DOI: Abstr D N gener 2010 Keyw T D N func prop were mapp Cros o and Saye D I D N the map estab map In Y ) no s LetA deno : http://dx.doi. D G Baza
ract : The pu
closed
map
ralized maps. S
0 Mathematics S
words: ND
The present
closed
,
tions in topo
perties [10]. D
e studied by
pings were i
sseley et.al [
open and
investigated
ed et. al [20
closed
m
n this paper
open
and
composition
s need not be
blish some pro
s.
n the whole
represent the
eparation axio
X
A the clo
oted by Cl(A)
.org/10.26483/
Interna
A NE
Purushott Department of Govt. P. G. Co ar, Raipur, Ch
urpose of this
ps by using N
Several properti
Subject: Classif open
maps, N
1. INTROD
authors in
open D N
ological space ifferent types various rese introduce and [5] initiated closed maps open semi ] devised the
maps and enlig
r, we introd
d ND closed of two ND
open D N
operties of N
paper (X,)
e non-empty
oms are assum
osure of
A
and Int(A) r
/ijarcs.v9i1.534
Volu
ational Jou
Av
EW CLAS
tam Jha Mathematics llege, Baloda hhattisgarh, Indpaper is to i
open D and es of these new
fication. Primar
closed D
N
DUCTION
ntroduced th
sets and N
es and study
of closed and
earchers. Gen
d studied by
and studied
. Misser et
n and semi e notion of
ghten its prope
duce new cl
d maps. . We
open
(resp
. (resp ND c
open D
N an
and (Y,)
topological sp
med, unless e
and interior
espectively. 47
ume 9, No.
urnal of Ad
RESE
vailable On
SS OF OP
dia
introduced and
d NDclosed w notions have b
ry: 54A05, 54C0
maps
he notion
continuou D
y some of th
d open mappin
neralized clos
Malghan [1
the notion
al.[49] defin
closed
map
open
D a
erties
asses of ma
also prove th
. D closed p N
) closed . We al nd NDclose
(simply X a
paces on whi
explicitly state
r of
A
will1, January
dvanced Re
EARCH PA
nline at ww
PEN AND
d study new c
d [10] sets an been discussed
05, 54C08; Sec
of us heir ngs sed 3]. of ned ps. and aps hat ) d lso ed and ich ed. be H se D of (i) (ii (ii (iv (v (v (v (v D (i) (ii
y-February
esearch in C
APER
ww.ijarcs.in
D CLOSE
G
lasses of open
nd establish rel
and the connec
ondary: 54C10,
Here we recoll
equel.
efinition 2.1.
f the space X ) preopen (Int(A C i) semiope
(A (C
Int
ii) open
close
v) open
close v) generaliz
(A)
C
and gen is gclo vi) semic
o semi vii) Dclo
ope D
viii) ND c
o D
N
efinition 2.2.
) cont
open set i) g-cont open set
y 2018
Computer
nfo
D MAPP
Manisha Department Govt. J. Y. ChhRaipur, Chh
n and closed f
lationships of t
tions between t
, 54D10.
lect all the de
2. PRELI
Let (X,) be X is said to be [14] if A
A. A))
en [11] if A
A.
(A)) .
n [16] if A ed if C(Int(C
[1] if A ed if Int(C(
closed d
ze ( b
U
whenever
open d neralize
osed . closed [17] if
open [18] if osed [20] if C
en if X\Ais closed [10] if open if X\A
A function f
tinuous [1] in Y is -op tinuous [4] [1 in Y is g-op
Science
ISSNPINGS
Shrivastava of Mathemati hattisgarh Col hattisgarh, Ind functions calledthese maps w
them are studied
efinitions whi
IMINARIES
e a topological e, (A)) (C Int a
A
C Int Int ( (C Int A (A)))
C .
(A (C Int ( C A (A))) Int (
briefly gclo r A Uand U
n (briefly
g
f Int(C(A)
Int ( * C
A
* (C Int ( *
C
closed
D .
f Int(C(Int
A is ND clo
(Y ) (X, : f
if the inver pen in X.
15] if the inv en in X .
N No. 0976-56
ics llege, dia
d NDopen with already ex
d.
ch will be us
l space. A sub
and preclosed
and semiclose
(A)))
t and
(A))) and A.
sed ) [12] if U is open in X
open
) if XA
)) and
(A))
t .
A (A)))
* an
A
(A))) and
osed.
)
Y, is said to
rse image of
verse image of
97
n and xisting
sed in
set A
if
ed if
Purushottam Jha International Journal of Advanced Research in Computer Science, 9 (1), Jan-Feb 2018,394-398
(iii) semi-continuous [19] if the inverse image of each
open set in Y is semi-open in X.
(iv) Dcontinuous [20] if the inverse image of each open set in Y is Dopen in X.
(v) ND continuous [10 ] if the inverse image of each open set in Y is ND open in X.
(vi) open [1] (resp. closed ) if the image of each open ( resp. closed ) set in X is open (resp.
closed
) in Y.
(vii) gopen [13] ( resp. gclosed ) if the image of each open ( resp. closed ) set in X is gopen ( resp.
closed
g
) in Y.(viii) semi-open ( resp. semi-closed ) [19] if the image of each open (resp. closed ) set in X is
open
-semi ( resp. semi-closed ) in Y.
(ix) D open [20] (resp. D closed ) if the image of each open ( resp. closed ) set in X is D open ( resp. D closed ) in Y.
Definition 2.3. [12] A topological space (X,) is said to
be T1/2 if every g-closedset is closed.
The intersection of all gclosed sets containing A [12] is called the gclosure of A and denoted by C
A and theerior
gint of A [12] is the union of all gopen sets contained in A and is denoted by Int
A . The intersection of all ND closed sets containing A [10] is called theclosure D
N of A and denoted by ND C
A and theerior D
N int of A [10 ] is the union of all NDopen sets contained in A and is denoted by ND Int
A .The collection of all ND closed (resp. closed ,
closed
-D , gclosed , closed , semi-closed) sets in )
(X, denoted by D C(X)
( resp., C(X) , DC(X) , )
(X
GC , C(X) , semiC(X) ). The collection of all open
-D ( resp. open , D open , gopen, open, open
-semi ) sets in (X,) denoted by DO(X) ( resp. O(X) , DO(X) , GO(X), O(X), semiO(X)). ( c.f. [2], [10] [17] ,[19], [20])
3.
Maps
Closed
β
D
N
We introduce the following definition
Definition 3.1. A map f :(X, ) (Y,) is said to be closed
D
N if f(V) is ND closed in Y for each closed set V in X.
Theorem 3.2. (i) Every closed map is closed
D
N .
(ii) Every gclosed map is ND closed.
(iii) Every semiclosed map is NDclosed.
(iv) Every D closed map is ND closed.
Proof. (i) The proof follows from the definition and from the Theorem 3.3 of [10] that every gclosed set is
closed D
N
(ii) The proof follows from the definition and from the Theorem 3.3 of [10] that every semi open set isND closed.
(iii) The proof follows from the definition and from the Theorem 3.3 of [10] that every open set is
closed D
N .
(iv) The proof follows from the definition and from the Theorem 3.3 of [10] that every Dclosed set is
closed D
N .
Remark. 3.3. (i) NDclosed map need not be closed . (see the Example 3.4 below)
(ii) ND closed map need not be gclosed. (see the Example 3.5 below)
(iii) NDclosed map need not be semiclosed (see the Example 3.6 below)
(iv) NDclosed map need not be D closed . (see the Example 3.7 below).
Example 3.4. Let X={a,b,c,d}Y ,
a,b,d
,{c,d},
d} b},{a, , {X,
=
and ={Y,, {a,b,d},{a}}
then (X,) and (Y,) be a topological spaces.
a,b,a,b,c
c , d , c, , Y,
C X C Y Y, ,c , b,c,d ,
b,c,c,d,b,d,a,c,d
,a,b,c
d , c, b, c , , Y, GC Y
, , , , , , }, )(Y ab ac b d
GO Y,, a,b,d , a , a,d
b,c c,d,b,d d ,b, c , d , c, b, , Y, C Y
,
a,c,d
,a,b,c
,a,d,a,b,a}, d , b , d c, c b, d}, {b, {c}, d}, c, {b, , {Y, = )
C(Y
D
N
Let f :(X, ) (Y,) be a map defined by f(a)d ,
c b
Purushottam Jha International Journal of Advanced Research in Computer Science, 9 (1), Jan-Feb 2018,394-398
)
(Y, . But map
f
is not closed, since b}{a, = d}) ({c,
f , which is not closed in Y.
Example 3.5. Let X={a,b,c} be any set with topology
a ,{a,b},
a,c} ,{X,
=
, then (X,) be a topological
space. Let Y={x,y,z}with topology ={Y,,{y,z},
y},then (Y,σ) be another topological space.
b} {c}}, c}, {b, , {X, =C(X)
,C(Y)={Y,,{x},
x,z},z}} {x, y}, {x, {x}, , {Y, = )
GC(Y
y}} {x, {y}, z}, {x, {z}, {x}, , {Y, = ) C(Y D
N .
Let f : X, Y, be the function defined by f(a)x,
z b f( )
and f(c)yis ND closed map, since f image of each open set in (X,) is NDclosed in (Y,). But map f is not gclosed, since f b,c y,z , which is
is not gclosed in Y. Example 3.6. Let X={a,b,c,d}
,
a, b,{a,b},
a,b,c
} ,{X,
=
and Y={1,2,3,4} ,
{2,3,4}} {2,4}, , {Y,
=
. Then (X,) and (Y,) be any two
topological space. C X X, , b,c,d , a,c,d ,
c,d, d},1 1,3 , , Y,
C Y .
1,2,1,2,3
,1,2,4
,1,3,4
1,4 , 1 , 1,3 , , Y,
GC Y ,
3,4 ,4 ,3 } {2,3},{2,3,4}, {2,4}, , {Y, = )
GO(Y ,
{3}} {1}, {1,3}, , {Y, = ) C(Y *
Semi ,
2,4,2,3,2,3,4,1,3,4
,3,4 }, 1,2,4 , 1,2,3 , 1,2 {1,4}, 1}, }, { 1,3 { , Y, { = ) C(Y D
N
.
Define a function f :(X, ) (Y,)
by, f(a)2, f(b)3,
4 ) (c
f , f(d)1, which is a NDclosed map, since
f
image of each closed set isNDclosed in (Y,). But f is not semiclosedmap, since {1,3,4}
= d}) c, ({b,
f , which is not semiclosed in (Y,).
Example 3.7. Let X={x,y,z} and X, ,y,z , z , then
)
(X, be a topological space. C Y Y, ,x , x,y
. Let
t} s, {r, =
Y and Y, ,s,t , s , t , then (Y,) be a
topological space. C Y Y, ,r , r,t , r,s ,
} {
)
(Y Y , ,r , r,t , r,s
GC ,
t s , t , s, , Y,
) (Y GO
} { , ,r , r,t , r,s
C Y
D Y ,
t s , t , s, , Y, DO(Y)
} { , , r , r,t ,r,s , s , t Y
D
N Y ,
t r, s , r, t , s , t , s, , Y, O Y D
N .
Let function f: X, Y, defined by
f
(
x
)
s
,t y
f( ) , f(z)r is NDclosed map, since the image of each closed set in X is NDclosed in Y , but f is not
closed
D , Since f(
x,y ={s,t} , which is notclosed D in Y .
Interrelationship
From the above discussions and known results, we have the
following implications.
Remark. 3.8. The composition of two NDclosed maps need not be NDclosed in general. This is shown by the following example.
Example 3.9. Let X=Y= Z={b,c,d}be the sets with the
topology ={X,,{c},{c,d}}, ={Y,,{b,c}}and
{d}} d}, {b, , {Z,
=
, respectively. Then (X,) , (Y,) and
)
(Z, be the topological spaces. C(X)={X,,{b,d},{b}} ,
} {d} , {Y, =
C(Y)
, C(Z)={Z,,{b,c},{c}}. We define a map
) (Y, ) (X, :
f as f(b)d , f(c)b and f(d)c the
map g :(Y,) (Z,) as g(b)c , g(c)d and g(d)b .
Then f and
g
are NDclosed maps, but their composition gof :(X, ) (Z,) is not D closed. For, if
b dA , be any closed set in (X,) and
d} {b, = d}) g({c, = ) d}) ({b, g( = (A) o
g f f , which is not a
closed D
Purushottam Jha International Journal of Advanced Research in Computer Science, 9 (1), Jan-Feb 2018,394-398
Theorem 3.10. If f :(X,) (Y,) is a closed map and )
(Z, ) (Y, :
g is D closed map, then their
composition gof :(X, ) (Z,) is ND closed map.
Proof. Let G be any closed set in (X,) . Since f is a closed map, f(G) is closed in (Y,) . Since g is
closed D
N map, g(f(G)) is ND closed set in (Z,)
. Therefore gof(G)=g(f(G))is ND closed set in (Z, ).
Theorem 3.11. If the space is T1/2 , then every
closed
D ( resp. NDclosed ) set is closed ( resp. closed).
Proof. Let
A
be any Dclosed( resp. NDclosed) subset of the space X, then we have (C(Int(C(A))) A ( resp.Int(C(Int(A))) A). Since X is T1/2space,every
g
closed
set is closed, consequently(A) C = (A)
C . Thus, we get
) A ) (A)) ( (C resp. ( A )) (A) (C (
CInt Int Int
Theorem 3.12. Let f :(X,) (Y,) and )
(Z, ) (Y, :
g be any two mappings such that their
composition gof :(X, ) (Z,)is ND closed mapping. Then the following statements are true :
1. If f is continuous map and surjective, then g is closed
D
N
mapping
2. If g is N D irresolute map and injective, then f is closed
D
N mapping.
3. If f is gcontinuous map , surjective and (X,) is space
T1/2 , then g is ND closed mapping.
Proof: 1. Let A be any closed set in (Y,). Since f is continuous, f-1(A) is ND closed in (X,)and therefore
) ) ( f ( o
g f -1 A is NDclosed in (Y,). Since f is surjective, g is NDclosed map.
2. Let F be any closed set in (X,). Since gof is closed
D
N mapping, gof(F) is ND closed in )
(Z, . Since g is ND irresolutemap, g-1(gof(F)) is closed
D
N in. (Y,). Since f is injective, f is closed
D
N mapping.
3. Let G be any closed set in (Y,). Since f is continuous
g , f-1(G) is gclosed in (X,). Since
)
(X, is T1/2space, f-1(G) is closed in (X,)
consequently gf(f-1(G)) is ND closed in (Z,). i.e. )
(
gG is NDclosed in (Z,), since f is surjective. It
Theorem 3.13. Let f :(X,) (Y,) be any closed
D
N
mapping. Then )) ( ( )) (
(f A f Cl A
Closure D
N .
Proof: Suppose f is NDclosed map and let A be
any subset of X. Since Cl(A) is the closed set in (X,),
)) ( (Cl A
f is NDclosed in (Y,). We have
)) ( ( )
(A f Cl A
f , therefore by Theorem 3.10 of [10]
) 1 ( ))) ( ( ( ))
(
(
Closure f A D Closure f Cl A
D N
N
. Since f(Cl(A))is ND closed,
) ( ( )) ( (
(f Cl A f Cl A
Closure D
N , we have from (1),
))) ( ( ( )) (
(f A f Cl A
Closure D
N
.
Remark 3.14. However, the converse of the above
Theorem 3.13 need not be true by the following example.
Example 3.15. Let X={a,b,c}, ={X,,{a,b},{a}}and
{1,2,3} =
Y , ={Y,,{2},{2,3}}. Then (X,)and (Y,)be
topological spaces. C(X)={X,,{b,c},{c}} ,
{1}} {1,3}, , {Y, = )
C(Y .
{b}} c}, {b, {a}, c}, {a, {c}, , {X, = C(X)
D ,
{3}} {2}, {1}, {1,2}, {1.3}, , {Y, = C(Y)
D . Let
) (Y, ) (X,
:
f be a function defined by f(a)1,
2 ) (b
f , f(c)3. Then
))) ( ( ( )) (
(f A f Cl A Closure
D
N for every subset A of
X . But f is not a NDclosedmapping, since
} 3 , 2 { }) , ({b c
f , which is not ND closed in (Y,) .
4. N
D
Open
Map
Definition 4.1. A function f :(X, ) (Y,) is said to be open
D
N if f(V) is NDopenin Y for each open set V in X.
Theorem 4.2. (i) Every open map is ND open.
(ii) Every gopen map isND open.
(iii) Every semiopen isNDopen.
(iv) Every Dopen map isND open.
Proof;- It is obvious.
Theorem 4.3. Let f :(X,) (Y,) be any bijective map, then the following statements are equivalent;
(i)f1 is ND continuous .
(ii) f is ND open map.
(iii) f is ND closed map.
Proof. (i)
(ii): Let V be any open set in (X,). By assumption (f-1)-1(V)=f(V) is ND open in (Y,) .Purushottam Jha International Journal of Advanced Research in Computer Science, 9 (1), Jan-Feb 2018,394-398
(ii)
(iii): Let G be any closed set in (X,). Thenc
G is open set in (X,), therefore by assumption
c c) (f(G))
f(G is ND open in (Y,) ,
consequently f(G) is ND closed in (Y,) .
Hence the map f isNDclosed.
(iii)
(iv): Let G be any closed set in (X, ). Then by assumption f(G) is ND closed in (Y,)and therefore f(G)=(f-1)-1(G) is ND closed in
)
(Y, , therefore f1 is D continuous
N .
Theorem 4.4. If a map f :(X,) (Y,) is ND open, then f(Int(A)) ND Int(f(A)) for every subset A of
) (X, .
Proof. Let f :(X,) (Y,) be any ND open map and suppose A be any open set in (X,), then Int(A)is open in (X,) . Therefore f(Int(A)) is ND open in
)
(Y, and therefore NDInt(f(Int(A)))f(Int(A))(1). Since Int(A) A , f(Int(A))f(A) , consequently
)) ( ( )))
( (
(f Int A D Int f A Int
D N
N . By (1), we have
)) ( ( ))
(
(Int A D Int f A
f N
.
Remark 4.5. However, the converse of the above
Theorem 4.4 need not be true by the following example.
Example 4.6. Let X={a,b,c}, ={X,,{a,b},{a}}and
{1,2,3} =
Y , ={Y,,{2},{2,3}}. Then (X,)and )
(Y, be any two topological spaces. {1,2}} {1,3}, {2,3}, {3}, {2}, , {Y, = O(Y)
D
.
Define a
function f :(X,) (Y,) by
f
(
a
)
1
, f(b)2 ,3 ) (c
f . Then f(Int(A)) ND Int(f(A)) for every
subset
A
of(X,
)
. But the mapf
is not ND open.Conclusion-The notion of lower separation axioms ,
closed graphs and strongly closed graphs, which are defined
in terms D open and D closedsets [9] can also be
established by using ND open and ND closedsets.
REFERENCES
[1]. M.E. Abd El-Monsef, S.N. El-Deeb, R.A. Mahmoud,
open
sets and continuous mappings, Bull. Fac. Sci. Assiut Univ., 12(1983), 77-90.
[2]. M.E. Abd El-Monsef, R.A. Mahmoud, E.R. Lashin,
closure
and interior, J. Fac. Ed. Ain Shams Univ., 10(1986), 235-245.
[3]. D. Andrijevic, Semi-preopen sets, Math.Vesnik, 38, 1(1986), 24-32.
[4]. K. Balachandran, P. Sundaram, H. Maki, On generalized continuous maps in topological spaces, Mem Fac Sci Kochi Univ (Math), 12, (1991), 513.
[5]. S.G. Crosseley, S.K. Hilderbrand, semi-topological properties, fund math, 74, (1972) , 233-254.
[6]. W. Dunham, T1/2 spaces , Kyngpook Math. J., 17,
(1977), 161-169.
[7]. W. Dunham, N. Levine, Further results on generalized closed sets in topology, Kyungpook Math J., 20, (1980), 169-75. [8]. W. Dunham, A new closure operator for non
ies Topo
T1 log , Kungpook Math .J., 22, (1982), 55-60. [9]. P. Jha, M. Shrivastava, Some functions defined by
closed
D set, International Journal of Advanced Research in Computer Science, 8 , 7( 2017), 887-895.
[10]. P. Jha, M. Shrivastava, A new generalized continuity by using generalized-closure operator, International Journal of Mathematics and its Application .(Accepted)
[11]. N. Levine, Semiopen sets and semicontinuity in topological space , Amer. Math. Monthly. 70 , (1963), 36-41.
[12]. N. Levine, Generalized closed sets in topology, Rend. Circ. Mat. Palermo(2), 19, (1970), 89-96.
[13].S. R. Malghan, Generalized closed maps, J. Karnatak Univ. Sci., 27, (1982), 828.
[14].A. S. Mashhour, M. E. Abd El-Monsef, S. N. El-Deeb, On precontinuous and weak precontinous mappings, Proc. Math. Phys. Soc. Egypt, 53, (1982), 47-53.
[15].B. M. Munshi, D. S. Bassan, g-continuous mappings, Vidya J. Gujarat Univ., 24, (1981), 63-68.
[16].O. Njstad, On some classes of nearly open sets, Pacific J. Math , 15, (1965), 961-70.
[17]. A. Robert, S. P. Missier, On semiclosed sets, Asian Journal of Current Engineering and Maths, 4(2012), 173-176. [18]. S.P.Missier, P.A.Rodrigo, Some notions of nearly open sets
in topological spaces, International Journal of Mathematical Archive, 12, 4(2013), 12-18.
[19].S.P. Missier, A. Robert, Functions Associated with
open semi
Sets, International Journal of Modern Science and Engineering, 1, 2( 2014), 39-46.
