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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 03, March 2019, pp. 650-659. Article ID: IJMET_10_03_068
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=3
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
CONTINUITY ON N-ARY SPACES
P.Thangavelu
Retired Professsor, Karunya University, Coimbatore.
R.Seethalakshmi
Dept. of Mathematics, Jaya collage of arts and Science, Thiruninravuir - India.
M.Kamaraj
Department of mathematics, Govt. Arts and Science Collage, Sivakasi- Tamilnadu, India.
ABSTRACT
Continuous functions play a dominant role in analysis and homotopy theory. They
have applications to image processing, signal processing, information, statistics,
engineering and technology. Recently topologists studied the continuous like functions
between two different topological structures. For example, semi continuity between a
topological structure, α-continuity between a topology and an α-topology.
Nithyanantha Jothi and Thangavelu introduced the concept of binary topology in
2011. Recently the authors extended the notion of binary topology to n-ary topology
where n˃1 an integer. In this paper continuous like functions are defined between a
topological and an n-ary topological structures and their basic properties are
studied.
Keywords: n-ary topology, n-ary open, n-ary closed and n-ary continuity. MSC 2010:
54A05, 54A99.
Cite this Article: P.Thangavelu, R.Seethalakshmi and M.Kamaraj, Continuity on N-
Ary Spaces, International Journal of Mechanical Engineering and Technology, 10(3),
2019, pp. 650-659.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=3
1. INTRODUCTION
Nithyanantha Jothi and Thangavelu [5-10] introduced the concept of a binary topology and
studied the corresponding closure and interior operators in binary topological spaces.
Following this topologists studied the notion of binary topology in soft topological,
generalized topological and supra topological settings {1-4]. The The second and the third
authors [11] discussed nearly open sets in binary topological spaces and also they extended
binary topology to n-ary topology [12] and discussed n-ary closed sets [13]
Continuity on N-Ary Spaces
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2. PRELIMINARIES
Let X1, X2, X3,…,Xn be the non empty sets. Then P(X1)P(X2)…P(Xn) is the Cartesian
product of the power sets P(X1), P(X2) ,…,P(Xn). Examples can be constructed to show that
the two notions „product of power sets‟ and „power set of the products‟ are different. Any
typical element in P(X1)P(X2)…P(Xn) is of the form (A1,A2,…,An) where AiXi for
i{1,2,3,…,n}. Suppose (A1,A2,…,An) and (B1,B2,…,Bn) are any two members in
P(X1)P(X2)…P(Xn). Throughout this paper we use the following notations and
terminologies.
(X1, X2,…,Xn) is an n-ary absolute set and (, , ,…,) is an n-ary null set or void
set or empty set in
P(X1)P(X2) …P(Xn).
(A1,A2,…,An)(B1, B2,…,Bn) if AiBi for every i{1,2,3,…,n} and
(A1,A2,…,An)(B1,B2,…,Bn) if AiBi for some i{1,2,3,…,n}.
Equivalently (A1,A2,…,An)=(B1, B2,…,Bn) if Ai=Bi for every i{1,2,3,…,n}. If AiBi for
each i{1,2,3,…,n} then we say that (A1,A2,…,An) is absolutely not equal to (B1, B2,…,Bn).
Let xiXi and AiXi for every i{1,2,3,…,n}. Then (x1, x2,…,xn)(A1, A2,…, An) if xiAi
for every i{1,2,3,…,n}. Let T P(X1)P(X2)…P(Xn).
Definition 2.1: T is an n-ary topology on (X1, X2, X3, …, Xn) if the following axioms are
satisfied.
(i). (, , ,…,) = T
(ii). (X1, X2, X3… Xn) T
(iii). If (A1, A2… An), (B1, B2, …, Bn) T then
(A1, A2… An)(B1, B2,…,Bn) T
(iv). If (A1, A2… An) T for each  then
)A..., ,A, (A n21
T .
If T is an n-ary topology then the pair (X, T ) is called an n-ary topological space where X
=(X1, X2, …, Xn). The element x=(x1,x2,…,xn)X is called an n-ary point of (X, T ) and the
members A=(A1,A2,…,An) of P(X1)P(X2)…P(Xn). Are called the n-ary sets of (X, T).
The members of T are called the n-ary open sets in (X, T). It is noteworthy to see that
product topology on X1X2…Xn and n-ary topology on (X1, X2, …, Xn) are independent
concepts as any open set in product topology is a subset of X1X2X3…Xn and an open set
in an n-ary topology is a member of P(X1)P(X2)…P(Xn).
Lemma 2.2: Let T be an n-ary topology on (X1, X2,…,Xn). T 1 ={A1: (A1, A2,…,A2) T
} and T 2 , T 3 , …, T n can be similarly defined. Then T 1 , T 2 ,, …, T n are topologies on X1,
X2, …,Xn respectively.
Lemma 2.3: Let 1, 2, …,n be the topologies on X1, X2, …,Xn respectively.
Let T = 12 …n = {(A1, A2,…,A2): Aii }. Then T is an n-ary topology on X=(X1,
X2,…,Xn). Moreover T i = i for every i{1,2,3,…,n}.
P.Thangavelu, R.Seethalakshmi and M.Kamaraj
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Definition 2.4: Let (A1, A2,…,An) be an element of P(X1)P(X2)…P(Xn). Then (A1,
A2,…,An) is n-ary closed in (X1, X2, X3,…,Xn ; T ) if (A1, A2,…,An)c = (X\A1,
X\A2,…,X\An) = XA is n-ary open in (X, T ).
For the notations and terminologies that are used here the reader may consult [11,12, 13].
3. N-ARY CONTINUOUS FUNCTIONS
Let X =(X1, X2, X3, …, Xn) , f:XYi be a function and V be a subset of Yi . Then f-1(V)
need not be an n-ary element in X as seen in the following examples.
Example 3.1: Let X1= {a1,b1,c1}, X2= {a2,b2,c2} and Y1= {1,2,3}. Then X =(X1, X2). Let
A be an n-ary element in X. Define f:XY1 by f(x1,x2) =1 when (x1,x2) A and f(x1,x2) =2
when (x1,x2) ɇ A. Then it is easy to see that f-1({1}) = A, f-1({2}) = XA, f-1({3}) = that
implies f-1(V) is an n-ary element of X for every subset V of Y1.
Example 3.2: Let X =(X1, X2, …, Xn) and f:XX1 be a function defined byf(x1, x2, …,
xn) = x1 . Then it is easy to see that f-1({x1}) =({x1}, X2, X3, …, Xn) that implies f-1(V) is an
n-ary element of X for every subset V of X1.
Example 3.3: Let X1= {1,2,3}, X2= {4,5,6} and Y1= {0,1,2}. Then X =(X1, X2). Define
f:XY1 by f(x1,x2) = x1+3 x2 where +3 is the addition modulo 3. Then it is easy to see
thatf-1({0}) = { (1,5), (2,4),(3,6) }; f-1({1}) = { (1,6), (2,5),(3,4) } and f-1({2}) = { (1,4),
(2,6),(3,5) } that implies f-1(V) is not an n-ary element of X for every subset V of Y1.
Example 3.4: Let X1= {1,2,3}, X2= {4,5,6} and Y1= {1,2,3}. Then X =(X1, X2). Define
f:XY1 by f(x1,x2) = g.c.d ( x1 ,x2 ). Then it is easy to see that f-1({1}) = { (1,4), (1,5),(1,6),
(2,5),(3,4), (3,5), }; f-1({2}) = { (2,4),(2,6) } = ({2}, {4,6}) and f-1({3}) = { (3,6)}= ({3},
{6}) that implies f-1(V) is an n-ary element of X for some subsets V of Y1.
Thus we infer from the above examples that f-1(V) may be an n-ary element of X under
the function f:XY1. This leads to the following definitions.
Definition 3.5: The function f:XY1 is an n-ary function if f-1(V) is an n-ary element in
X for every subset V of Y1.
Definition 3.6: The function f:XY1 is n-ary continuous from (X, T) to (Y1, 1) if for
every V1, f-1(V)T whenever f-1(V) is an n-ary element in X. f:XY1 is known as (T ,
1)-continuous from X to Y1.
Examples 3.7:
(i).Every constant function f:XY1 is (I , 1)-continuous from X to Y1.
(ii). Every n-ary function f:XY1 is (D , 1)-continuous from X to Y1..
(iii). Every n-ary function f:XY1 is (T , 1)-continuous from X to Y1 ,where
T={}{A : aA} is the a-inclusion n-ary topology, 1 is p-inclusion topology and f(a) =p.
(iv). Every n-ary function f:XY1 is (T , 1)-continuous from X to Y1 ,where T={}{
A : BA }is the B-inclusion n-ary topology, 1 is the p-inclusion topology and f(b)=p for
every bB.
(v). Every n-ary function f:XY1 is (T , 1)-continuous from X to Y1 ,where T={}{
A : aɇA } is the a-exclusion n-ary topology, 1 is p-exclusion topology and f(a)p.
Notation: For each j{1,2,3,…,n} let Aj = ( (Aj)1, (Aj)2, …, (Aj)n ) where (Aj)j = Aj and
(Aj)i = Xi for ij.
Continuity on N-Ary Spaces
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Proposition 3.8: Let T be an n-ary topology on X such that AiT for every AT. Then
the projection pi:XXi is (T, T i)-continuous from (X, T) to (Xi, T i) for each i{1,2,3,…,n}.
Proof: Let AiT i and AT with Ai=ith coordinate of A and AiT i. Then
(pi)-1(Ai) = Ai T . This shows that pi:XXi is (T, T i)-continuous from (X, T) to (Xi, T i ).
The condition that ”AiT “ cannot be dropped from the above proposition as shown
below.
Example 3.9: Let X = (X1,X2) and A = (A1, A2). If T = {(,), A, (A1, X2), (X1, A2),
(X1,X2) then the projections p1:XX1 and p2:XX2 are 2-ary continuous. If T = {(,),
A, (X1,X2) then the projections p1:XX1 and p2:XX2 are not 2-ary continuous.
Theorem 3.10: Let f:XY1 be an n-ary function. The following are equivalent.
(i).The function f is (T , 1)-continuous from X to Y1 .
(ii).The inverse image of each closed set in (Y1, 1 ) is n-ary closed in (X,T).
(iii). The inverse image of each basic open set in (Y1, 1 ) is n-ary open in (X,T).
(iv). The inverse image of each subbasic open set in (Y1, 1 ) is n-ary open in (X,T).
(v). For each n-ary point x in X and each nbd W off (x) in (Y1, 1 ) there is an n-ary nbd V
of x in (X,T) such that f(V(x)) W(f(x)).
(vi). For every n-ary element A of X, f(n-ClA) ))Cl(f(A)).
(vii). For every subset W of Y1, n-Cl (f-1(W)) f-1(ClW)
Proof: Let f be (T , 1)-continuous from X to Y1. Let F be a closed set in (Y1, 1) so that
G = Y1\F is open in (Y1, 1) . Since f is (T , 1)-continuous, f-1(G) is n-ary open in (X,T) . Let
f-1(G) = (A1, A2, A3, …, An) so that
f-1(G) = f-1(Y1\F) = X f-1(F) = (A1, A2, …, An) is n-ary open in (X,T). Therefore X
(X f-1(F) )= X (A1, A2, …, An) is n-ary closed in (X,T) so that f-1(F) ) is n-ary closed in
(X,T). This proves (i)(ii). Similarly we can prove that (ii)(i).Also (i) (iii) and (iii)
(iv) follow easily from the fact that basic open sets in (Y1, 1) are open sets and subbasic
open sets are basic open sets.
Now suppose (i) holds. Let xX and W be a nbd off(x) in (Y1, 1). Since f is (T , 1)-
continuous, f-1(W) is n-ary open in (X,T) with x f-1(W). Therefore there is an n-ary open set
V in (X,T) with xVf-1(W) that implies f(V)f(f-1(W)) W. This proves (i) (v).
Conversely suppose (v) holds. If W is an open set in (Y1, 1) and if x f-1(W) then f(x)W
which implies by (v) that there is an n-ary open set V in (X,T) with xV and f(V)W so
xVf-1(W) that implies f-1(W) is n-ary open in (X,T). This proves (v) (i).
Now suppose (ii) holds. Let A be an n-ary element of X. Then Cl(f(A)) is closed in (Y1,
1). By using (ii), f-1(Cl(f(A)) ) is n-ary closed in (X,T). Since A f-1(f(A)) f-
1(Cl(f(A))) and since f-1(Cl(f(A)) ) is n-ary closed in (X,T) we have n-ClA f-1(Cl(f(A))) so
that f(n-ClA)f (f-1(Cl(f(A))) that implies f(n-ClA) ))Cl(f(A)). This proves (ii) (vi).
Suppose (vi) holds.
Let W be a subset of Y1. Then f-1(ClW) is an n-ary element of X so that by taking A=f-
1(ClW) in (vi) we get f(n-Cl(f-1(ClW)))Cl(f(f-1(ClW))) ClW that implies f(n-Cl(f-
1(W)))f(n-Cl(f-1(ClW)) )ClW so that
P.Thangavelu, R.Seethalakshmi and M.Kamaraj
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f-1(f(n-Cl(f-1(W)) ))f-1(ClW). Therefore n-Cl (f-1(W))f-1(ClW). This proves (vi) (vii).
Now suppose (vii) holds. Then if F is closed in (Y1, 1) then by taking W = F in (vii) we get n-
Cl (f-1(F)) f-1(ClF) = f-1(F) that implies f-1(F) is n-ary closed in (X,T). This proves that (vii)
(ii).
4. N-ARY IRRESOLUTNESS
Let X =(X1, X2, X3, …, Xn) and Y =(Y1, Y2, Y3, …, Yn) . If f:XY is a function then for
every n-ary element V of Y, f-1(V) need not be an n-ary element in X as seen in the following
examples.
Example 4.1:Let X1= {a1,b1,c1}, X2= {a2,b2,c2} , and Y1= {1,2,3}=Y2. Then X =(X1, X2)
and. Y = (Y1, Y2). Let A be an n-ary element in X. Define f:XY1 by f(x1,x2) = (1,1) when
(x1,x2) A and f(x1,x2) =(2, 2) when (x1,x2) A. Then it is easy to see that f-1({(1,1)}) = A is
an n-ary element, f-1({(2,2)}) = Ac is not an n-ary element and f-1({(i,j)}) = is an n-ary
element when (i,j) (1,1), (2,2). Define g:XY by g(x1,x2) = (1,1) when (x1,x2) A,
g(x1,x2) =(1, 2) when (x1,x2) (X1\A1, A2), g(x1,x2) =(2, 1) when (x1,x2) (A1, X2\A2) and
g(x1,x2) =(3, 3) when (x1,x2) (X1\A1, X2\A2)
Then it is easy to see that g-1({(1,1)})=A, g-1({(3,3)})=(X1\A1, X2\A2), g-1({(1,2)})=
(X1\A1, A2) and g-1({(2,1)}) = (A1, X2\A2) that implies g-1(V) is an n-ary element of X for
every n-ary element V of Y.
Example 4.2: Let X =(X1, X2, X3, …, Xn) and Y =(Y1, Y2, Y3, …, Yn). Let fi:XiYi be
the given functions. Define f:XY by f(x1, x2, …, xn) = (f1(x1), f2(x2), …, fn(xn)) . Then it is
easy to see that
f-1(V) =((f1)-1(V1), (f2)-1(V2), …, (fn)-1(Vn)) that implies f-1(V) is an n-ary element of X
for every n-ary element V of Y.
Thus we infer from the above examples that f-1(V) may be an n-ary element of X under
the function f:XY1. This leads to the following definitions.
Definition 4.3: The function f:XY is an n-ary function if f-1(V) is an n-ary element in
X for every n-ary element V of Y.
Let (X, TX) and (Y, TY ) be any two n-ary spaces.
Definition 4.4: f: X Y is n-ary irresolute from (X,TX) to (Y, TY ) if for every VTY, f-
1(V) TX whenever f-1(V) is an n-ary element in X. It is also called f is a (TX,TY)-irresolute
function from X to Y.
Example 4.5: Let g:XY be a function as defined in Example 5.2.1. Let TX ={(,),
(A1, A2), (X1, X2)} and TY ={(,), ({1}, {1}), (Y1, Y2)}. Then g is (TX,TY)-irresolute from
X to Y.
Proposition 4.6: Let f:XY be a function as defined in Example 5.2.2. Let ViTY
whenever Vi (TY)i . Then f is (TX, TY)-irresolute from X to Y if and only if each fi is ((TX)i,
(TY)i) is continuous from Xi to Yi.
Proof: Suppose f is (TX, TY)-irresolute from X to Y. Let Vi (TY)i . Then ViTY . Since f
is (TX, TY)-irresolute from X to Y, f-1(Vi) TX that implies (fi)-1(Vi) (TX)i . This proves that
fi is ((TX)i, (TY)i) is continuous from Xi to Yi. Conversely let each fi be ((TX)i, (TY)i) be
continuous from Xi to Yi. Let VTY. Then Vi(TY)i . Since fi is ((TX)i, (TY)i) is continuous
from Xi to Yi, (fi)-1(Vi) (TX)i that implies f-1(V)TX .