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CHAPTER 4
g*p- LOCALLY CLOSED SETS IN TOPOLOGICAL SPACES
§4.1 Introduction
The notion of a locally closed set in a topological space was
introduced by Kuratowski and Sierpinski [53]. According to Bourbaki [15],
a subset A of a topological space X is called locally closed in X if it is the
intersection of an open set and a closed set in X. Ganster and Reilly [43]
used locally closed sets to define LC- Continuity and LC-irresoluteness.
Balachandran, Sundaram and Maki [10] introduced the concept of
generalized locally closed sets in topological spaces and investigated some
of their properties. Recently Sheik John [106] introduced the three new
classes of sets denoted by co-LC(X, x), co-LC*(X, x) and co-LC**(X, x), each
of which contains LC(X, x). Also various authors like, Arockiarani [3],
Gnanambal [47], Park and Park [93] and Veera Kumar [123] have
introduced regular-generalized locally closed sets, a-locally closed sets,
semi-generalized locally closed sets and g*-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 G*PLC(X,x),
G*PLC*(X,x) and G*PLC**(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 g*lc-set. Also we introduce g*p-submaximal spaces and obtain
some of their properties.
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In section 3, we introduce the concepts of G*PLC-continuous,
G*PLC*-continuous and G*PLC* *-continuous functions which are weaker
than LC-continuous functions.
Moreover in the last section, we define G*PCL-irresolute, G*PLC*irresolute and G*PLC* * -irresolute functions which are also weaker than LCirresolute functions in topological spaces and study some of their properties.
We recall the following definitions, which are prerequisites for our
present work
Definition 4.1.1: A subset A of a topological space (X,x) is called
(i)
locally closed (briefly lc) set [43] if A=UnV where U is open and
V is closed in (X,x).
(ii)
generalized locally closed (briefly glc) set [10] if A=UnV where U
is g-open and V is g-closed in (X,x).
(iii)
strongly generalized locally closed (briefly g*lc [96]) set [123] if A
= Un
V where U is strongly g-open (g*-open) and V is strongly
g-closed (g*-closed) in (X,x).
(iv)
a-locally closed (briefly ale) set [47] if A = UnV where U is
a-open and V is a-closed in (X,x).
(v)
P-locally closed (briefly pic) set [48] if A = UnV where U is
P-open and V is p-closed in (X,x).
(vi)
co-locally closed (briefly cole) [106] if A = U n V where U is co­
open and V is co-closed in (X,x).
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Definition 4.1.2: A topological space (X,x) is called a
(i)
submaximal space [29] if every if every dense subset of (X,t) is
open in (X,x).
(ii)
g-submaximal space [10] if every if every dense subset of (X,x) is
g-open in (X,t).
(iii)
g*-submaximal space [123] if every if every dense subset of (X,x)
is g*-open in (X,t).
(iv)
door space [30] if every subset of (X,x) is either open or closed in
(X,T).
Ganster and Reilly [43] have introduced the weaker forms of
continuous function known as locally continuous functions denoted by LCcontinuous. Later Balachandran et al [10] and Sundaram [112] introduced
and studied generalized locally continuous and semi-generalized locally
continuous functions in topological spaces. Later on Gnanambal [47], Park
and Park [93], Veerakumar [123] and Sheik John [106] have introduced alocally
continuous, generalized semi locally continuous, g* - locally
continuous and co-locally continuous functions in topological spaces.
We give some definitions of locally closed continuous functions from
various authors.
Definition 4.1.3 .A function f: (X, t) —»(Y, a) is called
i)
LC- continuous [43] if f *(G) is locally closed set in (X, x) for each
closed set G of (Y, a).
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ii)
GLC-continuous [10] if f'(G) is generalized locally closed (briefly
glc) set in (X, x) for each closed set G of (Y, a).
iii)
co-LC-continuous [106] if f'(G) is 0) - locally closed (briefly a)-lc) set
in (X, x) for each closed set G of (Y, a).
iv)
G*LC-continuous [123] if f '(G) is g*- locally closed (briefly g*lc) set
in (X, x) for each closed set G of (Y,
v)
g).
oeLC-continuous [47] if f l(G) is a-locally closed (briefly ale) set in
(X, x) for each closed set G of (Y, a).
Definition 4.1.4.A function f: (X, x) —> (Y, a) is called
i)
LC-irresolute [43] if f1 (G) is locally closed set in (X, x) for locally
closed set G of (Y, a).
ii)
GLC-irresolute [10] if f '(G) is generalized locally closed set in (X, x)
for each generalized locally closed G of (Y, a).
iii)
G*LC-irresolute [123] if f!(G) is g*- locally closed set in (X, x) for
each g*- locally closed G of (Y, a).
iv)
(O-LC-irresolute [106] if f'(G) is co-lc-set in (X, x) for each co-lc-set G
of (Y, a).
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§4.2 g*p- Locally Closed Sets
In this section we introduce three weaker forms of locally closed sets
denoted by G*PLC(X,x), G*PLC*(X,x) and G*PLC**(X,x) each of which
contains LC(X,x) and obtain some of their properties and also their
relationships with glc-set, a-lc-set and g*lc-set. Also we introduce g*psubmaximal spaces and obtain some of their properties.
Definition 4.2.1: A subset A of a topological space (X, t) is called a g*-pre
locally closed set (briefly g*plc-set) if A = S n F where S is g*p-open and F
is g*p-closed.
?
The class of all g*-pre locally closed sets in (X,x) is denoted by
G*PLC(X,x).
Definition 4.2.2: A subset A of a topological space (X,x) is said to be
G*PLC*-set if there exist g*p-open set S and a closed set F of (X,x) such
that A = S n F.
Definition 4.2.3: A subset A of a topological space (X,x) is said to be
G*PLC**-set if there exist an open set S and a g*p-closed set F of (X,x)
such that A = Sn F.
Theorem 4.2.4: If a subset A of (X,x) is locally closed then it is G*PLC
(X,x), G*PLC*(X,x) and G*PLC**(X,x) set.
Proof: Let A = P n Q, where P is open and Q is closed in (X, x). Since
every open set is g*p-open and every closed set is g*p-closed, A is G*PLC
(X,x), G*PLC*(X,x) and G*PLC**(X,x).
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The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.5: Let X = {a, b, c} and x = {0, X, {a}, {a, b}}. Then the
locally closed sets are: X,0, {a}, {b}, {c}, {a, b}, {b, c} and G*PLC(X,x) =
G*PLC*(X,x) = G*PLC**(X,x) = X, 0, {a}, {b}, {c}, {a, b}, {b, c}, {a, c}.
Here {a, c} is G*PLC (X,x), G*PLC*(X,x), G*PLC**(X,x) but not LC-set.
Theorem 4.2.6: If a subset A of (X,x) is G*PLC*-set then it is G*PLC-set.
Proof: Let A be a G*PLC*-set. Let P be a g*p-open set in (X,x) and Q be a
closed set in (X,x). Since A is G*PLC*-set by definition, A = P n Q. Since
every closed set is g*p-closed, A is G*PLC-set.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.7: Let X = {a, b, c, d} and x = {0, X, {b}, {c, d}, {b, c, d}}.
Then the subset {a, c, d} e G*PLC(X, x) but {a, c, d} g G*PLC*(X, x).
Theorem 4.2.8: Every G*PLC**(X, x) is G*PLC (X, x).
Proof: The proof follows from the definitions 4.2.1 and 4.2.3.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.9: Let X = {a, b, c} and x = {0, X, {a}}. Then the subset {a, b}
e G*PLC(X, x) but {a, b} <2 G*PLC**(X, x).
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Theorem 4.2.10: If a subset A of (X,t) is a-lc set, then A is G*PLC-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 g*p-open and every a-closed set is g*p-closed,
Therefore S is G*PLC-set in (X,x).
The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.11: Let X = {a, b, c} and x = {X, <j>, {a}, {b, c}}. Then the a-lc
sets are: X, 0, {a}, {b, c} and G*PLC-sets are: P(X). Here {a, b} is G*PLCset but not an a-lc set.
Theorem 4.2.12: Every g*-dosed set is g*p-closed set but not conversely.
Proof: Let A be a g*-closed of (X,x). Let U be a g-open set in (X,x) such
that A c U. Then cl(A) c U. But pcl(A) c cl(A) is always true. Therefore
pcl(A) c U. Hence A is g*p-closed set in (X,x).
Example 4.2.13: Let X = {a, b, c} and x = {X, ((), {a}, {b, c}}. Then the
subset {a, c} is g*p-closed set but not a g*-closed set in (X,x)
Theorem 4.2.14: Every g*lc-set (resp. g*lc*-set, g*le**-set) is g*plc-set
(resp. g*plc*-set, g*plc**-set).
Proof: Since every g*-closed set is g*p-closed and every g*-open set is g*popen, the proof follows.
The converse of the above theorem need not be true as seen from the
following example.
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Example 4.2.15: In Example 4.2.13, g*lc(X,x) = g*lc* (X, x) = g*lc**(X, x)
= X, <j), {a}, {b, c} and g*plc (X, x) = g*plc* (X, x) = g*plc** (X, x)= P(X).
Here the subset {a, b} is g*plc-set (resp. g*plc*-set, g*plc**-set) but not a
g*lc-set (resp. g*lc*-set, g*le!f!*-set)
Remark 4.2.16: The concept of glc*-sets (resp. glc**-sets) and g*plc*-sets
(resp. g*plc**-sets) are independent as seen from the following examples.
Example 4.2.17: Let X = {a, b, c, d} and x = {X, <]>, {a}, (a, d}, {a, b, d},
{a, c, d}. Then glc*-sets are: X, <J>, {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}, {c},
{d}, {a, b}, {a, c}, {b, c}, {b, d}, {c, d}, (a, b, d}, {a, c, d}, {b, c, d} and
g*plc*-sets = g*ple**-sets = P(X). Here the subset {a, b, c} is g*plc*-set
(resp. g*plc**-set) but not a glc*-set (resp. glc**-set).
Example 4.2.18: Let X = {a, b, c, d} and x = {X, <)>, {a}, (a, b}}. Then glc*sets = glc**-sets = P(X) and g*plc*-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
g*plc**-sets are: X, (J), {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
g*plc*-set (resp. g*plc**-set).
Theorem 4.2.19: If A eG*PLC (X,x) and B is g*p-open set in (X,x), then
A n B g G*PLC (X,x).
Proof: Since A
g
G*PLC (X,x), there exist a g*p-open set P and a g*p-
closed set Q such that A = P n Q. Now A n B = (P n Q) n B =(PnB)n
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Q. Since P n B is g*p-open and Q is g*p-closed, it follows that AnBe
G*PLC (X,x).
Remark 4.2.20: Union of two G*PLC-sets (resp. G*PLC*-sets, G*PLC**sets) need not be a G*PLC-set (resp. G*PLC*-set, G*PLC**-set) as seen
from the following example.
Example 4.2.21: In Example 4.2.18, the subsets {a} and {c, d} are two
G*PLC (X,t) (resp. G*PLC* (X,x), G*PLC** (X,x)) sets but their union {a}
u (c, d} = {a, c, d} is not G*PLC (X,x) -set (resp. G*PLC*(X,x),
G*PLC** (X,x)- set).
Remark 4.2.22: From the above results we get the following implications.
g*plc**-set
A
g*lc**-set
a-lc-set
where A —►B (resp. A
B) represents A implies B but not conversely
(resp. A and B are independent).
no
Theorem 4.2.23: For a subset A of (X,x), the following are equivalent:
1)
A G G*PLC*(X, x)
2)
A = P n pci (A) for some g*p-open set P.
3)
pci (A)-A is g*p-closed.
4)
A u (X-pcl(A)) is g*p-open.
Proof: (1) => (2):- Let Ae G*PLC* (X,x). Then there exists a g*p-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 g*p-open and pcl(A) is closed.
P npcl(A) G G*PLC* (X,x), which implies that Ag C,*PLC* (X,x).
(3) =► (4) :- Let F = pcl(A)-A. Then F is g*p-closed by the assumption and
X - F = X n (pcl(A) - A)c = A u (X-pcl(A)).But X-F is g*p-open. This
shows that A u (X - pcl(A)) is g*p-open .
(4) =» (3):- Let U = A u (X-pcl(A)). Since U is g*p-open, X-U is g*pclosed. X - U = X- (A u (X - pcl(A))) = pcl(A) n (X-A) =pcl(A) - A. Thus
pcl(A) - A is g*p-closed set.
(4) =» (2):- Let P = A u (X - pcl(A)) Thus P is g*p-open . We prove that
A = Pn pcl(A) for some g*p-open set P.Pn pcl(A) =(Au (X-pcl(A)))
n pcl(A) = (pcl(A) n A) u (pcl( A) n (X - pcl(A))) = A u <|> = A. Therefore
A = P n pci (A).
in
(2) => (4):- Let A = P n pci (A) for some g*p-open set P. Then we prove
that A u (X-pcl(A)) is g*p-open. Now A u (X-pcl(A)) = (P n pci (A)) u
(X - pcl(A)) = Pn (pci (A) u (X - pcl(A))) = P, which is g*p-open. Thus
Au(X- pcl(A)) is g*p-open .
Theorem 4.2.24: If A, Be G*PLC (X,t), then A n B e G*PLC (X,x).
Proof: From the assumptions, there exist g*p-open sets P and Q such that
A = P n pci (A) and B = Q n pci (B). Then AnB = (PnQ)n (pci (A) n
pcl(B)). Since P n Q is g*p-open set and pci (A) n pci (B) is closed.
Therefore A n B 6 G*PLC (X,x).
Theorem 4.2.25: If A e G*PLC (X,t) and B is g*p-closed set in (X,x), then
AnBs G*PLC (X,x).
Proof: Since A e G*PLC (X,x), there exist a g*p-open set P and a g*pclosed set Q such that A = P n Q. Now AnB = (PnQ)nB = P n(Q n
B). Since P is g*p-open and Q n B is g*p-closed, AnBs G*PLC (X,x).
Theorem 4.2.26: If A e G*PLC*(X,x) and B is g*p-open (or closed) set in
(X,t), then A n B e G*PLC*(X,x).
Proof:
Since A e G*PLC*(X,x), there exist a g*p-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 g*p-open and Q is closed, it follows that AnBs G*PLC*
(X,t).
In this case B being a closed set, we have AnB = (PnQ)nB =
Pn(Q n B). Since P is g*p-open set and Q n B is closed, AnBs
G*PLC* (X,x).
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Theorem 4.2.27: If A g G*PLC**(X,t) and B is g*p-closed (resp. open) set
in (X,x), then A n B
g
G*PLC**(X,x).
Proof: Since A g G*PLC**(X,t), there exist an open set P and a g*p-closed
set Q such that A = P n Q. Now AnB = (PnQ)nB = P n(Q n B). Since
P is open and QnB is g*p-closed, A n B
g
G*PLC**(X,x).
In this case 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 g*p-closed, A n B G
G*PLC**(X,x).
Theorem 4.2.28: Let (X,x) and (Y, cr) be topological spaces.
1) If A G G*PLC (X,x) and B G G*PLC (Y, a), then AxBg G*PLC
(X x Y, x x a)
2) If A G G*PLC*(X,x) and B G G*PLC*(Y, a), then AxBg G*PLC*
(X x Y, x x a).
3) If A
g
G*PLC**(X,x) and B
g
G*PLC** (Y, a), then AxBg
G*PLC**(X x Y, x x a).
Proof: 1) Let A
g
G*PLC (X,x) and B
g
G*PLC (Y, a). Then there exist
g*p-open sets M and M1 of (X,x) and (Y, a) and g*p-closed sets N and N 1
ofX and Y respectively such that A= M n N and B = M,nNl.
Then AxB = (MxM/)n(NxN/) holds. Hence AxBg G*PLC (X x Y,
xx a).
2) Let A G G*PLC*(X,x) and B G G*PLC*(Y, a). Then there exist g*popen sets K and K;of (X,x) and (Y, o) and g*p-closed sets L and L1 of (X,x)
and (Y, a) respectively such that A=KnL and B = K /nL/.
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Then AxB = (KxK/)n(LxL/) holds. Hence AxBe G*PLC*(X x Y,
xx
ct).
3) Let A e G*PLC** (X,x) and B e G*PLC**(Y, a). Then there exist
open sets W and W1 of (X,x) and (Y, a) and g*p-closed sets V and V 1 of
(X,t) and (Y, o) respectively such that A= WnV and B = W,nV/.
Then AxB = (WxW/)n(VxV/) holds. Hence AxB e G*PLC**(XxY,
xx a).
Definition 4.2.29: A topological space (X,x) is said to be g*p-submaximal
if every dense subset in it is g*p-open.
Theorem 4.2.30: Every submaximal space is g*p-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 g*p-open and so A is g*p-open.
Therefore (X,x) is g*p-submaximal.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.31: In the Example 4.2.13, the space (X,x) is g*p-submaximal
but not submaximal. 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 4.2.32: Every g*-submaximal space is g*p-submaximal.
Proof: Let (X,x) be a g*-submaximal space and A be a dense subset of
(X,x). Then A is g*-open. But every g*-open set is g*p-open and so A is
g*p-open. Therefore (X,x) is g*p-submaximal.
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The converse of the above theorem need not be true as seen from the
following example.
Example 4.2.33: In the Example 4.2.13, the space (X,x) is g*p-submaximal
but not a g*-submaximal.
Remark 4.2.34: g-submaximals and g*p-submaximals are independent as
seen from the following examples.
Example 4.2.35: In the Example 4.2.9, the space (X,x) is g-submaximal but
not a g*p-submaximal, since for the subset {a, c} is dense in (X,x) which is
not a g*p-open set in (X,x) but it is g-open in (X,x).
Example 4.2.36: Let X = {a, b, e} and x = {X, <j), {a, b}}. Then the space
(X,x) is g*p-submaximal but not a g-submaximal, since the subset {b, c} is
dense in (X,x) which is not a g-open set in (X,x) but it is g*p-open in (X,x).
Theorem 4.2.37: A topological space (X,x) is g*p-submaximal if and only if
G*PLC*(X,x) = P(X).
Proof: Necessity: Let Ae P(X). Let U = A u (X - pcl(A)). Then pcl(U) = X.
Since (X,x) is g*p-submaximal, U is g*p-open. By Theorem 4.2.23, A
e
G*PLC*(X,x) and so P(X) = G*PLC*(X,x).
Sufficiency: Let A be a dense subset of (X,x). Then Au(X - pcl(A)) =
Auct>=A. Since A E G*PLC*(X,x), by Theorem 4.2.23, A is g*p-open in
(X,x). Hence (X,x) is g*p-submaximal.
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Theorem 4.2.38: For a subset S of (X,x), if Se G*PLC**(X,x), then there
exists an open set P such that S = Pnpcl*(S), where pcl*(S) is the g*pclosure of S (that is the intersection of all g*p-closed subsets of (X,x) that
contains S).
Proof: Let Se G*PLC**(X,x). Then there exist an open set P and a g*pclosed 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) c P n F = S.
Thus S - Pnpcl*(S).
§ 4.3 G*PLC- Continuous Functions in Topological Spaces
In this section we introduce the concept of G*PLC-continuous,
G*PLC*-continuous and G*PLC**-continuous functions which are weaker
than LC-continuous functions.
Definition 4.3.1: A function f: (X, x) —» (Y, a) is called G*PLC-continuous
(resp. G*PLC*-continuous, G*PLC**-continuous) iff_1(V) e G*PLC(X, x)
(resp. f-1(V) e G*PLC*(X,x), f'_1(V) e G*PLC**(X, x)) for each open set
V of(Y, g)).
Example 4.3.2: Let X = Y = {a, b, c}. Let x={X,
{a}} and ct ={Y,(1),
{a, b}}. Then G*PLC(X, x) = G*PLC*(X, x) = G*PLC**(X, x) = P(X) and
the identity function f: (X, x) —» (Y, ct) is G*PLC-continuous, G*PLC*continuous and G*PLC**-continuous.
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Theorem 4.3.3: Let f: (X, x) —> (Y, a) be a function. Then we have the
following
1) If f is LC-continuous, then f is G*PLC-continuous, G*PLC*continuous and G*PLC**-continuous.
2) If f is G*PLC*-continuous or G*PLC**-continuous function, then f is
G*PLC-continuous.
Proof: 1) Suppose that f is LC-continuous. Let V be an open set of (X, x) ,
Then f _1(V) is locally closed in (X, x). Since every locally closed set is
G*PLC-set, G*PLC*-set and G*PLC**-set, it follows that f is G*PLCeontinuous, G*PLC*-continuous and G*PLC**-continuous.
2) Let f: (X, x) —» (Y, a) be a G*PLC*-continuous or G*PCL**-continuous
function. Since every G*PLC*-set is G*PLC-set and every G*PLC**-set is
G*PLC-set. Therefore the proof follows.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.3.4: Let X = Y = {a, b, e}. Let x= {X,<|>, {a}} and a = P(Y) =
{Y, (|>,{a},{b},{c},{a, b},{b, c},{a, c}}. Let f: (X, x) -» (Y,
function. Now locally closed sets of (X, x) are: X,
a)
be the identity
{a}, {b, c}.
G*PLC*(X, x) = G*PLC**(X, x) = G*PLC(X, x) = P(X) and Locally closed
sets of (Y, ct) are : P(Y), G*PLC*(Y, o) = G*PLC**(Y, a) = G*PLC(Y, a)
= P(Y). Then f is not LC-continuous, since for the open set {a, c},
r*({a, c}) = {a, c} is not locally closed in X, but it is G*PLC*-continuous,
G*PLC**-continuous.
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Example 4.3.5: Let X = Y = {a, b, c}. Let x= {X, <J>, {a, b}} and a = {Y, 0,
{a, c}}. Then the identity function f: (X, x) —> (Y, a) is G*PCL-continuous
but not G*PLC**-continuous, since for the open set {a, b} of (Y, a),
f _1({a, b}) = {a, b} is not G*PLC**-set in X but it is G*PLC-set in X,
Theorem 4.3.6: If f: (X, x) —> (Y, a) is G*PLC-continuous and g: (Y, cr) —»
(Z, r\) is continuous, then gof: (X,x) —> (Z,rj) is G*PLC-continuous fimction.
Proof: Let F be a closed set in (Z,T|). Then g _1 (F) is closed set in (Y,o),
since g is continuous. And then f ~‘(g _1 (F)) is G*PLC-set in (X,x) as f is
G*PLC-continuous. Thus gof is G*PLC-continuous function.
Theorem 4.3.7: If f: (X,x) —> (Y,ct) and g: (Y,a) —> (Z,ri) be any two
functions Then
i) gof is G*PLC*-continuous if f is G*PLC*-continuous and g is
continuous.
ii) gof is G*PLC**-continuous if f is G*PLC**-continuous and g is
continuous.
Proof: The proof is similar to that of Theorem 4.3.6.
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§4.4 G*PLC- Irresolute Functions in Topological Spaces
In this section, we define G*PCL-irresolute, G*PLC*-irresolute and
G*PLC**-irresolute
functions
which
are weaker than
LC-irresolute
functions in topological spaces and study some of their properties.
Definition 4.4.1: A function f: (X, x) —» (Y, a) is called G*PLC-irresolute
(resp. G*PLC*-irresolute, G*PLC**-irresolute) if f -1(V) e G*PLC(X, x)
(resp. f _1(V) sG*PLC*(X, t), f _1(V) e G*PLC**(X, x)) for each V e
G*PLC(Y, a) (resp. V e G*PLC*(Y, a), V e G*PLC**(Y, a)).
Example 4.4.2: In Example 4.3.2, the function f is G*PLC-irresolute.
Theorem 4.4.3: If a function f: (X, x) —» (Y, a) is LC-irresolute, then f is
G*PLC-irresolute, G*PLC*-irresolute and G*PLC**-irresolute.
Proof: Suppose that f is LC-irresolute. Let V be a locally closed set of
(X, x). Then f _1(V) is locally closed in (X, x). Since every locally closed set
is G*PLC-set, G*PLC*-set and G*PLC**-set, it follows that f is G*PLCirresolute, G*PLC*- irresolute and G*PLC**- irresolute.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.4.4: Let X = Y = {a, b, c}. Let x= (cj), {a}, X} and
g
=
P(Y) =
{()), Y,{a},{b},{c},{a, b},{b, c},{a, c}}. Let f: (X, x) —»(Y, o) be the identity
function. Now locally closed sets of (X, x) are: X, <]>, {a}, {b, c}.
G*PLC*(X, x) = G*PLC**(X, x) = G*PLC(X, x) = P(X) and Locally closed
sets of (Y,
a)
are : P(Y) G*PLC*(Y, a) = G*PLC**(Y, a) = G*PLC(Y, a)
= P(Y). Then f is not LC-irresolute, since for the locally closed set {a, c},
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f'({a, c}) = {a, c} is not locally closed in X, However f is G*PLCirresolute, G*PLC*- irresolute, G*PLC**- irresolute.
Theorem 4.4.5: Let f: (X, x) —» (Y, a) be a function. If f is G*PLCirresolute (resp. G*PLC*-irresolute, G*PLC* *-irresolute), then f is G*PLCcontinuous (resp. G*PLC*-continuous, G*PLC**-continuous).
Proof: Since every LC-set is G*PLC-set, G*PLC*-set and G*PLC**-set,
the proof follows.
The converse of the above theorem need not be true as seen from the
following example.
Example 4.4.6: Let X = Y = {a, b, c, d}. Let x= {X, 0, {a}, {a, b}} and c
={Y,0, {a, b}}. Define a function f: (X, x) —> (Y, a) by f(a) = a, f(b) = c,
f(c) = b and f(d) = c. Then f is G*PLC-continuous and G*PLC*-continuous
but not G*PLC-irresolute and G*PLC*-irresolute, since for the G*PLC-set
(resp. G*PLC*-set) {b, c} in (Y, a), f ~’({b, c}) = {c, d} is not G*PLC-set
(resp. G*PLC*-set) in (X, x).
Example 4.4.7: In Example 4.4.6, the function f is G*PLC**-continuous
but not G*PLC**-irresolute, since for the G*PLC**-set {a, b} in
(Y, a), f-1({a, b) = {a, c} is not G*PLC**-set in (X, x).
Theorem 4.4.8: If f: (X, x) -» (Y, a) and g: (Y, a) —> (Z, r|) be any two
functions. Then
1)
gof: (X,x) —> (Z, p) is G*PLC-irresolute if g is G*PLC-irresolute and
f is G*PLC-irresolute.
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2)
gof: (X,x) —> (Z,T|) is G*PLC-continuous if g is G*PLC-continuous
and f is G*PLC-irresolute.
Proof: 1) Let FeG*PLC (Z,rj). Then g_1(F) is G*PLC-set in (Y,cr) since g
is G*PLC -irresolute. As f is G*PLC-irresolute, f_1(g_l (F)) is G*PLC-set
in (X,x). That is (gof)_1(F) <=G*PLC (X,x). Thus gof is G*PLC-irresolute.
2) Let F be a closed set in (Z,r|). Then g-1(F) is G*PLC- set in (Y,a) since g
is G*PLC-continuous. Again since f is G*PLC-irresolute, L^g-^F)) is
G*PLC-set in (X,x). Thus gof is G*PLC-continuous function.
Theorem 4.4.9: If f: (X,x) —> (Y,a) and g: (Y,ct) —> (Z,r\) be any two
functions. Then
1)
gof is G*PLC*-irresolute if f and g are G*PLC*-irresolute.
2)
gof is G*PLC**-irresolute if f and g are G*PLC**-irresolute.
3)
gof is G*PLC*-continuous if f is G*PLC*-irresolute and g is
G*PLC*-continuous.
4)
gof is G*PLC**-continuous if f is G*PLC * * -irresolute and g is
G*PLC* “"-continuous.
Proof: The proof is similar to that of Theorems 4.3.6 and 4.4.8.
121