715
Internat. J. Math. & Math. Sci.
Vol. 9 No 4 (1986) 715-720
CHARACTERIZATIONS OF SOME NEAR-CONTINUOUS
FUNCTIONS AND NEAR-OPEN FUNCTIONS
C.W. BAKER
Department of MatLiematics
Indiana University Southeast
New Albany, Indiana 47150
(Received April 7, 1986)
ABSTRACT.
A subset
N
0-nelghborhood of
of a topological space is defined to be a
U
x if there exists an open set
x E U CI
such that
to characterize the following types of functions:
U
N.
This concept is used
weakly continuous, 0-contlnuous,
strongly 0-continuous, almost strongly 0-cbntinuous, weakly S-continuous, weakly open
Additional characterizations are given for weakly
and almost open functions.
The concept of 0-neighborhood is also used to define the fol-
-continuous functions.
0-open, strongly 0-open, almost strongly 0-open, and
lowing types of open maps:
weakly
-open
functions.
KEY WORDS AND PHRASES. O-neighborhood, weakly continuous function, O-continuous function, strongly O-continuous function, almost strongly O-continuous function, weakly
S-continuous function, weakly open function, almost open fnction, O-open function,
strongly O-open function, almost strongly O-open function, weakly S-open function.
19B0 AMS SUBJECT CLASSIFICATION CODE. 54C10.
I.
INTRODUCTION.
Near-contlnulty has been investigated by many authors including Levlne [I], Long
and Herrlngton [2], Noirl [3], and Rose [4].
Rose [5]
and Singal and Singal [6].
Near-openness has been developed by
The purpose of this note is to characterize sev-
eral types of near-continuity and near-openness in terms of the concept of 0-
neighborhood.
These characterizations clarify both the nature of these functions and
Additional characterizations of weak S-contlnulty are
the relationships among them.
given.
The concept of 0-neighborhood also leads to the definition of several new
types of near-open functions.
2.
DEFINITIONS AND NOTATION.
The symbols
X
Y
and
denote
sumed unless explicitly stated.
U
U
and the interior of
U
topological spaces with no separation axioms as-
Let
U
be a subset of a space
are denoted by
CI U
is said to be regular open (regular closed) if
0-closure (-closure) (Velicko [7]) of
U
and
U
Int U
X.
Int CI U (U
is the set of all
The closure of
respectively.
x
in
The set
CI Int U).
X
The
such that
every closed neighborhood (the interior of every closed neighborhood) of x
intersects
C.W. BAKER
716
the collection of all
X
space
e-open
e-open (-open)
respec-
A set is
ClU).
(u
For a given
-open
sets and the collection of all
with the
X
The space
form topologies.
CIU
and
if its complement is e-closed (-closed).
e-open (6-open)
said to be
U
(6-closed) if
is called e-closed
U
The set
tively.
CleU
CleU
are denoted by
U
The e-closure and the -closure of
U.
sets both
topology will be signified
X e (X).
s
by
DEFINITION I,
A function
X
f:
Y
is said to be weakly continuous (Levine
[I])
(e-continuous (Fomin [8]), strongly 0-continuous (Long and Herrington [2]), almost
strongly e-continuous (Noiri and Kang [9]), weakly -continuous (Baker [I0])) if for
each x e X
U
hood
V
and each open neighborhood
of
f(U)
such that
x
Int CI V, f(Int CI U)
CI V
f(x), there exists an open neighbor-
(f(Cl U)
f(Cl U)
CI V,
X
f:
Y
(f(U)
N (x
CI U
A subset
N
Int CI U
X
of a space
x in X
a point
e U
U
f(U)
X,
of
Int CI f(U)).
DEFINITION 3.
ne.ighborhood) of
f(Cl U)
is said to be weakly open (Rose [5])
(almost open (Rose [5])) provided that for each open subset
Int f(Cl U)
V,
CI V).
A function
DEFINITION 2.
of
is said to be a e-neighborhood (6-
if there exists an open set
U
such that
x
N).
Note that a e-neighborhood is not necessarily a neighborhood in the e-topology, but
a -neighborhood is a neighborhood in the -topology.
3.
NEAR-CONTINUOUS FUNCTIONS.
The main results can be paraphrased as follows:
,,f-i
(e-neighborhood)
neighborhood";
e-nelghborhood";
neighborhood)
weak continuity corresponds to
e-continuity corresponds to
,,f-I
strong e-continuity corresponds to "f
e-neighborhood"; almost strong e-continuity corresponds to’ -I
hood)
(e-
-I
(neighbor-
nelghborhoo
e-nelghborhood", and weak -contlnuity corresponds to ,,f-I (e-nelghborhood)
-nelghborhood
THEOREM I. A function f: X
Y is weakly continuous if and only if for each
x
in
X
and each
Assume
PROOF.
such that
N
of
f-l(N)
f(x),
x e X
Let
is weakly continuous.
is a neighborhood of
Then there exists an open set
V
and let
such that
N
f(U) CI V N.
f-l(N)
x e U
Thus
f-l(N)
and hence
x.
be a
CI V
f(x) e V
is weakly continuous, there exists an open neighborhood
f
Since
f
f(x).
neighborhood of
N.
e-nelghborhood
U
of
x
is a neighborhood
of x.
Assume for each
neighborhood of
CI V
x.
x e X
Let
and each e-neighborhood
x e X
U
for which
U
x
V
f-l(cl
f(x),
is a e-neighborhood of
an open set
and let
f
-i
(Cl V)
N
of
x
that
f-l(N)
be an open neighborhood of
V)
is a neighborhood of
and
f(U)
Cl V
x.
is a
f(x).
Since
Thus there is
which proves
f
is
weakly continuous.
THEOREM 2.
X
A function
and each e-nelghborhood
X
f:
N
of
Y
f(x),
is e-continuous if and only if for each
f-l(N)
is a e-neighborhood of
x.
x
in
CHARACTERIZATIONS OF SOME NEAR-CONTINUOUS FUNCTIONS
PROOF.
Assume
neighborhood’
f(x).
of
Y
X
f:
f(Cl U) ! C1 V N.
neighborhood of
U C1 U f-l(N)
g
x
X
in
and hence
and for each 0-neighborhood
Let
x.
x
X
g
f-l(c1
f(x),
is a 0-neighborhood of
CI V
and thus
f
V
and let
Hence there exists an open set U for which
f(Cl U)
f(x) e V =CIV
for which
U
of
f-l(N)
such th. t
x
is a 0-
x.
is a 0-neighborhood of
C1 V
x
be a 0-
N
and let
f, there exists an open neighborhood
Thus
Assume for each
Since
V
Then there exists an open set
By the 0-continuity of
N.
x e X
Let
is 0-continuous.
717
of
f(x)
f
that
V)
is a
0-neighborhood of
f-l(c I
C1 U
V).
-i
(N)
f(x).
be an open neighborhood of
U
x
N
x.
That is,
is 0-continuous.
The proof of the following theorem is similar to that of Theorem 2 and is omitted.
THEOREM 3.
each
X
x in
A function
x
A funciton
X
in
X
f:
is strongly 0-continuous if and only if for
N
and each neighborhood
THEOREM 4.
for each
Y
X
f:
f(x),
of
Y
f
-I
(N)
is a 0-neighborhood of
x.
is almost strongly 0-continuous if and only if
N
and each -neighborhood
of f(x),
f-l(N)
is a 0-neighborhood
of x.
PROOF.
N
Assume
-i
(N)
U
which proves that
Assume for each
0-neighborhood of
U
Hence there is an open set
Int C1 V
f(Cl U)
THEOREM 5.
N
of
f-l(Int
f(x)
f
X
N
Y
x e U
f
-I
!Cl U ! f (Int C1 V).
f
-I (N)
be a space and
Then
x
intersects
H} and
CI
H
{x e X:
every -neighobrhood of
x
intersects
H}.
The proof follows easily from the definitions.
For
X
f:
Y
(a)
f:
(b)
For each
Hi
X,
(c)
For each
K
Y, C1 6
(d)
f:
Y
X
X
S
the following statements are equivalent:
is weakly -continuous.
Y
x.
That is,
f(Cl H)!
f-1(K
x.
[9] for almost strongly 0-continuous functions.
H ! X.
o
THEOREM 6.
Since
These results are analogous
every 0-neighborhood of
(b)
is a
The following theorem
{x e X:
C1
(N)
f(x).
is a -neighborhood of
H
(a)
-I
is weakly -continuous if and only if for each
f(x),
of
to those obtained by Noiri and Kang in
X
that
is a 0-neighborhood of
C1 V)
gives additional characterizations of weak -continuity.
Let
x e U C1 U
x.
The proof of this theorem is similar to that of Theorem 4.
LEMMA.
Then
is almost strongly 0-continuous.
f:
and each 0-neighborhood
and let
such that
be an open neighborhood of
V
and let
such that
and hence
A function
Int C1 Vc N.
is a 0-neighborhood of
f(x),
is a 6-neighborhood of
Int C1 V
x e X
(N)
x e X
Let
f(Cl U)
and each -neighborhood
X
x
x.
f
-i
x e X
is almost strongly 0-continuous, there exists
f
for which
x
V
Then there exists an open set
Since
of
Let
is almost strongly 0-continuous.
f(x).
V Int CI V N.
f(x) e
an open neighborhood
! f
Y
X
f:
be a -neighborhood of
C1
e
f(H).
f-i (Cl e
is weakly continuous.
K).
C.W. BAKER
718
C16
f-l(N)
in
H
.
Let
(c).
y
Let
(d).
x
0
f-i
X
(a).
Y
s
x
CI
g
f(Cl6 f-l(K))
f-l(N
H,
=
H)
f(Cl
CI
n
.
H
f(f-l(K))
! CI 0
That is,
f(H).
0
CI
0
K.
Thus
.
CI V)
X
f-I (CI0
Hence x
CI V).
(Y
0
U
Thus there is a neighborhood
CI V.
f(Int CI U)
Then
CI V
Since
of
CI V).
(y
such that
x
Int CI U
Since
is
is a
is weakly continuous.
Y
x
CI
f(x)
f(x).
be an open neighborhood of
V
-open
C1 V
and hence
W
set
U
Then there is a regular open set
f(U) ! f(W)
f(x).
be an open neighborhood of
V
and let
is weakly continuous, there exists a
f(Int CI U)
4.
Thus
Cl V).
(y
X
Let
CI V.
f(W)
f(H).
and let
f(x),
(y
f:
regular open,
X
g
CI f-i
x
(Int CI U)
(d)
0
Since
By (b)
K ! Y.
a 0-neighborhood of
By (c)
CI
g
x.
By Theorem 5
f(x).
be a 0-neighborhood of
N
x
f-l(cl0 K).
CI f-l(K)
(c)
Hence
Then there exists an
H).
f(Cl
y
and let
Let
is a -neighborhood of
N n f(H) #
(b)
f(x).
y
such that
HX
Let
(a) => (b).
PROOF.
W.
U
g
x
for which
f:
Since
such that
x
containing
Then
is weakly -continuous.
f
NEAR-OPEN FUNCTIONS.
In this section weak openness and almost openness are characterized in terms of
the concept of 0-neighborhood.
THEOREM 7.
A function
PROOF.
Assume
of
x.
ly
open
U
x
Assume for each
x
such that
neighborhood of
f(x).
Thus
U
be an open set in
Int f(Cl U)
f(U)
X.
N
of
x
Suppose
x
is a neighborhood of
f(Cl U)
x,
CI U ! N. Since f is weakf(N) is a neighborhood of f(x).
U
Hence
and each 0-neighborhood
Let
is a 0-neighborhood of
and
f
be a 0-neighborhood
N
and let
g
X
in
x
X
g
f(U) ! Int f(Cl U) ! Int f(N).
f(x)
Int f(Cl U).
Let
is weakly open.
f
X
x
f(x).
is a neighborhood of
x, f(N)
Then there is an open set
of
is weakly open if and only if for each
Y
X
f:
N
and each 0-neighborhood
f(x).
that
f(N)
is a
U.
Since
CI U
Hence
f(x)
is weakly open.
The proof of the following theorem is similar and is omitted.
THEOREM 8.
A function f:
and each neighborhood
Theorem
N
of
X
Y
is almost open if and only if for each
x, CI f(N)
x
g
X
f(x).
is a 0-neighborhood of
and the characterizations of near-continuous functions in Section 3
suggest the following definitions of near-open functions.
DEFINITION 4.
A function f:
X
Y
is said to be 0-open
almost strongly 0-open, weakly -open) if for each
(neighborhood, -neighborhood, -neighborhood) N
x
of
(-neighborhood, 0-neighboehood, -neighborhood) of
g
X
x,
(strongly
0-open,
and each 0-neighborhood
f()
is a 0-neighborhood
f(x).
The following theorems characterize these near-open functions in terms of the closure and
interior operators. Since the proofs are all similar, only the first theorem is proved.
THEOREM 9.
A function f:
each open neighborhood
that
CI V ! f(Cl U).
U
of
X
Y is 0-open if and only if for each
x, there exists an open neighborhood
V
x
of
X
and
f(x) such
719
CHARACTERIZATIONS OF SOME NEAR-CONTINUOUS FUNCTIONS
PROOF.
borhood of
V
set
Assume
x.
such that
Y
X
f:
Assume that for each
V
x e X
be a 0-neighborhood of
V
There exists an open set
THEOREM I0.
X
x
f(x)
THEOREM ii.
x e X
f(x)
X
f(x)
X
!Cl
V
there exists
f(N).
! f(Cl U)
N
and let
CI U
x e U
N.
f(N)
Hence
0-open if and only if for each
is strongly
Y
U
of
Y
X
V
x, there exists an open neighborhood
of
U
0-open if and only if for each
is almost strongly
of
there exists an open neighborhood
x
V
of
V
of
! f(Int CI U).
A function f:
X
and each open neighborhood
such that
f(x) e V
for which
x
x e X
f(U).
A function f:
C1 V
U
of
Let
is 0-open.
f
and each open neighborhood
such that
THEOREM 12.
x
and
A function f:
CI V
f(Cl U).
CI V
for which
and each open neighborhood
such that
U
and each open neighborhood
such that
f(x)
is a 0-neighborhood of
be an open neigh-
f(x), there exists an open
Then there is an open set
x.
U
and let
f(Cl U).
f(x)
of
x e X
Let
is a 0-neighborhood of
CI V
f(x) e V
an open neighborhood
0-open.
is
f(Cl U)
Since
Int CI V
is weakly 6-open if and only if for each
Y
U
of
We have the following implications:
0-open => 0-open => weak
x, there exists an open neighborhood
f(Cl U).
6-open
implications are not reversible.
{a, b}, T
EXAMPLE I. Let X
{a} {a, b}}. The inclusion mapping:
<= st. 0-open => almost st.
almost open
=> weak open.
The following examples show that these
,
{X,
{a}},
(X, T I)
Y, T
{a, b, c}, and
Y
2)
T
2
,
={Y,
is weak open but not weak
-open.
In the next example the space
EXAMPLE 2.
T
2
{Y,
(X, T I)
(X,
TI)
be as in Example I.
Let
Y
{a, b, c, d}. and
{a}, {c}, {a, b}, {a, c}, {a, b, c}, {a, c, d}}.
(Y,
T2)
EXAMPLE 3.
(Y, T 2)
Let
(Y, T 2) is from Example 2.2 in Noiri and Kang [9].
is weak
Let
S-open, but
(Y, T 2)
The inclusion mapping:
not 0-open.
be as in Example 2.
The identity mapping:
(Y, T 2)
is 0-open but not almost strongly 0-open.
EXAMPLE 4.
Let
{c}, {a, c}, {a, b}}.
X
{a, b, c},
T
,
{X,
The identity mapping:
{a}, {a, c}}
(X, T I)
(X,
T2)
and
T2
{X,
,
{a},
is almost strongly
0-open and almost open, but not strongly 0-open.
REFERENCES
I.
2.
3.
LEVINE, Norman, A Decomposition of Continuity in Topological Spaces, Amer. Math.
Monthly 68(1961) 44-46.
LONG, Paul E. and HERRINGTON, Larry L., Strongly 0-Continuous Functions, J. Korean
Math. Soc. 18(1981) 21-28.
NOIRI, Takashi, On S-Continuous Functions, J. Korean Math. Soc. 16(1980) 161-166.
4.
ROSE, David A., Weak Continuity and Almost Continuity, Internat. J. Math. Math.
Sci. 7(1984) 311-318.
5.
ROSE, David A., Weak Openness and Almost Openness, Internat. J. Math. Math. Sci.
7(1984) 35-40.
C.W. BAKER
720
6.
SINGAL, M. K. and SINGAL, Asha Rani, Almost Continuos Mappings, Yokohama Math. J.
16(1968) 63-73.
7.
VELICKO, N. V., H-Closed Topological Spaces, Amer. Math. Soc. Transl. 78(1968)
103-118.
8.
FOMIN, S., Extensions of Topological Spaces, Ann. Math. 44(1943) 471-480.
9.
N01RI, T. and KANG, Sin Min, On Almost Strongly e-Continuous Functions, Indian
J. Pure Appl. Math 15(1984) 1-8.
I0.
BAKER, C. W., On Super-Continuous Functions, Bull. Korean Math. Soc., 22(1985)
17-22.