Download generalised irresolute map

STRONGLY -GENERALISED IRRESOLUTE MAPS,
STRONGLY SEMI -GENERALISED IRRESOLUTE MAPS
AND ALMOST -GENERALISED IRRESOLUTE MAPS
R.Devi and K.Bhuvaneswari
Department of Mathematics, Kongunadu Arts and Science College, Coimbatore-641 029
Abstract: The objective of the present paper is to introduce three new classes of functions
called strongly -generalised irresolute functions, strongly semi -generalised irresolute
functions and almost -generalised irresolute functions in a topological space. We obtain
some characterizations of this classes and several properties and investigate the relationships
with other classes of functions between topological spaces.
Keywords: Strongly -generalised irresolute functions; strongly semi  -generalised irresolute
functions; almost -generalised irresolute functions.
Introduction: O. Njåstad[14] and N. Levine [6] introduce the notion of -open sets and
generalised closed sets in topological spaces respectively .H.Maki et al [10] defined the
concept of generalised - closed sets and -generalised closed sets and investigated their
topological properties . G. LoFaro [8] defined and investigated the notion of a strong form of
 irresolute functions. In section 1 we define and investigate the notions of new classes of
function namely strongly -generalised irresolute function . In section 2 we introduce strongly
semi -generalised irresolute function and almost -generalised irresolute function. In section
3 we study final remarks.
Preliminaries: Throughout this paper a space means a topological space which lacks any
separation axioms unless explicitly stated. Let ( X, ) and ( Y,) be topological spaces.
Definition(i).[10]
A function f : (X, )  (Y, )
is said
to be  - generalised
-1
irresolute if f (V) is - generalised open in X for every -generalised open set V of Y.
(ii).[8] A function f : (X, )  (Y, ) is said to be strongly  - irresolute if f -1 (V) is open
in X for every - open set V of Y.
(iii).[17] A function f : (X, )  (Y, ) is said to be g* -continuous if f -1 (V) is g* closed
set of ( X,  ) for every closed set V of (Y,  ).
(iv)[1]A function f : (X, )  (Y, ) is said to be  - continuous(resp. -irresolute ) if
f -1 (V) is -open in ( X,  ) for every open (resp.-open) set V of (Y,  ).
(v)[16] A function f: ( X, ) (Y , ) is said to be strongly generalised -irresolute if f -1 (V)
is open in X for every generalised -open set V of Y.
(vi) [2] . A topological space ( X, ) is said to be an Tb –space if every  -generalised closed
set in it is closed.
1.
STRONGLY -GENERALISED IRRESOLUTE MAP
Definition 1.1 A function f : (X, ) (Y, ) is said to be strongly -generalised irresolute
if f -1 (V) is open in X for every - generalised open( written as g-open) set V of Y.
Theorem 2.1 If f: (X, ) (Y, ) is strongly -generalised irresolute map then
(i) f is -generalised irresolute map.
(ii) f is strongly -irresolute map.
(iii) f is g* -continuous map.
(iv) f is strongly generalised - irresolute map.
Proof : (i) Let V be - generalised open set in Y. Since f is strongly -generalised
irresolute map, f -1 (V) is open in X , and by [10] it is -generalised -open in X.
(ii) Let V be -open set in Y, by [10] it is -generalised open in Y since f is strongly generalised irresolute map, f -1 (V) is open in X. and hence it is -open in X.
(iii) Let V be open set in Y and hence - generalised - open set in Y. Since f is strongly
-generalised irresolute map, f -1 (V) is open in X and by [17] it is g* open in X.
(iv) Let V be generalised  -open set in Y , by [2] it is -generalised-open in Y. Since f is
strongly -generalised irresolute map, f -1 (V) is open in X. and hence it is -generalised
-open in X.
Example 3.1 : The converse of the above theorem need not be true. (i) Let X = Y = {a ,b, c }
and let  = { X,,{a},{a, b}} and  = { Y, ,{a}}. Let f: (X, ) (Y, ) be defined as
identity map. The mapping f is -generalised irresolute map but not strongly -generalised
irresolute map. Since the inverse image of every -generalised open set in Y is generalised open in X, but the inverse image of -generalised open set {a,c } is {a,c},
which is not open in X (ii) Let X = Y = {a, b, c} with  = { X,,{a},{a,b}} and  = { Y,
,{a, c}}.Define a function f: (X, ) (Y, ) by setting f(a) =a, f(b) =c and f(c) =b. Then f is
strongly -irresolute map but not strongly -generalised irresolute map. (iii) Let X = Y =
{a, b, c } With  = { X,,{a},{a, c}} and  = { Y, ,{a},{b},{a, b}}. Let f: (X, ) (Y, )
be defined by f(a) = a, f(b) = c and f(c) = b. In this example. the mapping f is g* continuous
but not strongly -generalised irresolute map. Since the inverse image of -generalised
open set is {b} in Y is {c}, which is not open in X..(iv) Let X = Y = {a, b, c } and
let
 = { X,,{b}, {b,c},{a, b}} and  = { Y, ,{a},{a,b}} .Define a function f: (X, ) (Y, )
by setting f(a) =b, f(b)=a and f(c)=c. Then f is strongly generalised -irresolute but not
strongly -generalised irresolute. Since the inverse image of all generalised - open set is
open in X. but the inverse image of -generalised open set {b} is {a} which is not open in X.
Theorem 2.2 If a map f: (X, ) (Y, ), is continuous and Y is Tb , then f is strongly generalised irresolute map.
Proof : Let V be -generalised open set in Y. since Y is Tb, V is open in Y. Hence
f -1 (V) is open in X. Therefore f is strongly -generalised irresolute map.
Theorem 2.3 Let f : (X, ) (Y, ),
g : (Y, ) (Z, ) be any two functions, then the
composition g o f : (X, ) (Z, ) is
(i) Strongly -irresolute if f is strongly -generalised irresolute and g is -irresolute map.
(ii) Continuous if f is strongly -generalised irresolute and g is -generalised irresolute
map.
(iii) g* continuous if f is g*-irresolute and g is strongly -generalised irresolute map.
From the Theorems 2.1 and Examples 3.1 we obtain the following diagram
-generalised irresolute map
Strongly-irresolute
map
Strongly -generalised
irresolute map
g*-
continuous
Strongly generalised irresolute map
Theorem 2.4 The following are equivalent for a function f: (X, ) (Y, ) :
(a) f is strongly -generalised irresolute;
(b) For each x X and each -generalised open set V of Y containing f(x) , there exists
a open set U of X containing x such that f(U) V;
(c) f -1(V) Int (f -1(V )) for each -generalised open set V of Y;
(d) f -1(F) is closed in X for every -generalised open set F of Y;
proof: (a)  (b) Let x  X and let V be any -generalised open set of Y containing f(x) . By
definition 1.1 f -1(V) is open in X and contains x. Set U = f -1(V) , then by (a) U is open in X
containing x and f(U) V.
(b)  (c) Let V be any -generalised open set of Y and x f -1(V) -----------(1)
By (b) there exist an open set U of X containing x such that f(U) V. Thus we have x  U
 Int (U) Int (f -1(V)) ----(2) and hence from (1) and (2)
f -1(V) Int (f -1(V)).
(c)  (d) Let F be any -generalised closed subset of Y. Set V = Y – F, then V is generalised open in Y. By (c) we obtain f -1(V)  Int (f -1(V)) and hence f -1(F) = X – f -1(Y F) = X - f -1(V) is closed in X.
(d)  (a) Let V be any -generalised open in Y set F = Y – V, by (d) f -1(F) is closed in X that
is f -1(F) = f -1(Y - V) = X - f -1(V) is closed in X and hence f -1(V) is open in X. Therefore f is
strongly -generalised irresolute.
Theorem 2.5
A function f: (X,) (Y ,) is strongly -generalised irresolute if the
graph function g: X  X  Y denoted by g(x) = ( x, f(x) ) for each x X is strongly generalised irresolute.
Proof : Let x X and V be any -generalised open set in Y containing f(x) . Then X  V
be g-open set of X  Y containing g(x) by lemma [2]. Since g is strongly -generalised
irresolute , there exist an open set U of X containing x such that g(U)  X  V and hence f(U)
V. Thus f is strongly -generalised irresolute by Theorem 2.4.
Theorem 2.6 If
f : (X,) (Y,) is strongly -generalised irresolute , then
the
restriction f/A :(A, /A) (Y,) is strongly -generalised irresolute.
Proof : Let V be any -generalised open set of Y. Since f is strongly -generalised irresolute,
f -1 (V) is open in X. Since A is open in X , (f / A) –1 (V) = A  f –1 (V) is open in A and
hence f/A is strongly -generalised irresolute.
Theorem 2.7 Let f: (X, ) (Y, ) be a function and {Ai / i  } be a cover of X by
open sets of ( X,  ). Then f is strongly -generalised irresolute, if f / A i : (A i , / A i)
(Y,) is strongly -generalised irresolute for each i  .
Proof : Let V be -generalised open set of Y. Since f /A i is strongly -generalised
irresolute map, ( f / A i ) -1 (V ) is open in A i . Since A i is open in X, ( f / A i ) -1 (V ) is
open in X for every i . Therefore
f -1 (V) = X  f -1(V)
=  {A i  f -1(V): i }
=  { (f / A i) -1 (V) : i  } is open in X, because the union of open
sets is a open set. Hence f is strongly  - generalised irresolute.
Definition 1.2 (i)A collection { A i / i   } of -generalised open sets in a topological space
X is called -generalised open cover of a subset S if S   { Ai / i   } holds.
(ii)A topological space ( X,  ) is -generalised compact if every -generalised open cover of
X has a finite sub cover.
(iii) A subset S of a topological space X is said to be -generalised compact relative to X, if for
every collection { A i / i   } of -generalised open subset of X such that S  { Ai / i   }
there exists a finite subset 0 of  such that S   { A i / i  0 }
(iv) A subset S of a topological space X is said to be -generalised compact if S is generalised compact as a subspace of X.
Theorem 2.8 Let f: (X, ) (Y, ) be a surjective, strongly -generalised irresolute map. If X
is compact, then Y is -generalised compact.
Proof : Let { Ai / i  A } be a -generalised open cover of Y. Then {f –1 (Ai ) / i   } is a
open cover of X. Since X is compact, it has a finite sub cover say { f –1 (A1),
f –1 (A2), ……
f –1 (An)} surjectiveness of f is implies {A1, A2, …… An} is -generalised open cover of Y and
hence Y is -generalised compact.
Theorem 2.9 If a map f: (X, ) (Y, ) is strongly -generalised irresolute and a subset S of
X is compact relative X, then the image f(S) is -generalised compact relative to Y.
Proof : Let { A i / i  A } be a collection of -generalised open sets in Y such that f(S)  
{(Ai )/ i   }. Then S   {f –1 (A i ) / i   } where f –1 (A i ) is open in X for each i. Since S
is compact relative to X, there exists a finite sub collection {A1, A2, …… An} such that S 
 {f –1 (A i ) / i = 1 to n } that is f(S)   {A i / i = 1 to n }. Hence f(S) is -generalised
compact relative to Y.
2. STRONGLY SEMI -GENERALISED IRRESOLUTE MAP AND
ALMOST -GENERALISED IRRESOLUTE MAP
Definition 1.3 Let ( X, ) and ( Y, ) be two topological spaces and f: (X, )  (Y, )
(i) A function f is said to be strongly semi -generalised irresolute if f -1 (V) is semiopen in X for every -generalised open set V in ( Y, ).
(ii) A function f is said to be almost -generalised irresolute , if f -1 (V) is -open in ( X,)
for every -generalised open set V in ( Y, ).
Theorem 2.10(i) Every strongly -generalised function is strongly semi -generalised
irresolute.
(ii) Every strongly semi- generalised irresolute function is almost - generalised irresolute.
(iii) Every almost -generalised irresolute function is -continuous.
Example 3.2 The converse of Theorem 2.10(i) is not true, Let X = {a, b, c} and  = {, X,
{a}, {b} , {a, b} } The identity function f: ( X,) ( X, ) is strongly semi -generalised
irresolute. The function f is not strongly -generalised irresolute , because the inverse image
of { a,c} is { a, c} which is a -generalised open set in X but not open in( X, ) The converse
of Theorem 2.10 (ii) is not true .Let X= { a, b, c} =Y,  = { , {a}, { b, c }, X} and  = { ,
Y, {a}, { a, b} } . Define a function f: ( X, ) ( Y, ) by setting f(a) = b, f(b)= c, f(c)= a.
Then f is almost -generalised irresolute. The function f is not strongly semi -generalised
irresolute, because {a} is -generalised open in ( Y, ) and f - 1 ( {a} ) = {c} is not semi
open in ( X, ) . The converse of the Theorem 2.10 (iii) is not true. Let X= { a, b, c} and  = {
,
{ a} , X} . Define a function f: (X,) ( Y,) by setting f(a)= f(b)= b , f(c) =c. Then
f -1( V) is -open for every open set V. i.e., f is -continuous .How ever f is not almost generalised irresolute.
Theorem 2.11 Let f: (X, )  (Y, ), g : (Y, )  (Z, ) be any two functions, then the
composition g o f : (X, )  (Z, ) is
(i) Semi continuous if f is strongly semi -generalised irresolute and g is
-generalised
irresolute map.
(ii)  continuous if f is strongly semi -generalised irresolute and g is -generalised
irresolute map.
(iii) Semi continuous if f is semi continuous and pre-open and g is strongly semi generalised irresolute map.
Theorem 2.12 Let f: (X, ) Y, ), g : (Y, )  (Z, ) be any two functions, then the
composition g o f : (X, )  (Z, ) is Strongly semi -generalised irresolute
(i) if f is strongly -generalised irresolute and g is -generalised irresolute.
(ii) if f is strongly semi generalised -continuous and g is -generalised irresolute.
(iii) if f is strongly -pre irresolute and g is strongly -generalised irresolute.
(iv) if f is semi continuous and g is strongly -generalised irresolute.
(v) if f is -continuous and g is strongly -generalised irresolute.
(vi) if f is strongly semi -irresolute and g is strongly -generalised
irresolute.
Theorem 2.13 Let f: (X, )  (Y, ) be a function
(i) The following properties are equivalent
a) f is strongly semi  - generalised irresolute.
b) For each x  X, and each - generalised open set V of Y containing f(x) there exits a
semi-open set U of X containing x, such that f(U)  V
c) f -1 (V)  Cl (Int ( f -1(V))) for every -generalised open set V of Y
d) f -1(F) is semi-closed in X for every -generalised closed set F of Y
(ii) The following properties are equivalent:
a) f is almost -generalised irresolute.
b) For each x  X, and each - generalised open set V of Y containing f(x) there
exits a -open set U of X containing x, such that f(U)  V
c) f -1 (V)  Cl (Int(Cl( f -1(V)))) for every - generalised open set V of Y
d) f -1(F) is -closed in X for every generalised closed set F of Y
Proof: (i) (a)  (b) Let x  X and let V be an -generalised open set of Y containing
f(x).By definition 1.3 (i) f -1 (V) is semi-open in X and contains x. Set U = f -1(V) then by
(a) U is semi-open subset of X containing x and f(U)  V.
(b)  (c) Let V be any -generalised open set of Y and x  f -1(V). ----------------- (1)
By (b) there exits a semi-open set U of X containing x such that f(U)  V.
 U  f -1(V). Thus we have
x  U  Cl (Int (U))
 Cl (Int (f -1(V)))
---------------------- (2)
-1
and U  Cl (Int (f (V))) and hence
f -1(V)  Cl (Int (f -1(V))) [ by (1) and (2) ]
(c)  (d) Let F be any -generalised closed subset of Y. Set V = Y – F, then V is generalised open in Y. By (c) we obtain, f -1(V)  Cl (Int (f -1(V))) and hence f -1(F) = X –
f -1(Y – F)
= X – f -1(V) is semi-closed in X.
(d)  (a) Let V be any -generalised open in Y set F = Y – V, by (d) f -1(F) is semi - closed
in X that is f -1(F) = f -1(Y - V) = X - f -1(V) is semi - closed in X and hence f -1(V) is semi open in X. Therefore f is strongly semi - generalised irresolute.
(ii) (a)  (b) Let x  X and let V be an - generalised open set of Y containing f(x).By
definition 1.3(ii) , f -1 (V) is -open in X and contains x. Set U = f -1(V) then by (a) U is
-open subset of X containing x and f(U)  V.
(b)  (c) Let V be any -generalised open set of Y and x  f -1(V). ----------------- (1)
By (b) there exits a -open set U of X containing x such that f (U)  V.
U  f -1(V). Thus we have
x  U  Cl (Int(Cl(U)))
 Cl (Int (Cl (f -1(V))))
---------------------- (2)
-1
Using (1) and (2) f (V)  Cl (Int (f -1(V)))
(c)  (d) Let F be any -generalised closed subset of Y. Set V = Y – F, then V is generalised open in Y. By (c) we obtain, f -1(V)  Cl (Int (Cl (f -1(V)))) and hence
f -1(F) = X – f -1(Y – F)
= X – f -1(V) is -closed in X.
(d)  (a) Let V be any -generalised open in Y set F = Y – V, by (d) f -1(F) is - closed in
X that is
f -1(F) = f -1(Y - V) = X - f -1(V) is -closed in X and hence f -1(V) is -open in X. Therefore f
is almost -generalised -irresolute.
Theorem 2.14 (i) A function f: X  Y is strongly semi -generalised irresolute if the graph
function g: X X  Y denoted by g (x) = ( x, f(x) ) for each x X is strongly semi generalised irresolute.
(ii) A function f: X  Y is almost -generalised irresolute if the graph function g: X X  Y
denoted by g(x) = ( x, f(x) ) for each x X is almost -generalised irresolute.
Proof: Using Theorem 2.13 (i) and (ii) the proof is similar to 2.5.
Theorem 2.15 (i) If a function f: X  Y is strongly semi - -generalised irresolute. Then
P  f: X Y is strongly semi - -generalised irresolute for each  where, P is the
projection of  Y onto Y.
(ii) If a function f: X  Y is almost -generalised irresolute. Then P  f: X  Y is
almost -generalised irresolute for each  where P is the projection of  Y onto Y.
Proof: (i) Let V be any -generalised open set of Y. Since P is continuous and open it is generalised irresolute and hence P -1 (V) is -generalised open in  Y. Also f is strongly generalised irresolute f –1 (P -1 (V)) = (P  f) –1 (V) is semi open in Y. Hence (P  f) is
strongly semi -generalised irresolute for each   .
(ii) Similar to (i).
Theorem 2.16 If f: (X,)  (Y, ) is strongly semi -generalised irresolute and A is a pre
open subset of X. Then the restriction f/A: AY is strongly semi -generalised irresolute.
Proof: Let V be any -generalised open set of Y. Since f is strongly semi -generalised
irresolute. f -1 (V) is semi - open in X. By
Lemma 1.1[12] and
since A is open in X ,
(f / A) –1 (V) = A  f
generalised irresolute.
–1
(V) is semi - open in A and hence f/A is strongly semi - -
Theorem 2.17 Let f: (X,)  (Y,) be a function and {A :  } be a cover of X by
semi-open sets of ( X,  ). Then f is strongly semi--generalised irresolute, if f / A : A  Y
is strongly semi--generalised irresolute for each   .
Proof: Let V be generalised -open set of Y. Since f / A is strongly semi -generalised
irresolute map, (f / A ) -1 (V ) is semi-open in A. Since A is semi-open in X, by Theorem 5
of [15] f / A ) -1 (V) is semi-open in X for every  .
f -1 (V) = X  f -1(V)
=  {A  f -1(V):   }
=  {(f / A) -1 (V) :    } is semi-open in X, because the union of semi-open sets is
a semi-open set. Hence f is strongly semi- - generalised irresolute.
Theorem 2.18. If f: (X,)  (Y, ) is almost -generalised irresolute and A is an -open
subset of X. Then the restriction f/A: A Y is almost -generalised irresolute.
Proof: Let V be any - generalised open set of Y. Since f is almost -generalised irresolute,
f -1 (V) is  - open in X. By Lemma 3.2(b) of [13] and since A is -open in X, (f / A) –1 (V) =
A  f –1 (V) is - open in A and hence f/A is almost -generalised irresolute.
Theorem 2.19 Let f: (X,)  (Y,) be a function and {A:  } be a cover of X by -open
sets of (X, ). Then f is almost -generalised irresolute, if f / A : A. Y is almost generalised irresolute for each   
Proof: Let V be - generalised open set of Y. Since f / A is almost -generalised irresolute
map, (f / A) -1 (V) is -open in A. By Lemma3.3 (b) of [13] and since A  O(X), then
( f / A ) -1 (V ) is -open in X for every  . Therefore f -1 (V) = X  f -1(V) =  {A 
f -1(V):   } =  {(f / A) -1 (V):    } is -open in X, because the union of -open sets
is a -open set. Hence f is almost -generalised irresolute.
Theorem 2.20 If a function f: (X, )  (Y, ) is almost -generalised irresolute, then
f -1(B) is -closed in X for any nowhere dense set B of Y
Proof: Let B be any nowhere dense subset of Y. Then Y-B is -open in Y and hence by [10] it
is -generalised open in Y. Since f is almost -generalised
irresolute, then f -1(Y –B) =
-1
-1
X- f (B) is -open in X and hence f (B) is -closed in X.
Theorem 2.21 Let f: (X, )  (Y, ), g : (Y, )  (Z, ) be any two functions, then the
composition
g o f : (X, )  (Z, ) is almost -generalised irresolute
(i) if f is almost -generalised irresolute and g is -generalised irresolute.
(ii) if f is almost -irresolute and g is strongly -generalised irresolute.
(iii) if f is -irresolute and g is almost -generalised irresolute.
(iv) if f is -irresolute and g is strongly -generalised irresolute.
(v) if f is -continuous and g is strongly -generalised irresolute.
(vi) if f is -irresolute and g is strongly semi -generalised irresolute.
(vii) if f is almost-irresolute and g is strongly semi -generalised irresolute.
Theorem 2.22 Let (X,) be a sub maximal and extremely disconnected space. Then the
following are equivalent for a function f: (X,)  (Y,)
(i) f is strongly -generalised -irresolute
(ii) f is strongly semi -generalised irresolute
(iii) f is almost -generalised irresolute
Proof: This follows from the fact that if (X,) is sub maximal and extremely disconnected,
then  = SO(X) = O (X) [13]
3.FINAL REMARKS
Definition 1.4 A function f: (X,)  (Y,) is said to be
(i) Strongly -generalised c-homeomorphism if f is bijective, strongly -generalised irresolute
function and f –1 is strongly -generalised irresolute,
(ii) Strongly semi -generalised c- homeomorphism if f is bijective, strongly semi generalised irresolute and f –1 is strongly semi -generalised irresolute.
(iii) Almost -generalised c-homeomorphism if f is bijective, almost -generalised irresolute
function and f –1 is almost -generalised irresolute.
Theorem 2.23. If a bijection map f: X  Y is strongly -generalised c-homeomorphism then
it is -generalised c- homeomorphism.
Proof: Since a bijection map f is strongly -generalised c-homeomorphism, f and f –1 are
strongly -generalised –irresolute. By Theorem 2.1 f and f –1 are -generalised irresolute. Thus
f is -generalised c-homeomorphism.
Remark : Using Example 3.1 it is clear that -generalised c-homeomorphism need not be
strongly -generalised c-homeomorphism.
Theorem 2.24 If f: X  Y is strongly -generalised c-homeomorphism then it is generalised homeomorphism.
Proof: Since every strongly -generalised irresolute map is -generalised continuous the
proof follows.
Remark :(i) Since every open set is semi open and -open, every semi open set is -open, it is
clear that strongly -generalised c-homeomorphism is (a) Strongly semi -generalised chomeomorphism
(b) Almost -generalised c-homeomorphism,
(ii)Every Strongly semi -generalised c-homeomorphism is almost -generalised chomeomorphism
(iii) The family of all strongly -generalised c-homeomorphism from (X, ) on to itself is
denoted by strongly -generalised ch (X, ).
Theorem 2.25 Let X, Y and Z be topological spaces and f: X  Y, g: Y  Z be strongly generalised c-homeomorphisms then their composition g o f: X  Z is a strongly generalised c-homeomorphism.
Proof: Let V be a -generalised open set in Z. Consider (g o f) –1(V) = f –1 (g –1(V)) = f –1(U)
where U = g –1(V). As g is strongly -generalised c-homeomorphism, g is strongly generalised irresolute, g –1(V) is open in Y. by [10] g –1(V) is -generalised open in Y. Now
f is strongly -generalised c-homeomorphism, f is strongly -generalised irresolute, f –1(U) is
open in X. That is (g o f) –1(V) is open in X. Hence (g o f) is strongly -generalised irresolute.
Again for an -generalised open set A in X, consider (g o f)(A) = g(B) where B = f(A). As f is
strongly -generalised c-homeomorphism, f –1 is strongly -generalised irresolute. Therefore
f(A) is open in Y and hence -generalised open in Y. Now g is strongly -generalised chomeomorphism, g -1 is strongly -generalised irresolute, we have g (B) is open in Z. That is
(g o f)(A) is -generalised open in Z. Thus
(g o f) –1 is strongly -generalised irresolute.
Hence (g o f) is strongly -generalised c-homeomorphism.
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