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ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962
Volume-7, Issue-1, January-February 2017
International Journal of Engineering and Management Research
Page Number: 49-51
Nano Star Pre Generalized Continuous Functions in Nano Topological
Spaces
Nancy J E1, Vaiyomathi K2
PG Student, Nirmala College for Women, Coimbatore, Tamil Nadu, INDIA
2
Assistant Professor, Nirmala College for Women, Coimbatore, Tamil Nadu, INDIA
1
ABSTRACT
The purpose of this paper is to introduce, Nano
star pre generalized continuous function in Nano
Topological Spaces.
Keywords-- Nano topological space, Nano*pg closed,
Nano*pg continuous function
I.
INTRODUCTION
Continuous function is one of the main
concepts of Topology. Balachandran et al. [2] and
Mashouret al. [3] have introduced g-continuous and precontinuous function in topological spaces respectively.
Arokiarani [1] introduced generalized pre-continuous
functions and generalized pre-irresolute functions and
compared with various stronger forms of the same
functions. The notion of Nano topology was introduced (i)
by LellisThivagar[5] which was defined in terms of (ii)
approximations and boundary region of a subset of an (iii)
universe using an equivalence relation on it and he also (iv)
defined Nano closed sets, Nano-interior, Nano-closure,(v)
Nano continuous functions, Nano open mapping, Nano(vi)
closed mapping and Nano Homeomorphism. Nancy et(vii)
al[4] introduced and studied some properties of Nano
star pre generalized closed sets in Nano topological
spaces. In this paper, a new class of continuous functions(ix)
called Nano star pre generalized continuous function is(x)
introduced and some of its properties in terms of Ng –
closed sets, Ng-closure and Ng- interior are discussed.
II.
PRELIMINARIES
(i)
Definition:2.1:[6]A subset A of (U,
) is called(ii)
Nano pre generalized closed set (briefly Npg-closed) if
Npcl(A)⊆V whenever A⊆V and V is Nanopre open in(iii)
(U,
).
Definition:2.2[7] Let U be a non-empty finite set of
objects called the universe and R be an equivalence
relation on U named as the indiscernibility relation. Then
U is divided into disjoint equivalence classes. Elements
49
belonging to the same equivalence class are said to be
indiscernible with one another. The pair (U, R) is said to
be the approximation space. Let X⊆ U
1. The lower approximation of X with respect to R is the
set of all objects, which can be for certainly classified as
X with respect to R and it is denoted by
. That is
⊆ } , where R(x) denotes
⋃ {
the equivalence class determined by X U.
2. The upper approximation of X with respect to R is the
set of all objects, which can be possibly classified as X
with respect to R and it is denoted by
. That is
}
⋃ {
3. The boundary region of X with respect to R is the set
of all objects, which can be classified neither as X nor as
not-X with respect to R and it is denoted by
. That
is
Definition:2.3[7] If (U,R) is an approximation space and
X,Y⊆U, then
⊆ ⊆
⊆
⊆
(viii)
[
[
] .
⊆
]
⊆
Definition:2.4[7] Let U be non-empty, finite universe of
objects and R be an equivalence relation on U. Let X⊆U.
{
}. If
Let
satisfies the following axioms:
.
is in
.
The intersection of the elements of any finite sub
collection of
is in
Then
is a topology on U called the nano
topology on with respect to X. The elements of
are
called as nano-open sets.
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ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962
Definition:2.5[7]:A subset of (U,
) is called
Nanogeneralized closed set (briefly Ng-closed) if
Ncl(A)⊆V whenever A⊆V and V is Nano open set in
(U,
).
Definition:2.6[4] A subset A of (U,
) is called
Nano*pre generalized closed set (briefly N*pg closed) if
Nint(Npcl(A))⊆V whenever A⊆V and V is Nano pre
open in (U,
).
Definition:2.7[3] Let (U,
and (V,
be two
nano topological spaces. Then a mapping f:
(U,
(V,
is nano continuous function on
U if the inverse image of every nano open set in
(V,
is nano open in (U,
.
Definition:2.8[9] Let (U,
and (V,
be two
nano topological spaces. Then a mapping f:
(U,
(V,
is nano generalized continuous
function (shortly Ng-continuous) on U if the inverse
image of every nano open set in (V,
is nano
generalized open in (U,
.
III.
PROPERTIES OF NANO STAR
PRE GENERALIZED CONTINUOUS
FUNCTION IN NANO TOPOLOGICAL
SPACE
Definition:3.1 Let (U,
and (V,
be two
nano topological spaces. Then a mappingf:(U,
(V,
is nano*pre generalized continuous (shortly
N*pg continuous) function on U if the inverse image of
every nano open set in
(V,
is nano*pre
generalized open in (U,
.
Example:3.2
Let
U={a,b,c,d}
with
U/R={{a},{c},{b,d}}
and
X={a,b}.
Then
={U, {a,b,d},{b,d}}.
Then
nano* pre generalized open sets are:
{U, ,{a},{b},{c},{d},{a,b},{b,c},{c,d}{a,d},
{b,c},{a,c},{a,b,c},{a,b,d},{b,c,d}{a,c,d}}.Let
V={x,y,z,w} with V/R´={{x},{y,z},{w}} and Y={x,z}.
Then
{V, { } {
},{y,z}}
and
={ V, {y,z,w},{w},{x,w}} Define: f:
(U,
(V,
as f(a)=x, f(b)=y, f(c)=w,
{
}
{ }
f(d)=z Then
{ }
That is the inverse
image of every nano open set in V is nano* pre
generalized open sets in U. Therefore f is nano* pre
generalized continuous.
Theorem:3.3 Every nano continuous function is
nano*pre generalized continuous.
Proof: Let f: (U,
(V,
be nano
continuous on (U,
. Since f in nano continuous of
(U,
), the inverse image of everynano open set in
(V,
is nano open in (U,
. But every nano
open set is nano*pre generalized open set. Hence the
inverse image of every nano open set (V,
is
nano*pre generalized open in (U,
. Therefore, f is
nano*pre generalized continuous.
Remark:3.4 The converse of the above theorem is not
true as seen from the following example.
50
Example:3.5
Let
U={a,b,c}=V.
Then
={U, {a,c}} with U/R={{a,c},{b}} and X={a,c}
{ } { }} with V/R´={{b},{a,c}}
and
{
and Y={a,b} Define f: (U,
(V,
as
f(a)=c, f(b)=b, which is nano*pre generalized
continuous. But for the nano open set {a,c} in
(V,
its inverse
{
} is not nano
open in (U,
.Hence f is not nano continuous.
Theorem:3.6 Every nano pre continuous function is
nano*pre generalized continuous.
Proof: Let f: (U,
(V,
be nano pre
continuous on (U,
. Since f in nano pre continuous
of (U,
), the inverse image of every nano open set
in (V,
is nano open in (U,
. But every
nano pre open set is nano*pre generalized open set.
Hence the inverse image of every nano open set
(V,
is nano*pre generalized open in (U,
.
Therefore, f is nano*pre generalized continuous.
Remark:3.7 The converse of the above theorem is not
true as seen from the following example.
Example:3.8 Let U={a,b,c}=V. Then
={U, {a}}
with U/R={{a},{b,c}} and X={a} and
{ } { }}with V/R´={{b},{a,c}} and Y={a,b}
{
Define f: (U,
(V,
is an identity
mapping. Then f is nano*pre generalized continuous.
But for the nano closed set {b} in (V,
its inverse
{ } is not nano pre open in (U,
. Hence
f is not nano pre continuous.
Theorem:3.9 Every nano generalized
continuous
function is nano*pre generalized continuous.
Proof: Let f: (U,
(V,
be nano
generalized continuous on (U,
. Since f in nano
generalized continuous of (U,
), the inverse image
of every nano open set in (V,
is nano open in
(U,
.But every nano generalized open set is
nano*pre generalized open set.Hence the inverse image
of every nano generalized open set (V,
is
nano*pre generalized open in (U,
.Therefore, f is
nano*pre generalized continuous.
Remark:3.10 The converse of the above theorem is not
true as seen from the following example.
Example:3.11Let
U={a,b,c}=V.
Then
={U, {a,c}} with U/R={{a,c},{b}} and X={a,c}
{ } { }} with V/R´={{b},{a,c}}
and
{
and Y={a,b} Define f: (U,
(V,
as
f(a)=c, f(b)=b, which is nano*pre generalized
continuous. But for the nano open set {a,c} in
(V,
its inverse
{ } is not nano
generalized open in (U,
.Hence f is not nano
generalized continuous.
Theorem:3.12 A function f: (U,
(V,
is
nano*pre generalized continuous iff the inverse image of
every nano open set in
(V,
is nano*pre
generalized open in (U,
.
Proof: Let f: (U,
(V,
be nano*pre
generalized continuous and F be nano open in
(V,
. Then
is nano closed in (V,
. Since
f is nano*pre generalized continuous.
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ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962
is nano*pre generalized closed in
.But
Therefore,
is nano*pre generalized open in (U,
.Thus the inverse image of every nano open set in
(V,
is nano*pre generalized open in (U,
if
f is nano*pre generalized continuous on (U,
.
Conversely, Assume that
is nano*pre
generalized open in (U,
for each nano open set in
(V,
Let G be a nano closed set in (V,
.
Then
is nano open in (V,
and by assumption,
is nano*pre generalized continuous on
(U,
. Since
=
we have
is nano*pre generalized continuous on (U,
.
Therefore f nano*pre generalized continuous.
(U,
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