characterizations of some near-continuous functions and near-open ...

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:

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.