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CONTINUITY ON N-ARY SPACES

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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 NAry Spaces, International Journal of Mechanical Engineering and Technology, 10(3),
2019, pp. 650-659.
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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]
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Continuity on N-Ary Spaces
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 AiXi 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 AiBi for every i{1,2,3,…,n} and
(A1,A2,…,An)(B1,B2,…,Bn) if AiBi 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 AiBi for
each i{1,2,3,…,n} then we say that (A1,A2,…,An) is absolutely not equal to (B1, B2,…,Bn).
Let xiXi and AiXi for every i{1,2,3,…,n}. Then (x1, x2,…,xn)(A1, A2,…, An) if xiAi
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
 (A1 , A 2 , ..., A n )
 
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 X1X2…Xn and n-ary topology on (X1, X2, …, Xn) are independent
concepts as any open set in product topology is a subset of X1X2X3…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}.
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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) = X⃠A 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:XYi 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:XY1 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}) = X⃠A, 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:XX1 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:XY1 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:XY1 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:XY1. This leads to the following definitions.
Definition 3.5: The function f:XY1 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:XY1 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:XY1 is known as (T ,
1)-continuous from X to Y1.
Examples 3.7:
(i).Every constant function f:XY1 is (I , 1)-continuous from X to Y1.
(ii). Every n-ary function f:XY1 is (D , 1)-continuous from X to Y1..
(iii). Every n-ary function f:XY1 is (T , 1)-continuous from X to Y1 ,where
T={}{A : aA} is the a-inclusion n-ary topology, 1 is p-inclusion topology and f(a) =p.
(iv). Every n-ary function f:XY1 is (T , 1)-continuous from X to Y1 ,where T={}{
A : BA }is the B-inclusion n-ary topology, 1 is the p-inclusion topology and f(b)=p for
every bB.
(v). Every n-ary function f:XY1 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
j
(A )i = Xi for ij.
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Proposition 3.8: Let T be an n-ary topology on X such that AiT for every AT. Then
the projection pi:XXi is (T, T i)-continuous from (X, T) to (Xi, T i) for each i{1,2,3,…,n}.
Proof: Let AiT i and AT with Ai=ith coordinate of A and AiT i. Then
(pi)-1(Ai) = Ai T . This shows that pi:XXi is (T, T i)-continuous from (X, T) to (Xi, T i ).
The condition that ”AiT “ 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:XX1 and p2:XX2 are 2-ary continuous. If T = {(,),
A, (X1,X2) then the projections p1:XX1 and p2:XX2 are not 2-ary continuous.
Theorem 3.10: Let f:XY1 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) = X⃠ f-1(F) = (A1, A2, …, An) is n-ary open in (X,T). Therefore X⃠
(X⃠ f-1(F) )= X⃠ (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 xX 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 xVf-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 xV and f(V)W so
xVf-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))  f1
(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=f1
(ClW) in (vi) we get f(n-Cl(f-1(ClW)))Cl(f(f-1(ClW))) ClW that implies f(n-Cl(f1
(W)))f(n-Cl(f-1(ClW)) )ClW so that
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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 nCl (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:XY 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:XY1 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:XY 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:XiYi be
the given functions. Define f:XY 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:XY1. This leads to the following definitions.
Definition 4.3: The function f:XY 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 VTY, f1
(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:XY 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:XY be a function as defined in Example 5.2.2. Let ViTY
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 ViTY . 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 VTY. 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 .
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Theorem 4.7: Let f:XY be an n-ary function. The following are equivalent.
(i).The function f is (TX,TY)- irresolute from X to Y. .
(ii).The inverse image of each n-ary closed set in (Y, TY ) is n-ary closed in (X,TX).
(iii). The inverse image of each basic n-ary open set in (Y, TY ) is n-ary open in (X,TX)
(iv). The inverse image of each subbasic n-ay open set in (Y, TY ) is n-ary open in (X,TX)
(v). For each n-ary point x in X and each n-ary nbd W of f(x) in (Y, TY ) there is an n-ary
nbd V of x in (X,TX) such that f(V(x))  W(f(x)).
(vi). For every n-ary element A of X, f(n-ClA) )) n-Cl(f(A)).
(vii). For every n-ary subset B of Y , n-Cl (f-1(B)) f-1(n-ClB)
Proof: Let f be (TX,TY)- irresolute from X to Y. Let F be n-ary closed in (Y, TY) so that G
= Y⃠F is n-ary open in (Y, TY). Since f is (TX,TY)- irresolute, f-1(G) is n-ary open in (X, TX)
.
f-1(G) = f-1(Y⃠F) = X⃠ f-1(F) is n-ary open in (X, TX) that implies f-1(F) ) is n-ary
closed in (X, TX). This proves (i)(ii). Similarly we can prove that (ii)(i). Also (i) (iii)
and (iii) (iv) follow easily from the fact that n-ary basic open sets in (Y, TY) are n-ary
open sets and n-ary subbasic open sets are n-ary basic open sets.
Now suppose (i) holds. Let xX and W be a nbd of f(x) in (Y, TY). Since f is (TX,TY)irresolute, f-1(W) is n-ary open in (X, TX) with xf-1(W). Therefore there is an n-ary open set
V in (X, TX) with xVf-1(W) that implies f(V)f(f-1(W)) W. This proves (i) (v).
Conversely suppose (v) holds. If W is an n-ary open set in (Y, TY) 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, TX) with xV and
f(V)W so xVf-1(W) that implies f-1(W) is n-ary open in (X, TX). This proves (v) (i).
Now suppose (ii) holds. Let A be an n-ary element of X. Then Cl(f(A)) is n-ary closed in
(Y, TY). By using (ii), f-1(n-Cl(f(A))) is n-ary closed in (X, TX).
Since A  f-1(f(A) )  f-1(n-Cl(f(A)) ) and since f-1(n-Cl(f(A)) ) is n-ary closed in (X, TX)
we have n-Cl A  f-1(n-Cl(f(A)) ) so that f(n-Cl A) f (f-1(n-Cl(f(A))) that implies f(n-ClA)
))n-Cl(f(A)). This proves (ii) (vi). Suppose (vi) holds.
Let W be an n-ary member of Y. Then f-1(n-ClW) is an n-ary element of X so that by
taking
A=f-1(n-ClW) in (vi) we get f(n-Cl(f-1(n-ClW)))n-Cl(f(f-1(ClW)))n-ClW that implies
f(n-Cl(f-1(W)))f(n-Cl(f-1(n-ClW)) )n-ClW so that f-1(f(n-Cl(f-1(W)) ))f-1(n-ClW).
Therefore
n-Cl (f-1(W))f-1(n-ClW). This proves (vi) (vii). Now suppose (vii) holds. Then if F is
n-ary closed in (Y, TY) then by taking W = F in (vii) we get n-Cl (f-1(F)) f-1(n-ClF) = f-1(F)
that implies f-1(F) is n-ary closed in (X, TX). This proves that (vii) (ii).
5. THE OPEN AND CLOSED FUNCTIONS ON N-ARY SPACES
Let (X,TX) and (Y, TY ) be the n-ary spaces and let (Yi, i) be a topological space. Let
f:XY1 and g: X Y be the functions .
Definition 5.1:The function f:XY1 is
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(i).a (T X , 1)-open function from X to Y1 if for every U T X, f(U)1 and
(ii). a (T X , 1)-closed function from X to Y1 if for every n-ary closed element A in (X, T
X }, f(U) is closed in (Y1, 1).
Definition 5.2:The function g: X Y is
(i).a (TX,TY)-open function from X to Y if for every UTX, g(U) TY whenever g(U) is
an n-ary element in Y and
(ii).a (TX, TY)-closed function from X to Y if for every n-ary closed element A in (X,TX),
g(A) is n-ary closed in (Y, TY ) whenever g(A) is an n-ary element in Y.
Theorem 5.3:Let f:XY1 be a function. The following are equivalent.
(i).The function f is a (T , 1)-open function from X to Y1 .
(ii). For each n-ary point x in X and each n-ary nbd U of x in (X, T) ) there is a nbd V of
f(x) in (Y1, 1) such that f(x)V f(U).
(iii). For every n-ary element A of X, f(n-IntA) Int(f(A)).
Proof: Suppose (i) holds. Let x be an n-ary point in X and U be an n-ary nbd of x in
(X,T). Since f is a (T , 1)-open function from X to Y1 , f(U) is open in (Y1, 1) with
f(x)f(U) so that there is a nbd V of f(x) in (Y1, 1) such that f(x) Vf(U). This proves
(i)(ii). Suppose (ii) holds. Let U T and y f(U). Then f(x) =y for some x U. Then by
using (ii), there is an open set V1 with y=f(x) Vf(U) that implies f(U) is open in (Y1,
1). This proves (ii)(i). Again to prove (i)(iii) we assume (i). Then for any n-ary
element A of X, n-IntA is n-ary open in (X,T) so that by using (i), f(n-IntA) is open in (Y1,
1) . Since f(n-IntA) (f(A)) and since f(n-IntA) is open, it follows that f(n-IntA) Int(f(A)).
This proves (i)(iii). Now suppose (iii) holds. If U is an n-ary open set in (X,T) then by
taking A=U in (iii) we get f(n-IntU)Int(f(U)) that implies f(U)Int(f(U)) so that f(U) is
open in (Y1, 1). This proves (iii)(i).
Theorem 5.4: Let f:XY be a function such that f maps an n-ary element in X to an nary element in Y. The following are equivalent.
(i).The function f is a (TX, TY)-open function from X to Y.
(ii). For each n-ary point x in X and each n-ary nbd U of x in (X,T X) ) there is an n-ary
nbd V of f(x) in (Y, TY ) such that f(x)V f(U).
(iii). For every n-ary element A of X, f(n-IntA) ))n-Int(f(A)).
Proof: Suppose (i) holds. Let x be an n-ary point in X and U be an n-ary nbd of x in
(X,T). Since f is a (TX,TY)-open function from X to Y , f(U) is n-ary open in (Y, TY) with
f(x) f(U) so that there is an n-ary nbd V of f(x) in (Y, TY) such that f(x) Vf(U). This
proves (i)(ii). Suppose (ii) holds. Let U TX and
y f(U). Then f(x) =y for some
xU. Then by using (ii), there is an n-ary open set V TY with y=f(x)Vf(U) that implies
f(U) is n-ary open in (Y, TY). This proves (ii)(i). Now to prove (i)(iii) we assume (i).
Then for any n-ary element A of X, n-IntA is n-ary open in (X, TX) so that by using (i), f(nIntA) is n-ary open in (Y, TY). Since f(n-IntA) f(A) and since f(n-IntA) is n-ary open, it
follows that
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f(n-IntA) n-Int(f(A)). This proves (i)(iii). Now suppose (iii) holds. If U is an n-ary
open set in (X, TX) then by taking A=U in (iii) we get f(n-IntU)n-Int(f(U)) that implies
f(U)n-Int(f(U)) so that f(U) is n-ary open in (Y, TY). This proves (iii)(i).
Theorem 5.5: Let f:XY1 be a function. The following are equivalent.
(i).The function f is a (T , 1)-closed function from X to Y1 .
(ii). For every n-ary element A of X, Cl (f(A))f(n-Cl A).
Proof: Suppose (i) holds. Let A be an n-ary element of X. Since f is a (T , 1)-closed
function from X to Y1 , f(n-Cl A) is closed in (Y1, 1). Again since f(A)f(n-Cl A) we have
Cl (f(A))f(n-Cl A). This proves (i)(ii). Cnversely suppose (ii) holds. Then for any n-ary
closed set F of (X, TX), by taking A=F in (ii) we get Cl(f(F))f(n-Cl F) that implies Cl
(f(F))f( F) so that f(F) is closed in (Y1, 1). This proves (ii)(i).
Theorem 5.6: Let f:XY be a function such that f maps an n-ary element in X to an nary element in Y. Then the function f is a (TX,TY)- closed function from X to Y if and only if
for every n-ary element A of X,
n-Cl (f(A))f(n-Cl A).
Proof: Suppose f is a (TX,TY)- closed function from X to Y. Let A be an n-ary element of
X. Since f is a (TX,TY)-closed function from X to Y, f(n-Cl A) is n-ary closed in (Y, TY).
Again since
f(A)f(n-Cl A) we have n-Cl (f(A))f(n-Cl A). Conversely suppose n-Cl (f(A))f(n-Cl
A) ) holds for every n-ary element A of X. Then for any n-ary closed set F of (X, TX), by
taking A=F , we get n-Cl (f(F))f(n-Cl F) that implies n-Cl (f(F))f( F) so that f(F) is n-ary
closed in (Y, TY). This proves f is a (TX,TY)-closed function from X to Y .
6. N-ARY HOMEOMORPHISM
Definition 6.1: Let f: X Y, be an n-ary function such that f maps an n-ary element in X to
an n-ary element in Y and be bijective. Then f: X  Y is a (TX, TY)-homeomorphism if f is
(TX, TY)-irresolute and f-1 is (TY, TX)-irresolute .
Theorem 6.2: Let f:XY be an n-ary function such that f maps an n-ary element in X to
an n-ary element in Y and be bijective. The following are equivalent.
(i).The function f is a (TX, TY)-homeomorphism
(ii). f is (TX, TY )-irresolute and (TX, TY)-open.
(iii). f is (TX, TY)- irresolute and (TX, TY)-closed.
(iv).For every n-ary element A of X, f(n-Cl A) = n-Cl f(A).
Proof:. Suppose (i) holds. Since f-1 is (TY, TX)-irresolute, for every n-ary open set V in
(X,TX), g-1(V) is n-ary open in (Y ,, TY ) where g = f-1 that implies f(V) is n-ary open in (Y,,
TY ). This proves (i)(ii). Conversely suppose (ii) holds.
Let A be n-ary open in (X, TX). Since f is (TX , TY)-open, f(A) is n-ary open in (Y,, TY )
that implies f-1 is (TY, TX)-irresolute. This proves (ii)(i).
Suppose (ii) holds. For every n-ary closed set F in (X,TX), X⃠F is n-ary open in (X,TX)
that implies f(X⃠F) is n-ary open in (Y ,, TY ) so that Y⃠ f(F) is n-ary open in (Y ,, TY )
which implies f(F) is n-ary closed in (Y,, TY ). This proves (ii)(iii). Similarly we can prove
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that (iii)(ii). Now to prove (iii)(iv) we assume that (iii) holds. Let A be an n-ary element
of X. Since f is (TX, TY)- irresolute, f(n-Cl A) n-Cl f(A) and since f is (TX, TY)- closed nCl f(A) f(n-Cl A) that implies f(n-Cl A) = n-Cl f(A) which proves (iv). Now suppose (iv)
holds. Then obviously f is n-ary irresolute and n-ary closed. This proves (iii)  (iv).
Notations: Let (Xi, i) and (Yi , i ) be the topological spaces. Then (X,TX) and (Y,, TY
) are n-ary spaces where 12 …n = TX and 12 …n = TY . If fi :Xi Yi is a
function then f =(f1,f2,…,fn):XY is an n-ary function from X to Y defined by f(x) =
(f1(x1), f2(x2),…, fn(xn)) for every n-ary point x of X.
Lemma 6.3: f:XY is bijective iff each fi is bijective
Theorem 6.4 : The function f:XY
(i).is (TX, TY )-irresolute if and only if each fi is continuous.
(ii).is (TX, TY )-open if and only if each fi is open .
(iii).is (TX, TY )-closed if and only if each fi is closed .
Proof: Clearly f maps an n-ary element in X to an n-ary element in Y and f-1(V) is an nary element in X for every n-ary element V in Y. Further
f-1(V) = ((f1)-1(V1), (f2)-1(V2), …, (fn)-1(Vn)). This shows that f:XY is (TX, TY ) irresolute if and only if each fi is continuous. Also for every n-ary element A in X, f(A) =
(f1(A1), f2(A2), …, fn(An)). This shows that f is
(TX, TY )-open if and only if each fi is
open and f is (TX, TY )-closed if and only if each fi is closed .
Theorem 6.5: Let f:XY be a bijective n-ary function. Then the function f is a (TX, TY ) homeomorphism if and only if each fi is a (i,i )-homeomorphism .
Proof: Follows from Theorem 6.4 and Theorem 6.2.
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