Download CHAPTER 6 sgp- LOCALLY CLOSED SETS IN TOPOLOGICAL

CHAPTER 6
sgp- LOCALLY CLOSED SETS IN TOPOLOGICAL
SPACES
6.1 Introduction
The notion of a locally closed set in a topological space was introduced
by Kuratowski and Sierpinski [46]. According to Bourbaki [12], a subset A of
a topological space X is called locally closed in X if it is the intersection of an
open set in X and a closed set in X. Ganster and Reilly [39] used locally
closed sets to define LC- Continuity and LC-irresoluteness. Balachandran,
Sundaram and Maki [8] introduced the concept of generalized locally closed
sets in topological spaces and investigated some of their properties. Recently
Sheik John [88] introduced the three new class of sets denoted by co-LC(X, x),
co-LC*(X, x) and co-LC**(X, x) and each of which contains LC(X, x). Also
various authors like Gnanambal [43] and Park and Park [81] have introduced
a-locally closed and semi generalized locally closed sets respectively in
topological spaces.
This chapter contains four sections. In section 2 of this chapter, we
introduce three weaker forms of locally closed sets denoted by SGPLC(X,x),
SGPLC*(X,x) and SGPLC**(X,x) each of which contains LC(X,x) and
obtained some of their properties and also their relationships with glc-set,
a-lc-set and colc-set. Also we introduce sgp-submaximal spaces and study
some of their properties.
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In section 3 of this chapter, we introduce the concept of SGPLCcontinuous, SGPLC*-continuous and SGPLC**-continuous functions which
are weaker than LC-continuous functions.
In the last section of this chapter, we define SGPCL-irresolute,
SGPLC*-irresolute and SGPLC**-irresolute functions which also weaker
than LC-irresolute functions in topological spaces and study some of their
properties.
6.2 sgp-Localty Closed Sets
In this section we introduce sgp-locally closed sets and sgpsubmaximal and study some of their properties.
Definition 6.2.1: A subset A of a topological space (X, x) is called a semigeneralized-pre locally closed set (briefly sgplc-set) if A = S n F where S is
sgp-open and F is sgp-closed.
The class of all semi-generalized-pre locally closed sets in (X,x) is denoted by
SGPLC(X,t).
Definition 6.2.2: A subset A of a topological space (X,x) is said to be
SGPLC*-set if there exist sgp-open set S and a closed set F of (X,x) such that
A = SnF.
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Definition 6.2.3:
A subset A of a topological space (X,t) is said to be
SGPLC**-set if there exist an open set S and a sgp-closed set F of (X,x) such
that A = Sn F.
Theorem 6.2.4: If a subset A of (X,t) is locally closed then it is SGPLC
(X,t), SGPLC*(X,t) and SGPLC**(X,x).
Proof: Let A = P n Q, where P is open and Q is closed in (X, x). Since every
open set is sgp-open and every closed set is sgp-closed, Therefore A is
SGPLC (X,x), SGPLC*(X,x) and SGPLC**(X,x).
The converse of the above theorem need not be true as seen from the
following example.
Example 6.2.5: Let X = {a, b, c} and x = {<(>, X, {a, b}}. Then the locally
closed sets are: X, <(>, {c}, {a, b} and SGPLC(X,x) = SGPLC*(X,x) =
SGPLC* *(X,x) = X,
{a}, {b}, {c}, {a, b}, {b, c}, {a, c}. Here {a, c} is
SGPLC (X,x), SGPLC*(X,x), SGPLC**(X,x) but not LC-seL
Theorem 6.2.6: If a subset A of (X,x) is SGPLC*-set then it is SGPLC-set.
Proof: Let A be a SGPLC*-set. Let P be a sgp-open set in (X,x) and Q be a
closed set in (X,x). Since A is SGPLC*-set by definition, A = P n Q. Since
every closed set is sgp-closed. Therefore A is SGPLC-set.
The converse of the above theorem need not be true aes seen from the
following example.
Example 6.2.7: Let X = {a, b, c, d} and x = {<|>, X, {b}, {c, d}, {b, c, d}}.
Then the subset {a, c, d} e SGPLC(X, x) but {a, c, d} g SGPLC*(X, x).
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Theorem 6.2.8: Every SGPLC**(X, x) is SGPLC (X, x).
Proof: The proof follows from the definitions 6.2.1 and 6.2.3.
The converse of the above theorem need not be true as seen from the
following example.
Example 6.2.9: Let X = {a, b, c} and x = {<{>, X, {a}}. Then the subset {a, b}
€ SGPLC(X, x) but {a, b} e SGPLC* *(X, x).
Theorem 4.2.10: If a subset A of (X,x) is a-lc set then A is SGPLC-set.
Proof: Let A = P n Q, where P is a-open and Q is a-closed in (X, x). Since
every a-open set is sgp-open and every a-closed set is sgp-closed, Therefore
S is SGPLC-set in (X,x).
The converse of the above theorem need not be true as seen from the
following example.
Example 6.2.11: Let X = {a, b, c} and x = {X, <|>, {a}, {b, e}}. Then the a-lc
sets are: X, <h, {a}, {b, c} and SGPLC-sets are: P(X). Here {a, b} is SGPLCset but not an a-lc set.
Theorem 6.2.12: Every ©LC-set (resp. ©LC*-set, ©LC**-set) is SGPLC-set
(resp. SGPLC*-set, SGPLC**-set).
Proof: Since every ©-closed set is sgp-closed and every ©-open set is sgpopen, the proof follows.
The converse of the above theorem need not be true as seen from the
following example.
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Example 6.2.13: In Example 6.2.9, ©lc(X,x) = ©LC* (X, x) = coLC**(X, x) =
X, (j>, {a}, {b, c} and SGPLC(X, x) = SGPLC* (X, x) = SGPLC** (X, x)=
P(X). Here the subset {a, b} is SGPLC-set (resp. SGPLC*-set, SGPLC**-set)
but not a ©LC-set (resp. ©LC*-set, ©LC**-set)
Remark 6.2.14: The concept of GLC*-sets (resp. GLC**-sets) and SGPLC*sets (resp. SGPLC**-sets) are independent as seen from the following
examples.
Example 6.2.15: Let X = {a, b, c, d} and x = {X, <{>, {a}, {a, d}, {a, b, d},
{a, c, d}. Then GLC*-sets are: X, <|>, {a},{b}, {c}, {d}, (a, d}, {b, c}, {b, d},
(c, d}, {a, b, d}, (a, c, d}, {b, c, d} and GLC**-sets are: X, <(>, (a},{b}5 {c},
{d}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, (a, b, d}, {a, c, d}, {b, c, d} and
SGPLC*-sets = SGPLC**-sets = P(X). Here the subset {a, b, c} is SGPLC*set (resp. SGPLC**-set) but not a GLC*-set (resp. GLC**-set).
Example 6.2.16: Let X = {a, b, c, d} and x = {X, <j>, {a}, (a, b}}. Then
GLC*-sets = GLC**-sets = P(X) and SGPLC*-sets are: X, <J>, {a},{b}, {c},
{d}, {a, b}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}, {a, b, c}, {a, b, d}, {b, c, d}
and SGPLC**-sets are: X, <|), {a}, {b}, {c}, {d}, {a, b}, {b, c}, (b, d}, {c, d},
{b, c, d}. Here the subset {a, c, d} is GLC*-set (resp. GLC**-set) but not a
SGPLC*-set (resp. SGPLC**-set).
Remark 6.2.17: Union of two SGPLC-sets (resp. SGPLC*-sets, SGPLC**sets) need not be a SGPLC-set (resp. SGPLC*-set, SGPLC**-set) as seen
from the following example.
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Example 6.2.18: In Example 6.2.16, the subsets {a} and {c, d} are two
SGPLC (X,t) (resp. SGPLC* (X,x), SGPLC** (X,x)) sets but their union {a}
u {c, d} - {a, e, d} is not SGPLC(X,x)-set (resp. SGPLC*(X,x),
SGPLC** (X,x)- set).
Theorem 6.2.19: If A e SGPLC (X,x) and B is sgp-open set in (X,x), then
A n B e SGPLC (X,x).
Proof: Since A e SGPLC (X,x), there exist a sgp-open set P and a sgp-closed
set Q such that A = P n Q. Now A n B = (P n Q) n B =(PnB)nQ. Since
P n B is sgp-open and Q is sgp-closed, it follows that AnBe SGPLC (X,x).
Remark 6.2.20: From the above results we get the following implications.
Where A —► B (resp. A4-)-*-
B) represents A implies B but not conversely
(resp. A and B are independent).
Theorem 6.2.21: For a subset A of (X,x), the following are equivalent:
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1)
A g SGPLC*(X, x)
2)
A = P n pCl (A) for some sgp-open set P.
3)
pCl (A)~A is sgp-closed.
4)
A u (X-pCl(A)) is sgp-open.
Proof: (1) => (2):- Let A€ SGPLC* (X,x). Then there exists a sgp-open set P
and a closed set F of (X,x) such that A = Pn F. Since AcP and A c pCl(A)
. Therefore we have AcPn pCl(A).
Conversely, since pCl(A) cF, Pn pCl(A) cPnF=A, Which implies that
A = P n pCl(A).
(2) => (1):- Since P is sgp-open and pCl(A) is closed.
P n pCl(A) e SGPLC* (X,x). Which implies that Ae SGPLC* (X,x).
(3) => (4)
Let F = pCl(A)-A. Then F is sgp-closed by the assumption and
X - F = X n (pCl(A) - A)c = A u (X-pCl(A)). But X-F is sgp-open. This
shows that Au(X- pCl(A)) is sgp-open .
(4) => (3):- Let U = A u (X-pCl(A)). Since U is sgp-open, X-U is sgp-closed.
X - U = X- (A u (X - pCl(A))) = pCl(A) n (X-A) =pC(A) - A.
Thus pCl(A) - A is sgp-closed set.
(4) => (2):- Let P = A u (X - pCl(A)) Thus P is sgp-open . We prove that
A = Pn pCl(A) for some sgp-open set P.PnpCl(A) = (A u (X-pCl(A)))
n pCl(A) == (pCl(A) n A) u (pCl(A) n (X - pCl(A») = A u <j> = A.
Therefore A = PnpCl (A).
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(2) => (4):- Let A = P n pCl (A) for some sgp-open set P. Then we prove that
A u (X-pCl(A)) is sgp-open. Now A u (X-pCl(A)) = (P n pCl(A)) u
(X - pCl(A)) = P n (pCl(A) u (X - pCl(A))) = P. Which is sgp-open. Thus
Au(X-pCl(A)) is sgp-open .
Theorem 6.2.22: If A, Be SGPLC (X,x), then A n B e SGPLC (X,x).
Proof: From the assumptions, there exist sgp-open sets P and Q such that
A = P n pCl (A) and B = Q n pCl (B). Then A n B = (P n Q) n (pCl (A) n
pCl(B)). Since P n Q is sgp-open set and pCl (A) n pCl (B) is closed.
Therefore AnBe SGPLC (X,x).
Theorem 6.2.23: If A e SGPLC (X,x) and B is sgp-closed set in (X,x), then
AnBe SGPLC (X,x).
Proof: Since A e SGPLC (X,x), there exist a sgp-open set P and a sgp-closed
set Q such that A = PnQ. Now AnB = (PnQ)nB = P n(Q n B). Since
P is sgp-open and Q n B is sgp-closed, Therefore AnBe SGPLC (X,x).
Theorem 6.2.24: If A eSGPLC*(X,x) and B is sgp-open (or closed) set in
(X,x), then A n B e SGPLC*(X,x).
Proof:
Since A e SGPLC*(X,x), there exist a sgp-open set P and a closed
set Q such that A = P n Q. Now AnB = (PnQ)nB = (PnB)nQ. Since
P n B is sgp-open and Q is closed, it follows that AnBe SGPLC* (X,x).
In this case of B being a closed set, we have AnB = (PnQ)nB = Pn (Q
n B). Since P is sgp-open set and Q n B is closed. Thus AnBe SGPLC*
(X,x).
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Theorem 6.2.25: If A gSGPLC**(X,x) and B is sgp-closed (resp. open) set
in (X,t), then A n B
Proof: Since A
g
g
SGPLC**(X,x).
SGPLC* *(X,x), there exist an open set P and a sgp-closed
set Q such that A = PnQ. Now AnB = (PnQ)nB = P n(Q n B). Since
P is open and QnB is sgp-closed, Therefore A n B e SGPLC**(X,x).
In this case of B being an open set, we have A n B = (P n Q) nB = (P nB)
nQ. Since P nB is open and Q is sgp-closed, Thus A n B
g
SGPLC**(X,x).
Theorem 6.2.26: Let (X,x) and (Y, ct) be topological spaces.
1) If A € SGPLC (X,x) and B
g
SGPLC (Y, ct), then A x B
g
SGPLC
(X x Y, x x a )
2) If A
g
SGPLC*(X,x) and B
SGPLC*(Y, ct), then A x B
g
g
SGPLC*
(X x Y, x x a ).
3) If A
g
SGPLC**(X,t) and B
g
SGPLC** (Y, ct), then A x B
g
SGPLC**(X x Y, x x cr).
Proof: 1) Let A
g
SGPLC (X,x) and B
g
SGPLC (Y, cr). Then there exist
sgp-open sets M and M1 of (X,x) and (Y, a) and sgp-closed sets N and N1 of
X and Y respectively such that A= M n N and B = M1 n N1.
Then AxB = (MxM')n(Nx N') holds. Hence AxBg SGPLC (X x Y,
x x a ).
2) Let A
g
SGPLC*(X,x) and B
g
SGPLC*(Y, ct). Then there exist sgp-open
sets K and K1 of (X,x) and (Y, ct) and sgp-closed sets L and L 1 of X and Y
respectively such that A= K n L and B = K?nL/.
Then AxB = (KxK/)n(LxL/) holds. Hence AxBg SGPLC*(X x Y,
x x ct ).
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3) Let A e SGPLC** (X,x) and B e SGPLC**(Y, cr). Then there exist
open
sets W and W1 of (X,x) and (Y, a) and sgp-closed sets V and V ; of X and Y
respectively such that A- WnV and B = W;n V 1.
Then A x B = (W x W1) n (V x Vl) holds.
Hence AxBe SGPLC* *(X xY.ixo).
Definition 6.2.27: A topological space (X,x) is said to be sgp-submaximal if
every dense subset in it is sgp-open.
Theorem 6.2.28: Every submaximal space is sgp-submaximal.
Proof: Let (X,x) be a submaximal space and A be a dense subset of (X,x).
Then A is open. But every open set is sgp-open and so A is sgp-open.
Therefore (X,x) is sgp-submaximal.
The converse of the above theorem need not be true as seen from the
following example.
Example 6.2.29: In the Example 6.2-. 11, the space (X,x) is sgp-submaximal
but not submaximal, every dense subset is sgp-open. However the set A=
{a, b} is dense in (X,x), but it is not open in X. Therefore (X,x) is not
submaximal.
Theorem 6.2.30: Every ©-submaximal space is sgp-submaximal.
Proof: Let (X,x) be a ©-submaximal space and A be a dense subset of (X,x).
Then A is ©-open. But every ©-open set is sgp-open and so A is sgp-open.
Therefore (X,x) is sgp-submaximal.
The converse of the above theorem need not be true as seen from the
following example.
no
Example 6.2.31: Let X = {a, b, c} and x = {X, <j), {a}}. Then the space (X,x)
is sgp-submaximal but not an ra-submaximal.
Remark 6.2.32: g-submaximals and sgp-submaximals are independent as
seen from the following examples.
Example 6.2.33: In the Example 6.2.31, the space (X,x) is g-submaximal but
not a sgp-submaximal, because for the subset {a, c} is dense in (X,x) it is not
a sgp-open set in (X,x) but it is g-open in (X,x).
Example 6.2.34: Let X = {a, b, c} and x = {X, <(>, {a, b}}. Then the space
(X,x) is sgp-submaximal but not a g-submaximal, because for the subset {b,
c} is dense in (X,x) it is not a g-open set in (X,x) but it is sgp-open in (X,x).
Theorem 6.2.35: A topological space (X,x) is sgp-submaximal if and only if
SGPLC*(X,t) = P(X).
Proof: Necessity: Let Ae P(X). Let U = A u (X-pCl(A)). Then pCl(U) = X.
Since (X,x) is sgp-submaximal, U is sgp-open. By Theorem 4.2.23, A e
SGPLC*(X,x) and so P(X) = SGPLC*(X,x).
Sufficiency: Let A be a dense subset of (X,x). Then Au(X-pCl(A)) =
Au(()=A. Since A e SGPLC*(X,x), by Theorem 4.2.23, A is sgp-open in
(X,x). Hence (X,x) is sgp-submaximal.
Theorem 6.2.36: For a subset S of (X,x) , if Se SGPLC**(X,x), then there
exist an open set P such that S = PnpCl*(S), where pCl*(S) is the sgp-closure
of S (that is the intersection of all sgp-closed subsets of (X,x) that contains S).
Ill
Proof: Let Se SGPLC**(X,x), then there exist an open set P and a sgp-closed
set F of (X,x) such that S = PnF. Since ScP and S c pCl*(S), we have S c
PnpCl*(S). Since pCl*(S) c F, we have PnpCl*(S) cPnF = S. Thus S =
PnpCl*(S).
6.3 SGPLC- Continuous Functions in Topological Spaces
In this section we introduce the concept of SGPLC-continuous,
SGPLC*-continuous and SGPLC* *-continuous functions which are weaker
than LC-continuous functions.
Definition 6.3.1: A function f: (X, x) -> (Y, a) is called SGPLC-continuous
(resp. SGPLC*-continuous, SGPLC**-continuous) if f ~!(V) e SGPLC(X, x)
(resp. r'(V) e SGPLC*(X, x), f~\V) e SGPLC* *(X, x)) for each open set
V of (Y, ct)).
Example 6.3.2: Let X = Y = {a, b, c}. Let x={<(), {a}, X} and cr ={<j>, Y,
{a, b}}. Then SGPLC(X, x) = SGPLC*(X, x) = SGPLC**(X, x) = P(X) and
the identity function f: (X, x) -> (Y, ct) is SGPLC-continuous, SGPLC*continuous and SGPLC**-continuous.
Theorem 6.3.3: Let f: (X, x) -» (Y, ct) be a function. Then we have the
following
1) If f is LC-continuous, then it is SGPLC-continuous, SGPLC*continuous and SGPLC**-continuous.
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2) If f is SGPLC*-continuous or GSPLC**-continuous, then it is SGPLCcontinuous.
Proof: 1) Suppose that f is LC-continuous. Let V be an open set of (X, x) .
Then f _!(V) is locally closed in (X, x). Since every locally closed set is
SGPLC-set, SGPLC*-set and SGPLC**-set, it follows that f is SGPLCcontinuous, SGPLC*-continuous and SGPLC**-continuous.
2) Let f: (X, x) -> (Y, cr) be a SGPLC*-continuous or SGPCL**-continuous
function. Since every SGPLC*-set is SGPLC-set and every SGPLC**-set is
SGPLC-set. Therefore the proof follows.
The converse of the above theorem need not be true as seen from the follows
from the example.
Example 6.3.4: Let X = Y = {a, b, c}. Let x= {<(), {a}, X} and a = P(Y) = {<j>,
Y,{a},{b},{c},{a, b},{b, c},{a, c}}. Let f: (X, x) -> (Y, cr) be the identity
function. Now locally closed sets of (X, x) are: X, <|>, {a}, {b, c}.
SGPLC*(X, x) = SGPLC**(X, x) = SGPLC(X, x) = P(X) and Locally closed
sets of (Y, ct) are : P(Y) SGPLC*(Y, a) = SGPLC**(Y, a) = SGPLC(Y, a) =
P(Y). Then f is not LC-continuous, because for the open set {a, c}, f“'({a, c})
= {a, c} is not locally closed in X, but it is SGPLC*-continuous, SGPLC**continuous.
Example 6.3.5: Let X = Y = {a, b, c}. Let x= {<{>, {a}, X} and a = {((), {a, b},
Y}. Then the identity function f: (X, x) -> (Y, a) is SGPCL-continuous but
not SGPLC**-continuous because for the open set {a, b} of (Y, a),f !({a, b})
= {a, b} is not SGPLC**~set in X but it is SGPLC-set in X.
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•
Theorem 6.3.6: If f: (X, x) -» (Y, o) is SGPLC-continuous and g: (Y, a) —»
(Z. r|) is continuous, then gof: (X,x) -» (Z,r|) is SGPLC-continuous function.
Proof: Let F be a closed set in (Z,r|). Then g
(F) is closed set in (Y,a),
since g is continuous. And then f “'(g “1 (F)) is SGPLC-set in (X,x) as f is
SGPLC-continuous. Thus gof is SGPLC-continuous function.
Theorem 6.3.7: If f: (X,x) -» (Y,ci) and g: (Y,g) -> (Z,r|) be any two
functions Then
i) gof is SGPLC * -continuous if f is SGPLC*-continuous and g is
continuous.
ii) gof is SGPLC**-continuous if f is SGPLC**-continuous and g is
continuous.
Proof: The proof is similar to Theorem 6.3.6.
6.4 SGPLC- Irresolute Functions in Topological Spaces
In this section, we define SGPCL-irresolute, SGPLC*-irresolute and
SGPLC* *-irresolute functions which are weaker than LC-irresolute functions
in topological spaces and study some of their properties.
Definition 6.4.1: A function f: (X, x) -» (Y,
a)
is called SGPLC-irresolute
(resp. SGPLC*-irresolute, SGPLC**-irresolute) if f -,(V) e SGPLC(X, x)
(resp. f -‘(V) eSGPLC*(X, x), f "'(V) e SGPLC**(X, x)) for each V e
SGPLC(Y,o) (resp. V € SGPLC*(Y, a), V e SGPLC**(Y, a)).
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Example 6.4.2: In Example 6.3.2, the function f is SGPLC-irresolute.
Theorem 6.4.3: If a function f: (X, x) -» (Y, <r) is LC-irresolute, then f is
SGPLC-irresolute, SGPLC*-irresolute and SGPLC**-irresolute.
Proof: Suppose that f is LC-irresolute. Let V be a locally closed set of (X, x).
Then f “’(V) is locally closed in (X, x). Since every locally closed set is
SGPLC-set, SGPLC*-set and SGPLC**-set, it follows that f is SGPLCirresolute, SGPLC*- irresolute and SGPLC**- irresolute.
The converse of the above theorem need not be true as seen from the follows
from the example.
Example 6.4.4: Let X = Y = {a, b, c}. Let x= (<j>, {a}, X} and a = P(Y) = {<)>,
Y, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}}. Let f: (X, x) -» (Y, a) be the identity
function. Now locally closed sets of (X, x) are: X, <j), {a}, {b, c}.
SGPLC*(X, x) = SGPLC**(X, x) = SGPLC(X, x) = P(X) and Locally closed
sets of (Y, a) are : P(Y) SGPLC*(Y, o) = SGPLC**(Y, a) = SGPLC(Y, a) =
P(Y). Then f is not LC-irresolute, because for the locally closed set {a, c},
f'({a, c}) = {a, c} is not locally closed in X, but it is SGPLC- irresolute,
SGPLC*- irresolute, SGPLC**- irresolute.
Theorem 6.4.5: Let f: (X, x) -» (Y, a) be a function. Then if f is a SGPLCirresolute (resp. SGPLC*-irresolute, SGPLC**-irresolute), then it is SGPLCcontinuous (resp. SGPLC*-continuous, SGPLC**-continuous).
Proof:
The proof follows from the fact that every LC-set is SGPLC-set,
SGPLC*-set and SGPLC**-set.
The converse of the above theorem need not be true as seen from the
following example.
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Example 6.4.6: Let X = Y = {a, b, c, d}. Let x= |X, <j>, {a}, {a, b}} and ct
= {Y,<j>, {a, b}}. Define a function f: (X, t) -> (Y, ct) by f(a) = a, f(b) = c, f(c)
= b and f(d) = c. Then f is SGPLC-continuous and SGPLC*-continuous but
not SGPLC-irresolute and SGPLC*-irresolute because for every SGPLC-set
(resp. SGPLC*-set) {b, c} in (Y, ct), f _,({b, c}) = (c, d} is not SGPLCirresolute (resp. SGPLC*-irresolute) in (X, x).
Example 6.4.7: In Example 6.4.6, the function f is GPLC * * -continuous but
not SGPLC* *-irresolute, because for every SGPLCS**-set {a, b} of (Y, ct),
f ~‘({a, b) = {a, c} is not GSPLC** in (X, x).
Theorem 6.4.8: If f: (X, x) -» (Y, ct) and g: (Y, ct) -> (Z, rj) be any two
functions Then
1)
gof: (X,x) -> (Z, r)) is SGPLC-irresolute if g is SGPLC-irresolute and
f is SGPLC-irresolute.
2)
gof: (X,x) -» (Z,r|) is SGPLC-continuous if g is SGPLC-continuous
and f is SGPLC-irresolute.
Proof: 1) Let Fe SGPLC (Z,r|). Then g _1 (F) is SGPLC-set in (Y,ct), since g
is SGPLC -irresolute. As f is SGPLC-irresolute, f ~'(g _1 (F)) is SGPLC-set in
(X,x). That is (gof)"1 (F) € SGPLC (X,x). Thus gof is SGPLC-irresolute.
2) Let F be a closed set in (Z,rj). Then g
(F) is SGPLC- set in (Y,ct), since
g is SGPLC-continuous. Again since f is SGPLC-irresolute, r,(g"1(F)) is
SGPLC-set in (X,x). Thus gof is SGPLC-continuous function.
Theorem 6.4.9: If f: (X,x) -> (Y,ct) and g: (Y,ct) -> (Z,ti) be any two
functions Then
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1)
gof is SGPLC*-irresolute if f and g are SGPLC*-irresolute.
2)
gof is SGPLC**-irresolute if f and g are SGPLC**-irresolute.
3)
gof is SGPLC*-continuous if f is SGPLC*-irresolute and g is
SGPLC * -continuous.
4)
gof is SGPLC**-continuous if f is SGPLC**-irresolute and g is
SGPLC* *-continuous.
Proof: The proof is similar to Theorems 6.3.6 and 6.4.8.
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