
of
V
is weakly continuous if and only if for every open set
Y
X
f
-1
-i
(V) C Int(Cl(f
-i
(V)))
X
f
Y
is almost
V
for every open set
Y [7,
of
Therefore, almost weak continuity is implied by both weak continuity and
almost continuity.
From some remarks and definitions previously stated, we obtain the following
In Section 5, it will be shown that the four weak forms of continuity which
diagram.
are all weaker than weak continuity are respectively independent.
DIAGRAM
continuous
s-continuous
a.w.c I--
3.
continuous--semi
weakly
almost continuous
s.w.c.
f.c.
continuous
w.q.c.
ALMOST WEAKLY CONTINUOUS FUNCTIONS.
In this section, we obtain some characterizations of
functions and some
a.w.c,
relations between almost weak continuity and almost continuity (or weak continuity).
For a function
THEOREM 3.1.
(a) f
Y
X
f
the following are equivalent:
is a.w.c.
(b) Cl(Int(f
-I
x e X
(c) For each
a neighborhood of
PROOF.
-i
(V))) C f (CI(V))
V
for every open set
V
and each open set
containing
Y.
of
f(x), Cl(f
-I
(CI(V)))
is
x.
(b): Let
(a)
be an open set of
V
Y.
Then
CI(V)
Y
Y
is open in
and we have
X
f
C Int(Cl(f
Cl(Int(f
Therefore, we obtain
x e X
(c): Let
(b)
Y,
is open in
X-
a neighborhood of
(a): Let
CI(f-I(cI(V)))
f
-i
-i
-i
(Y
CI(V))
Cl(Int(f
CI(V)))))C X
(V)))
V
f
f
-i
-i
(V))).
(CI(V)).
an open set containing
f(x).
Since
Y
CI(V)
we have
-i
Cl(Int(f-l(Y
-i
Int(Cl(V)))C f (Y
(Y
x e
Therefore, we obtain
(c)
(CI(V))
(CI(Y
and
Int(Cl(f-l(cl(V))))
f
obtain
-i
-i
CI(V)))) C f-I(cI(Y
-i
X- f (V).
V)
f-l(v)C Int(Cl(f-l(cl(V))))
and hence
CI(V)))
CI(f-I(cI(V)))
is
x.
V
be any open set of
is a neighborhood of
-i
(V) C Int(Cl(f
(CI(V)))).
x.
Y
and
x e f
Therefore, x e
-i
(V).
Then
f(x) e V
Int(Cl(f-l(cl(V))))
and
and we
T. NOIRI
i00
[8] remarked that
Jankovit
functions into regular spaces are almost
a.w.c,
It will be shown in Example 5.8 (below) that an almost continuous
continuous.
Therefore, it is
function into a discrete space is not necessarily weakly continuous.
Y
not true in general that if
X
f
is a regular space and
Y
f
is a.w.c, then
is weakly continuous.
Rose [7] defined a function
X
f
Y
to be a/most open if for every open set
Y is almost open
X
X, f(U) C Int(Cl(f(U))) and showed that a function f
-i
-I
if and only if f (CI(V))C Cl(f (V)) for every open set V of Y.
U
of
If a function
THEOREM 3.2.
X
f
Y
is a.w.c, and almost open, then it is
almost continuous.
x e X
Let
PROOF.
we have
f-l(v)
x e
C
is a neighborhood of
and
V
f
and hence
x
COROLLARY 3.3 (Rose [7]).
C
By Theorem ii of [7]
f(x).
an open set containing
Int(Cl(f-l(el(V))))
Int(Cl(f-l(v)))-
CI(f-I(v))
Therefore,
is almost continuous.
Every weakly continuous and almost open function is
almost continuous.
An
and almost open function is not necessarily weakly continuous since the
a.w.c,
function in Example 5.8 (below) is almost continuous and almost open but not weakly
It will be shown in Examples 5.2 and 5.8 that semi continuity and almost
continuous.
Therefore, semi continuity does not
weak continuity are independent of each other.
However, we have
imply weak continuity.
THEOREM 3.4.
If a function
is a.w.c, and semi continuous, then it is
Y
X
f
weakly continuous.
Let
PROOF.
f-l(v)
g
SO(X)
f
hand, since
f
Since
f-I(cI(V)).
is semi continuous, we have
Cl(Int(f-l(v)))
CI(f-I(v))
[15, Lemma 2]. On the other
-i
-i
(V))) C f (CI(V)) and
Cl(Int(f
is aoW.C., by Theorem 3.1 we have
CI(f-I(v))
hence
Y.
be an open set of
V
and hence
It follows from Theorem 7 of [7] that
is weakly
f
continuous.
WEAKLY
4.
CONTINUOUS FUNCTIONS.
In this section, we obtain some characterizations of w.q.c, functions.
A function
THEOREM 4.1.
V
and each open set
PROOF.
f(x).
Put
f(GN) C CI(V).
{x},
G
f(x).
G C U
THEOREM 4.2.
f
-i
f
N e A}, then
x e U e SO(X)
x e X, U
Let
g
x
X
V
and
be the family of all open neighborhoods of
{GNI
G
Let
is w.q.c.
G
there exists an open set
There exists an
Int(A/’U).
such that
A
then we have
Sufficiency.
containing
Let
N e A
Then for each
G
Suppose that
Necessity.
containing
Put
x
g
X
such
x
f(U) C CI(V).
that
U
Y is w.q.c, if and only if for each
X
f(x), there exists U g SO(X) containing
f
containing
and
G
X
of
N
containing
Then, by Lemmas 1 and 4 of [15], G
f(G) C CI(V).
A function f
X
and
X.
NC
N
X
and
x e CI(G).
x
and
V
and
Let
f(U) C CI(V).
be an open set containing
A e SO(X)
in
# G
such that
is open in
an open set
x
f
This shows that
x
an open set
such that
is a nonempty
f(A) C CI(V).
open set of X
is w.q.c.
Y
is w.q.c, if and only if for every
f
is w.q.c.
F e RC(Y)
(F) e SO(X).
PROOF.
[5], we have
Suppose that
Necessity.
Sufficiency.
Let
F e RC(Y).
f-l(gl(Int(F)))C Cl(Int(f-l(el(Int(F)))))
f-l(F)
Therefore, we obtain
Let
f
-i
V
C
By Theorem 2 of
Cl(Int(f-l(F))).
(F) e SO(X).
be an open set of
Y.
Since
CI(V) e RC(Y), we have
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
f
-i
(CI(V)) e SO(X)
Theorem 2 of [5] that
It follows from
is w.q.c.
is w.q.c.
-i
(b) sCl(f
(CI(V)) C Cl(Int(f -i (CI(V)))).
For a function
THEOREM 4.3.
(a) f
f
-i
f
and hence
I01
(Int(Cl(B)))) E f
-i
-i
Y
X
f
(CI(B))
the following are equivalent:
for every subset
B
of
Y.
x
f-I(cI(B)).
-i
(Int(F))) C f (F) for every F e RC(Y).
-i
f (CI(V)) for every open set V of Y.
(V))
-i
-i
(e) f (V)
slnt(f (CI(V))) for every open set V of Y.
(c) sCl(f
(d) sCl(f
(a)
PROOF.
f(x)
Then
;
V( B
(b): Let
CI(B)
hence
x
containing
and hence
-i
x
B
CI(V)’ Int(Cl(B))
f(U) C/ CI(V).
such that
sCl(f-l(Int(Cl(B)))).
sCl(f
C
(c)
f
-i
(d)
(e): Let
CI(V)
Y
-i
V
(e)
(CI(B)).
sCl(f-l(Int(Cl(Int(F)))))
(Cl(Int(F)))
f
-i
(F).
V of Y, CI(V) e RC(Y) and by (c) we have
(V)) C sCl(f -i (Int(Cl(V)))) C f -i (CI(V)).
Y
be an open set of
-i
and
(a): Let
sInt(f
-i
x
sInt(f-l(cl(V))).
Then
-i
(CI(V)))
sCl(f (Y
CI(V))).
Y, by (d) we have
-i
sCl(f (Y
CI(V)))
CI(V)))C f (CI(Y
-i
-i
f (Y- Int(Cl(V)))C X- f (V).
f-l(v)
x
x e X
f
x
(CI(V))).
f-l(v) C slnt(f-l(cl(V))).
and hence
f(x).
V
be an open set containing
(V) C sInt(f -i (CI(V))) e SO(X).
and
-i
x e U e SO(X)
Then, we obtain
follows from Theorem 4.1 that
5.
-i
is open in
-i
Therefore, we obtain
U
U e SO(X)
U/’f-l(Int(Cl(B)))
Therefore, we have
x e X- sInt(f
Put
such that
(d): For an open set
sCl(f
Since
f(x)
containing
Thus, we obtain
-i
sCl(f-l(Int(F)))
Assume that
V
By Theorem 4.1, there exists
(Int(Cl(B)))) C f
F e RC(Y). By (b), we have
(c): Let
(b)
.
Y.
be a subset of
and there exists an open set
f
and
We have
f(U) C CI(V).
It
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
are independent of each other.
weakly continuous and
Y
It is shown in Theorem 2 of [i] that if
is regular then
f
is continuous.
f
X
Y
is
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.
{a, b, c}, T
{, X, {a}, {b}, {a, b}} and o
{, X,
(X, T)
(X, ) be the identity function. Then (X, ) is
Since {b, c} e SO(X, ), f is semi continuous and hence w.q.c.
EXAMPLE 5.2. Let
{a}, {b, c}}. Let f
a
rgular space.
However, f
is neither
REMARK 5.3.
a.w.c.
X
f.c.,
s.w.c,
nor a.w.c.
There exists a f.c. function which is neither w.q.c., s.w.c, nor
The following example is due to Long and Herrington [6].
T. NOIRI
102
{0, i} and T {, X, {i}}. Let Y {a, b, c} and
EXAMPLE 5.4. Let X
{, Y, {a}, {b}, {a, b}}. Define a function f
(X, T)
(Y, s) as follows: f(0)
a and f(1)
b. Then f is f.c. [6, Example 2]. However, f is neither w.q.c.,
s.w.c, nor a.w.c.
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
function is not necessarily
s.w.c,
weakly continuous even if the range is a regular space.
EXAMPLE 5.6.
topology for
X
Let
X
o
and
the identity function.
basis for
o
Then
(X, )
and
REMARK 5.7.
be the set of all real numbers,
the discrete topology for
is
f
I.
However, f
(X, T)
f
{{x}l
is s.w.c, since the set
T
the countable complement
Let
X.
x e X}
(X, o)
be
is an open
is neither w.q.c., f.c. nor a.w.c.
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
real numbers with the discrete topology and
f
X
f
Y
However, f
is almost continuous and hence a.w.c.
the identity function.
the
Then
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.
EXAMPLE 5.10. Let X
{a, b, c, d}
{a, b, c}, {b, c, d}}. Define a function
f(b)
d, f(c)
However, f
e
I
6.
f(d)
and
b
a.
(X, o)
(X, I)
f
f
Then
{, X, {b}, {c}, {b, c}, {a, b},
and
as follows: f(a)
.
is weakly continuous
[16, Example].
is neither semi continuous nor almost continuous since there exists
f-l({c})
such that
{a}
and
Int({a})
Int(Cl({a}))
c,
{c}
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].
results will be denoted only by numbers with the bracket
new results will be denoted by THEOREM,
6.1.
LEMMA, EXAMPLE
).
The known
In contrast to this,
etc.
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.
PROOF.
See the proof of Theorem 6.1.8 (below).
THEOREM 6.1.3.
The composition of two w.q.c. (resp. a.w.c.) functions is not
necessarily w.q.c. (resp. a.w.c.).
PROOF.
composition
In Example 2 of [18], f
gof
and
g
are weakly continuous.
However, the
is neither w.q.c, nor a.w.c.
In the sequel we investigate the behaviour of compositions in case one of two
functions is continuous.
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
THEOREM 6.1.4.
Y
g
Z
If
X
f
Y
(resp. almost continuous) and
is semi continuous
X
go f
is continuous, then
103
is semi continuous (resp. almost
Z
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
THEOREM 6.1.6.
go f
continuous, then
X
f
If
Y
(resp. a.w.c.) and
is w.q.c.
Z
Y
g
(resp. s.w.c., f.c.).
is weakly continuous
go f
is continuous, then
is weakly continuous (resp. s.w.c., f.c.) and
Y
X
f
Y
g
Z
is
(resp. a.w.c.).
is w.q.c.
First, by using Theorem 4.1 we show that go f is w.q.c. Let x e X
-i
g(f(x)). Then g (W) is an open set containing
-i
and there exists U e SO(X) containing x such that f(U) C Cl(g (W)).
PROOF.
W
and
f(x)
an open set containing
g
Since
(go f)
-i
go f
This shows that
THEOREM 6.1.7.
W
Let
is a.w.c.
hence we have
PROOF.
Y
Y
g
{a,
Z
(Y, )
g
g
and
go f
THEOREM 6.1.8.
Int(Cl((g
Next, we show
Y
is open in
-i
and
(CI(W)))).
Y
X
f
and
be the identity functions. Then f
-i
g ({b, c, d}) e SO(Y, s). The set
f)-l({b,
(go
and
c,
d})
SO(X, ).
Z
Z
is
Thus,
8
f)-l(cl({z}))))
@
be the usual
Let
the discrete topology.
(Z, 8)
(Y, )
and
is not necessarily a.w.c.
be the set of real numbers.
g
and
Y
X
f
of a continuous function
go f
Y
g
Let
However, g, f
{z} e 8.
for every
Hence
f
Then
be the identity functions.
is almost continuous by Example 5.8.
g
continuous and
since
Y
(Y, )
(X, T)
f
(Z, 8)
the indiscrete topology and
topology,
CI(W).
(W)
d}, T {, X, {a}, {b}, {a, b}, {a, c, d}},
{, Z, {a}, {b}, {a, b}, {b, c, d}}. Let
(Z, 8)
The composition
X
Let
-i
is not w.q.c, and hence not semi continuous.
an almost continuous function
PROOF.
8
is semi continuous since
is regular closed in
by Theorem 4.2
g
is not necessarily w.q.c.
b, c,
(X, )
{b, c, d}
Then
of a continuous function
go f
Z
and
continuous and
Z.
is a.w.c.
{, Y, {a}, {b}, {a, b}}
(Y, )
g(Cl(g-l(w)))C
(W))))) C Int(Cl((g= f)
(Cl(g
The composition
X
Let
f)(U)C
be an open set of
-i
-I
(W) C Int(Cl(f
a semi continuous function
f
(g
is continuous, we obtain
f
g
that
is
is not a.w.c.
is not almost
g, f
continuous.
The following is shown in Lemma i of [18].
(6.1.9)
g
f
If
THEOREM 6.1.10.
f.c.), then
is continuous and
E
basis
Z
is s.w.c.
W
of
Z, g
Y
g
Z
(W)
g, f
is f.c.
6.2
RESTRICTIONS.
THEOREM 6.2.1.
is continuous and
is weakly continuous, then
Cl((go
Cl(g
(W))
g
is open in
g
is continuous and
Y
Y
is s.w.c.
(CI(W))
for every
f-l(cl(g-l(w)))
f)-l(w)) C
f
g
Z
is s.w.c. (resp.
(resp. f.c.).
is continuous and
-i
-I
f
Suppose that
-i
Y
is s.w.c.
such that
continuous, we have
go f
Z
Suppose that
of
X
f
If
X
g f
PROOF.
set
Y
X
f
is weakly continuous.
and hence
(g
C
There exists an open
W
E.
Since
(g= f)-I(cI(W)).
f
is
Therefore,
is f.c. For every 8-open
-i
f) (W) is open in X. Hence
g
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
T. NOIRI
104
RC(X, ).
flA
The restriction
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
PROOF.
V
Let
4 of [20] we have
X
f
flA
then the restriction
A
Y
A
is weakly continuous and
Y
is a subset of
X,
is weakly continuous.
be an open set of Y. Since f is weakly continuous, by Theorem
-l
-1
Cl(f (V))Ci f (CI(V)). Therefore, we obtain
CIA((flA)-I(v)) CiA(f-l(v)hA)C Cl(f-l(v))(’ A C (flA)-l(cl(V)),
where
CIA(B)
B
denotes the closure of
Theorem 7] that
flA
It follows from [7,
A.
in the subspace
is weakly continuous.
The following are shown in [17, Theorem 4] and [6, Theorem 12].
(6.2.4)
The restriction of a s.w.c. (resp. f.c.) function to a subset is s.w.c.
(resp. f.c.).
THEOREM 6.2.5.
The restriction of an
function to a subset is not
a.w.c,
necessarily a.w.c.
In Example 3 of [19], f
PROOF.
flM
However, the restriction
M
R
R
R
is almost continuous and hence a.w.c.
0.
x
is not a.w.c, at
In the sequel we investigate the case of restrictions to open sets.
(6.2.6)
The following
3] and [19, Theorem 4].
are shown in [15, Theorem
The restriction of a semi continuous (resp. almost continuous) function
(resp. almost continuous).
to an open set is semi continuous
The following are immediate consequences of Theorem 6.2.3 and (6.2.4).
The restriction of a weakly continuous (resp. s.w.c., f.c.) function to
(6.2.7)
an open set is weakly continuous (resp. s.w.c., f.C.)o
THEOREM 6.2.8.
restriction
flA
PROOF.
Let
If
X
f
Y
A
x e A
Y
and
V
A
X, then the
U e SO(X)
f(x).
containing
containing
x
Since
f
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.
restriction
flA
PROOF.
Let
-I
If
A
V
X
f
Y
Y
is a.w.c, and
is open in
f
is a.w.c., we have
is a.w.c.
be an open set of
(CI(V)))).
X, then the
A
Since
A
Y.
Since
f
-i
(V)
is open, we obtain
(flA)-l(v) C A/’ Int(Cl(f-l(cl(V)))) IntA(A/CI(f-I(cI(V))))
ClntA(ACI(A f-I(cI(V)))) IntA(CIA((flAI-I(cI(V))))
where
IntA(B)
subspace
6.3.
and
CIA(B
A, respectively.
denote the interior and the closure of
This shows that
flA
B
in the
is a.w.c.
GRAPH FUNCTIONS.
Let
(x, f(x))
X
f
Y
for every
be a function.
A function
X
g
g(x)
Y, defined by
X
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
almost continuous, weakly continuous) if and only if
almost continuous, weakly continuous).
f
is semi continuous
f
is
f(U) C CI(V).
such that
is open in
f(A(’IU) C f(U)
Int(Cl(f
is open in
Y
be an open set of
w.q.c., by Theorem 4.1 there exists
Since
A
is w.q.c, and
is w.q.c.
(resp.
is semi continuous
(resp.
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.
Y
X
f
The graph function
PROOF.
Suppose that
Necessity.
f(x).
containing
X
Then
U e SO(X)
there exists
V
g(x).
containing
(x, f(x)) e U
W.
THEOREM 6.3.5.
Y
f
(B)
we obtain
Y
V C W.
W
and
V C Y
and
an open set
Therefore, we
be an open set
g(x)
such that
is w.q.c., by Theorem 4.1 there exists
The graph function
Suppose that
Necessity.
B
for every subset
-i
(V)))
Cl(Int(f
Put
U
U
I(’ U 2,
Y
X
X
g
g(x).
containing
f
Since
g
Y.
of
U
g
[15,
is w.q.c.
is a.w.c, if and only if
f
-i
g
-i
(X
B)
V
be an open set of Y. By Theorem 3.1,
-i
-i
f (CI(V)). It
V))
V)))C g (CI(X
(X
is a.w.c.
f
Let
is a.w.c.
x e X
There exists a basic open set
-i
Cl(f
is a.w.c., by Theorem 3.1
W
and
U
V
be an open set of
g(x) e
such that
(CI(V)))
is a neighborhood of
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
x
e SO(X)
2
x e U e SO(X)
then
In general, we have
is a.w.c.
Let
Cl(Int(g
Suppose that
Sufficiency.
X
x e X
Let
X
V).
It follows from Theorem 4.1 that
follows from Theorem 3.1 that
U
is w.q.c, by Theorem 4.1.
is w.q.c.
V
and by Theorem 4.1
is a.w.c.
PROOF.
-i
f
UIC
and
g(x)
g(U) C CI(X
such that
f(U 2) C CI(V).
g(U) C CI(W).
x e X
f
f
Since
such that
x
containing
Lemma i] and
X
is w.q.c, if and only if
Let
is w.q.c.
There exist open sets
V
I
g
x
containing
Sufficiency.
f
Y
X
X
is an open set containing
f(U) C CI(V) and hence
Suppose that
obtain
g
is w.q.c.
and
neighborhood of
6.4.
and hence
x
a
is a.w.c, by Theorem 3.1.
g
PRODUCT FUNCTIONS.
{XI
{YI
e V} be any two families of topological spaces
e e V} (resp. {Y
e e V})
V. The product space of
Y
is simply denoted by HX (resp. HY ). Let f
be a function for each
X
e e V. Let f
HY
HX
be the product function defined as follows: f({x })
e
for
The natural projection of
(resp. HYe) onto X
every
{x}
{f(xe)}
8
HY e
The
X (resp. q8
(resp. Ya) is denoted by
following are
Ya).
8
P8
shown in [15, Theorem 5], [14, Theorem 2.6] and [18, Theorem i].
HY
is semi continuous (resp. almost
(6.4.1) The function f
X
Y
is semi continuous (resp.
X
continuous, weakly continuous) if and only if f
almost continuous, weakly continuous) for each e e V.
Let
e V}
and
{Xel
with the same index set
HX=.
HX
HXe
The following two results are shown in Theorems 3 and 5 of [18].
(6.4.2)
If
f
If
f
X
Y
is s.w.c, for each
e e
V, then
f
HY
HX
is
s.w.c.
(6.4.3)
LEMMA 6.4.4.
If
funtion.
PROOF.
4.2.
Since
f((go f)
-I
ge f
Let
f
(F))
HY
HX
Let
X
f
X
Z
F e RC(Z).
go f
g
e V.
is f.c. for each
be an open continuous surjection and
is w.q.c., then
Since
Y
X
f
is f.c., then
Y
g
Y
Z
a
is w.q.c.
is w.q.c.,
(ge f)
-i
(F)
SO(X)
by Theorem
is an open sontinuous surjection, by Theorem 9 of [3] we obtain
-i
g (F) e SO(Y). It follows from Theorem 4.2 that g is w.q.c.
T. NOIRI
106
THEOREM 6.4.5.
is w.q.c, for each
PROOF.
f
The function
X
HY
f
is w.q.c.
Suppose that
is continuous, by Theorem 6.1.6
f6 P6
continuous surjection and by Lemma 6.4.4
Let
{xs} e HX
s
x
Vs
There exists a basic open set
V, say,
number of
f
Since
CI(V)
s
for
s
e2’
I,
i’ s2
qB
f8
is w.q.c.
V
n
SO(HX)n
x e U e
C
f(U)
n"
Put
U
H U
s
PROOF.
Sj
f
Since
(Us
-i
H
Ys
s#.
j
Y
is
PROOF.
Z
such that
xs
Y
Vs
J
containing
H
C
j=l
"C
f (U s)
X
CI(V
f
Y C CI(W).
#s.
j
is w.q.c.
be an open continuous surjection and
g
is a.w.c.
Since
g f
is a.w.c., then
Z.
be an open set of
f8
The function
for each
a.w.c,
-i
(CI(W)))) C Int(f
f
HX
a e
V.
Suppose that
Necessity.
Ys Y6
q6
hence
W,
and otherwise
g
-i
(W)
j
Z
Y
g
a
is a.w.c., we have
-i
(Cl(g
Int(Cl(g
-i
(CI(W))))).
-i
(CI(W)))).
This shows
is a.w.c.
X
and
Y
X
(W) .i Int(Cl((g f)
THEOREM 6.4.7.
f
f
X
W
Let
f(x).
where for a finite
s#.
is an open surjection, we obtain
g
that
Let
gf
(g. f)
P6
n
Therefore, it follows from Theorem 4.1 that
If
Ys
HYs Y8
is an open
n
j=l
function.
Y
[15, Theorem 2] and
H f
LEMMA 6.4.6.
HVsC
f(x) e
SO(Xs)
=1
then
X
q
be an open set containing
is open in
Us e
Since
Moreover,
is w.q.c.
W
and
V.
e
Let
f
such that
there exists
is w q c
f
V.
s e
Necessity.
Sufficiency.
is w.q.c, if and only if
f
HY
is a.w.c, if and only if
6 e V.
Let
is a.w.c.
is continuous, by Theorem 6.1.6
f8 P8
Since
f
q8
f
is a.w.c.
is a.w.c, and
is a.w.c, by Lemma 6.4.6.
Sufficiency.
{x
Let x
e X
V
exists a basic open set
and
W
be an open set containing
f(x).
There
such that
n
f(x) e
where
V
(.
Theorem 3.1
is open in
Y
VC
S.
W
V j=IH Vj
and
for
i, 2
j
Y
#a. ’
n.
Since
f
is a.w.c., by
CI(f-I(cI(VeJ ))) a neighborhood of x and
n
n X C CI(f-I(cI(W))).
)))
CI(f-I(cI(Va
e"
j=l
j
s#s.
is
j
-I
Therefore, Cl(f (CI(W))) is a neighborhood of x
It is well-known that a function f
HY
X
qB
f.: X
YB
is continuous for each
B
and
f
is a.w.c, by Theorem 3.1.
is continuous if and only if
We investigate if weak forms of
e V.
continuity have this property.
The following are shown in [15, Theorem 6] and [3, Example i0].
(6.4.8)
If a function
is semi continuous for each
THEOREM 6.4.9.
q6
X
YB
f
X
8 e V.
A function
f
HY
is semi continuous, then
q. f
X
Y
However, the converse is not true.
is almost continuous if and only if
X
IFf
is almost continuous for each
B
e V.
107
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
PROOF.
Since
Necessity.
is continuous, this is an immediate consequence
q8
of Theorem 6.1.4.
x e X
Let
Sufficiency.
i, 2
for
i, 2,
B
f(x)
VC
V
f(x) e
qB(f(x))
Since
V, Cl((q
e
W, where
Y-I (Ve))
Ve
HY.
in
Y
J
e
and
V8
f
q8o
is a neighborhood of
$
There
is open in
is a neighborhood of
f)
J
n
)-I(v ))
Cl((q.=
and
n
an open set containing
such that
and otherwise
n
almost continuous for each
for
W
and
V
exists a basic open set
is
for
x
X.
Moreover, we have
n
Cl((q.o f)-l(ve
j=l
Assume that
U
.
z
f-I(Hv)
shows that
CI(f-I(Nv)).
Consequently, Cl(f
-i
ek
f
f)_l(Vk
(W))
Cl(f-l(Hva )) C CI(f-I(w)).
(qk f)-l(Vk
and hence we obtain
is a neighborhood of
qs
X
f
A function
f
HYe
X
---n] Cl((qe.o
z
and hence
x
f
n).
k
f)
-i
This
(V.)).
is almost continuous.
2, 4 and 6 of [18].
is weakly continuous if and only if
8 e V.
is weakly continuous for each
Y8
(6.4.11)
is s.w.c, if
qD
f
X
Y8
is s.w.c, for
q8
f
X
Y
8 e V.
each
(6.4.12)
HYe
X
f
If a function
is f.c., then
is f.c. for
8 e V.
each
THEOREM 6.4.13.
s.w.c,
If a function
Since
Since
X2,
1
However,
HY
X
f
q=m
X
f
Y8
is
is w.q.c., then
q8
q
X
f
Y8
is
the converse is not true in general.
is continuous, by Theorem 6.1.6
is semi continuous for
X- X.
1
1
X
X
If a function
8 e V.
w.q.c, for each
i0 of [3], f.
is s.w.c., then
is continuous, this follows immediately from (6.1.5).
q8
THEOREM 6.4.14.
PROOF.
HYa
X
f
8 e V.
for each
PROOF.
f
NY
X
f
A function
k (i
such that
z
containing
for some
The following three results are shown in Theorems
(6.4.10)
U
There exists an open set
Therefore, U
Cl((q
z
)) C
i
(fl(x), f2(x))
defined as follows: f(x)
f
q8
i, 2.
is w.q.c.
In Example
However, a function
for every
x e X, is not
w.q.c.
A function
THEOREM 6.4.15.
f
Ya
X
is a.w.c, if and only if
The necessity follows from Theorem 6.1.6.
PROOF.
YB
X
qB=f
V.
for each
is a.w.c,
By using Theorem 3.1, we can
prove the sufficiency similarly to the proof of Sufficiency of Theorem 6.4.9.
CLOSED GRAPHS.
6.5.
For a function
X
y
f
X
f
Y
{(x, f(x))l
Y, the subset
X
Y
is continuous and
is Hausdorff then
shall investigate the behaviour of
G(f)
x
G(f).
and is denoted by
f
is called the graph of
G(f)
X}
of the product space
It is well known that if
is closed in
X
y.
We
in case the assumption "continuous" on
f
is replaced by one of seven weak forms of continuity.
THEOREM 6.5.1.
G(f)
If
is semi-closed in
PROOF.
[3], f
closed in
X
X
f
X
X
y
Y
is semi continuous and
is semi-closed in
X*
X*
is semi continuous and
because
is Hausdorff, then
but it is not necessarily closed.
By Theorem 3 of [21], G(f)
X*
Y
(1/2, O)
is Hausdorff.
e Cl(G(f)) -G(f).
X
y.
In Example 8 of
However, G(f)
is not
108
T. NOIRI
A w.q.c, function into a Hausdorff space need not have a
COROLLARY 6.5.2.
closed graph.
An almost continuous function into a Hausdorff space need not
THEOREM 6.5.3.
have a closed graph.
In Example I of [19], f
PROOF.
Hausdorff.
However, G(f)
R
R
is almost continuous and
R
is
G(f)
(p, -p) e CI(G(f))
is not closed since
for a
p.
positive integer
An
COROLLARY 6.5.4.
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)
Y
X
f
If
is weakly continuous and
Y
G(f)
is Hausdorff, then
is
closed.
The above result was improved by Baker [17] as follows:
(6.5.6)
If
Y
X
f
is s.w.c, and
Y
is Hausdorff, then
G(f)
is closed.
PRESERVATIONS OF CONNECTEDNESS AND HYPERCONNECTEDNESS.
6.6.
In this section we investigate if connected spaces and hyperconnected spaces are
A space
preserved under seven weak forms of continuity.
X
hyperconnected if every nonempty open set of
X
is dense in
is said to be
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].
(6.6.2)
Weakly continuous surjections preserve connected spaces.
Connectedness is not necessarily preserved under s.w.c.
THEOREM 6.6.3.
surjections.
PROOF.
be real numbers with the finite complement topology, Y
X
Let
f
numbers with the discrete topology and
s.w.c,
X
surjection and
is connected.
X
Y
the identity function.
However, Y
real
f
Then
is a
is not connected.
The following is an improvement of (6.6.2) [25, Corollary 3.7].
(6.6.4)
Connectedness is preserved under f.c. surjections.
Neither w.q.c, surjections nor a.w.c, surjections preserve
COROLLARY 6.6.5.
connected spaces in general.
PROOF.
This is an immediate consequence of
(6.6.1).
The following is shown in [26, Lemma 5.3].
(6.6.6)
Semi continuous surjections preserve hyperconnected spaces.
Almost continuous surjections need not preserve hyperconnected
THEOREM 6.6.7.
spaces.
PROOF.
In Example 5.8, f
hyperconnected.
However, Y
THEOREM 6.6.8.
Y
X
is an almost continuous surjection and
X
is
is not hyperconnected.
Weakly continuous surjectlons need not preserve hyperconnected
spaces.
PROOF.
{a, b, c},
X
Let
{a}, {b}, {a, b}}.
Let
f
T
(X, T)
a weakly continuous surjection and
{, X, {c}, {a, c}, {b, c}}
and
{, X,
(X, ) be the identity function. Then f
(X, ) is hyperconnected. However, (X, )
is
is
not hyperconnected.
COROLLARY 6.6.9.
f.c., w.q.c, or
PROOF.
a.w.c,
Hyperconnectedness is not necessarily preserved under s.w.c.,
surjections.
This follows immediately from Theorem 6.6.8.
PROPERTIES OF SOME WEAK FORMS OF CONTINUITY
109
SURJECTIONS WHICH IMPLY SET-CONNECTED FUNCTIONS.
6.7.
DEFINITION 6.7.1.
A
Let
.
said to be connected between
A C F
and
F/’AB
f(X)
provided that
B
and
A
and
A function
be subsets of a space
f
Y
X
X
F
is said to be set-connected
f(A)
is connected between
relative topology if
A space
X.
if there exists no clopen set
B
A
is connected between
f(B)
and
X
is
such that
[27]
with respect to the
B.
and
The following lemma is very useful in the sequel.
LEMMA 6.7.2 (Kwak [27]).
if
-i
f
(F)
A surjection
X
is a clopen set of
In Example 5.2, f
F
for every clopen set
of
Y.
is a semi continuous surjection but it is not
f-l({a})
set-connected since
(X, T).
is not closed in
An almost continuous surjcetion need not be set-connected.
THEOREM 6.7.4.
In Example 5.8, f
PROOF.
is set-connected if and only
A semi continuous surjcetion need not be set-connected.
THEOREM 6.7.3.
PROOF.
Y
X
f
is an almost continuous surjection but it is not
set-connected.
COROLLARY 6.7.5.
Neither w.q.c, surjections nor a.w.c, surjections are
set-connected in general.
PROOF.
This is an immediate consequence of Theorems 6.7.3 and 6.7.4.
The following is shown in [28, Theorem 3].
(6.7.6)
Every weakly
A
THEOREM 6.7.7.
PROOF.
continuous surjection is set-connected.
surjection need not be set-connected.
s.w.c,
(X, T)
In Example 5.6, f
f-l({x})
not set-connected since
(X, )
is a s.w.c, surjection but it is
is not open in
(X, Y)
{x}
for a clopen set
of
(x, o).
The following is shown in [25, Theorem 3.4].
(6.7.8)
7.
Every f.c. surjection is set-connected.
QUESTIONS.
In this section we sum up several questions concerning subweak continuity and
faint continuity.
Are the following statements for
QUESTION i.
i) A function is
X
f
functions true
if the graph function is s.w.c.
s.w.c,
2) Each function
s.w.c,
Y
is s.w.c, if the product function
f
X
HY
is s.w.c.
Are the following statements for f.c. functions true
QUESTION 2.
i) The composition of f.c. functions is f.c.
2) If a function is f.c., then the graph function is f.c.
3) If each
5) If
f
q8(C)
X
Y
X
f
4) If each
Y
f
X
is f.c., then
Y8
is f.c. and
Y
X
f
is f.c., then
f
X
Y
is f.c.
HYa
is f.c.
is Hausdorff, then
G(f)
is closed in
X
y.
Finally, the results obtained in Section 6 are summarized in the following table,
where
denotes the results already known.
T. NOIRI
110
TABLE
f:X
2
3
4
5
6
’7
8
9
i0
ii
12
13
s .c.
a.c.
w.c.
Z:P (-)
Y:P, g:Y
6.1.1
gof:X- Z:P
(-)
(-)
6.1.2
6.1.1
6.1.1
(+)
(+)
(+)
+
+
6.1.5
6.1.5
6.1.5
6.1.6
6.1.6
(+)
+
+
Z:P
6.1.7
6.1.8
6.1.9
6.1.10
6.1.10
6.1.7
6.1.8
Y:P, AC X
Y:P
(-)
+
6.2.1
6.2.2
6.2.3
(+)
6.2.4
6.2.4.
6.2.1
6.2.5
Y:P, A:open
Y:P
(+)
6.2.6
(+)
6.2.6
(+)
(+)
(+)
+
+
6.2.7
6.2.7
6.2.7
6.2.8.
6.2.9
Z:P
Y:C, g:Y
flA:A
f:X
flA:A
15
(+)
(+)
(+)
(+)
+
+
Y:P
6.3.1
6.3.1
6.3.1
6.3.3
6.3.4
6.3.5
Y:P
g:X
(+)
(+)
(+)
(+)
+
+
XY:P
6.3.1
6.3.1
6.3.1
6.3.2
6.3.4
6.3.5
XY:P
HY :P
:X / Y :P
f:HX
+af
(+)
(+)
(+)
(+)
+
+
6.4.1
6.4.1
6.4.1
6.4.3
6.4.5
6.4.7
()
(+)
+
+
6.4.2
6.4.5
6.4.7
(+)
(+)
HY :P
6.4.1
6.4.1
pa:X Ye:P
(+)
+
(+)
+
(+)
+
+
6.4.8
6.4.9
6.4.10
6.4.13
6.4.12
6.4.14
6.4.15
6.4.15
Y :P
f :X
+af:HX
HY :P
6. .i
+
(-)
+
(+)
(+)
:P
6.4.8
6.4.9
6.4.10
6.4.11
6.4.14
Y:P, Y:T2
G(f):closd
(+)
6.5.1
6.5.3
6.5.5
(+)
6.5.6
6.5.2
(-)
6.6.1
(-)
6.6.1
(+)
6.6.2
6.6.3
6.6.4
6.6.5
6.6.5
6.6.7
6.6.8
6.6.9
6.6.9
6.6.9
6.6.9
6.7.4
6.7.6
6.7.7
6.7.8
6.7.5
6.7.5
p f:X
f:X
Y :P
/SHY
Y:onto P,
f:X
X connected
Y:onto P
f:X
(+)
X :hyperconnected
6.6.6
Y :hyperconnected
f:X
(+)
f:X
g:X
Y:connected
14
6.1.3
+
gof:X
f:X
6.1.3
6.1.4
f:X
f:X
a.w.c.
+
Z:C
Z:P
gof:X
f :X
w.q.c.
6.1.4
Y:P, g:Y
f:X
f:X
f.c.
s.w.c,
(+)
(+)
Y:onto P
f:set-connected 6.7.3
(+)
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