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International Journal of Mathematics and Computer
Applications Research (IJMCAR)
ISSN 2249-6955
Vol. 3, Issue 2, Jun 2013, 249-258
© TJPRC Pvt. Ltd.
SOME NEW WEAKER FORMS OF LOCALLY CLOSED SETS IN
TOPOLOGICAL SPACES
P. G. PATIL
Department of Mathematics, SKSVM Agadi College of Engineering and Technology, Laxmeshwar, Karnataka, India
ABSTRACT
In this paper we introduce new weaker forms of locally closed sets called lc-sets, lc*-sets and lc**-sets and
study some of their properties. Also introduce, LC-continuous, LC*-continuous, LC**-continuous functions and their
irresolute maps and study their properties in topological spaces.
KEYWORDS: -Closed, Locally Closed, -Locally Closed
2000 AMS Mathematics Subject Classification: 54A05, 54G65
1. INTRODUCTION
Kuratowski and Sierpinski [11] introduced the notion of locally closed sets in topological spaces. According to
Bourbaki [6], a subset of a topological space (X,) is locally closed in (X,) if it is the intersection of an open set and a
closed set in (X,). Stone [14] has used the term FG for a locally closed subset. Ganster and Reilly [9] have introduced
locally closed sets, which are weaker forms of both closed and open sets. After that Balachandran et al [2,3], Gnanambal
[10], Arockiarani et al [1],Pusphalatha [12] and Sheik John[13] have introduced -locally closed, generalized locally
closed, semi locally closed, semi generalized locally closed, regular generalized locally closed, strongly locally closed and
 - locally closed sets and their continuous maps in topological space respectively. Recently as a generalization of closed
sets -closed sets and continuous maps were introduced and studied by Benchalli et al [4,5].A subset A of a topological
space (X,) is said to be -closed set if Cl(A)  U whenever A  U and U is -open.
2. PRELIMINARIES
Throughout the paper (X, ), (Y, ) and (Z, )( or simply X,Y and Z) represent topological spaces on which no
separation axioms are assumed unless otherwise mentioned. For a subset A of a space (X, ), Cl(A), Int(A), Cl(A) and Ac
denote the closure of A, the interior of A ,the
-closure of A and the compliment of A in X respectively.
We recall the following definitions, which are useful in the sequel.
Definition 2.1
A subset A of a topological space (X,) is called

A locally closed (briefly lc) [9] if A = U  V where U is open and V is closed in (X,).

Strongly generalized locally closed (briefly g*lc) [12] if A = U  V where U is strongly g-open and V is strongly gclosed in (X,).

-locally closed (briefly lc) [10] if A = U  V where U is -open and V is -closed in (X,).
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P. G. Patil

-locally closed (briefly lc) [13] if A = U  V where U is -open and V is -closed in (X,).
Definition 2.2
A topological space (X,) is said to be a

Sub maximal space [7] if every dense subset of (X,) is open in (X,).

Door space [8] if every subset of (X,) is either open or closed in (X,).

T-space[4] if every -closed set is closed
Definition 2.3
A map f: (X,)  (Y, ) is called

LC- continuous [9](resp. S*GLC-continuous [12], LC-continuous [10]) if f-1(G) is locally closed(resp. strongly
g- locally closed, -locally closed )set in (X, ) for each open set G of (Y, ).

LC-irresolute [9] if f-1 (G) is locally closed set in (X, ) for locally closed set G of (Y, ).
3. -LOCALLY CLOSED SETS IN TOPOLOGICAL SPACES
In this section we introduce the concept of -locally closed sets, lc*-sets and lc**-sets and study some of
their properties.
Definition 3.1
A subset A of a topological space (X,) is called -locally closed set (briefly lc-set) if A = G  F where G is
-open and F is -closed set in (X,).
The class of all -locally closed sets in (X,) is denoted by LC (X,).
Theorem 3.2
Every locally closed set is -locally closed set but converse need not be true.
Proof
From [1], every closed (resp.open) set is -closed (resp. -closed).Hence the proof
Example 3.3
Let X = {a, b, c} and  = {, {a}, X}. Then LC(X,) = P(X) and LC(X, ) = {,{a},{b, c},X}. The set A = {a,
b} is -locally closed set but not locally closed set in (X,).
Remark. 3.4
If A is -lc set (resp. strongly g-locally) in (X,), then A is -locally closed in (X,) but not conversely.
Example 3.5
Let X = {a, b, c} and  = {, {a, b}, X}. Then LC(X, ) = P(X) and LC(X, ) = {,{c},{a, b},X}. The set A
= {a, c} is -locally closed but not -lc set in (X,) and in example 3.3, the set A = {a, b} is -locally closed set but not
strongly g- locally closed set in (X,).
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Some New Weaker Forms of Locally Closed Sets in Topological Spaces
Theorem 3.6
Let (X,) be a T-space. Then

LC(X, ) = LC(X, ).

LC(X, )  GPRLC(X, ).
Theorem 3.7
If A is -locally closed set in (X,) and B is -open in
(X,), then A  B is -locally closed set in (X,).
Proof
Since A is -locally closed set there exists a -open set G and a -closed set H in (X,) such that A = G 
H. Now A  B = (G  H)  B = (G  B)  H, since G  B is -open. Hence A  B is -locally closed set in (X,).
Definition 3.8
A subset A of a topological space (X,) is called lc*-set (resp. -lc** set ) if A = P  Q where P is open(resp.open) and Q is closed(- closed) set in (X,).
The classe of lc*-sets(resp. -lc**) in (X,) is denoted by LC*(X,)(resp. LC**(X,)).
Remark 3.9

Every -closed set (resp.-open) is -lc set.

Every -lc* set is -lc set and every lc** set is -lc set

If every subset of (X,) is locally closed then it is lc* and -lc** in (X,).
Remark 3.10
LC(X,) (resp. LC*(X,), LC**(X,)) are independent from LC(X,) (resp. LC*(X,), LC**(X,)) sets
respectively as seen from the following examples.
Example 3.11
In Example 3.3, LC(X,) = LC*(X,) = LC**(X,) = {, {a}, {b, c}, X} and LC(X,) = LC*(X,) =
LC**(X,) = P(X). The set A = {a, c} is lc-set, lc*-set and lc*-set but not lc-set, lc*-set and lc**-set in
(X,) respectively.
Example 3.12
Let X = {a, b, c} and  = {, {a}, {b, c),X}. Then, LC(X,) = LC*(X,) = LC**(X,) =P(X) and LC(X,) =
LC*(X,) = LC**(X,) = {, {a}, {b, c}, X}.The set A = {a, b} is lc-set, lc*-set and lc**-set but not lc-set,
lc*-set and lc*-set in (X,) respectively.
Theorem 3.13
Let A be a subset of (X,). If A  LC**(X,), then their exists an open set P such that A = P  cl(A)
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Proof
Let A  LC**(X,). Then, there exist an open set P and -closed set F of (X,) such that A = P  F. Since A
 P and A  cl(A), we have A  P  F  cl(A). Since cl(A)  F, P  cl(A)  P  F=A. Thus A = P 
cl(A).
Theorem 3.14
If the collection of all -closed (or -open) subsets of (X,) is closed under finite intersections and if A and B
are subsets of (X,), such that A  LC(X,) and B is -open or -closed then A  B  LC(X,).
Proof
If A  LC (X,) then there exists -open set P and -closed set Q such that A = P  Q. Now A  B = (P 
Q)  B = (P  B)  Q is -locally closed set, since by hypothesis P  B is -open set. Hence A  B  LC(X,).
(Similarly for -closed set).
Theorem 3.15
For a subset A of (X, ) the following are equivalent:

A  LC(X, )

A = P  cl(A) for some -open set P.

cl(A) – A is -closed

A  (X  cl(A)) is -open
Proof
(i)  (ii): Let A  LC(X,). Then there exist - open set P and closed set F of (X,) such that A = P  F,
since
A  P and A  cl(A). Therefore A  P  cl(A).
Conversely, since cl(A)  F, P  cl(A) = P  F = A and
cl(A) is -closed set which implies that
A = P  cl(A).
(ii)  (i): Since cl(A) is -closed set and by hypothesis P is -open hence A = P  cl(A) is -locally
closed set in (X, ). Therefore A = P  cl(A)  LC(X,).
(iii)  (iv): Let F = cl(A) – A . Then by assumption F is -closed, X – F = X  (X  cl(A)) = A  (X 
cl(A)). But X – F is -open. This shows that A  (X  cl(A)) is -open.
(iv)  (iii): Let U = A  (X  cl(A)). Then by assumption U is -open,
– (A  (X  cl(A))) = cl (A) 
X  U is -closed. So X  U = X
(X  A) = cl(A) – A. Since X  U is -closed then cl(A) – A is -closed
set.
(iv)  (ii): Let P = A  (X  cl(A)). Thus P is -open by assumption. Now we prove that A = P  cl(A)
for some -open set P. Now
P  cl(A) = (A  (X  cl(A))  cl(A) = (cl(A)  A)  (cl(A)  (X
 cl(A))) = A   = A. Thus A = P  cl(A) for some -open set P.
Some New Weaker Forms of Locally Closed Sets in Topological Spaces
253
(ii)  (iv): Let A = P  cl(A) for some -open set P. Then we prove that A  (X  cl(A)) is -open.
Now A  (X  cl(A) = (P  cl(A))  (X  cl(A)) = P  (cl(A)  (X  cl(A))) = P  X = P, which is open. Thus A 
(X  cl(A)) is -open.
Theorem 3.16
For a subset A of (X, ) the following are equivalent:

A  LC*(X, )

A = P  cl(A) for some -open set P

cl(A) – A is -closed

A  (X – cl(A)) is -closed
Proof
Follows form the Theorem 3.15
Remark 3.17
As seen from the following example, it is not true that A LC*(X,) if and only if A  int(A(X – cl(A)), (cf.
Ganster and Reilly[9], Proposition 1(v)).
Example 3.18
Let X = {a, b, c} and  = {, {a}, {a, b}, X}.Let A = {a, c} in the topological space (X,), then (A(X – cl(A)) =
{a} is not containing A and A= {a, c} LC*(X,).
Theorem 3.19
Let A and Z be any two subsets of (X,) and let A  Z. Suppose that the collection of all -open subsets of (X,)
is closed under finite intersections. If Z is -open and A  LC*(Z, /Z), then A  LC*(X,).
Proof
If A  LC*(Z, /Z), then there exist -open set G in (Z, /Z) such that A = G  clz(A) where clz(A) = Z  cl(A).
Since G and Z are -open then G  Z is -open. This implies that A = G  (Z  cl(A) = (G  Z)  cl(A) is lc* set in (X,
). So A  LC*(X, ).
Remark 3.20
The following example shows that the assumption of the above theorem that is Z is -open in (X,) cannot be
removed.
Example 3.21
Let X = {a, b, c} and  = {, {a, b}, X}. Then LC*(X,) = {, {a}, {b}, {c}, {a, b}, X}. Let Z = A = {a, c}, is not
-open set and A  LC*(Z, /Z) but A  LC*(X,).
Theorem 3.22
If the collection of all -closed subsets of (X,) is closed under finite intersections, and if Z is -closed, open
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P. G. Patil
in (X,) and A  LC*(Z, /Z) then ALC(X,)
Proof
Let A LC*(Z, /Z). Then there exist -open set G in (Z, /Z) and closed set F in (Z, /Z) such that A = G  F.
Since F is closed in (Z, /Z), F = B  Z for some closed set B of (X,). So B and Z are -closed sets in (X,). Therefore F is
the intersection of -closed sets B and Z. So F is also -closed set in (X,). Therefore A = G (B  Z)  LC(X,)
Theorem 3.23
If Z is -closed, open subset of X and A  LC** (Z, /Z), then A  LC**(X,).
Theorem 3.24
Suppose that the collection of all -open sets of (X,) is closed under finite unions. Let A  LC*(X,) and B 
LC*(X,). If A and B are separated, then A  B  LC*(X,)
Proof
Let A and B  LC*(X,). Then there exists -open sets G and F of (X,) and by the Theorem 3.16 A = G  cl(A)
and
B = F cl(B). Take V = G  (X cl(B)) and W = F  (X cl(A)). Then A = V  cl(A) and B = W  cl(B) holds and V
 cl(B) =  and W  cl(A) = . Hence it follows that V and W are -open subsets of (X, ). Therefore A  B = (V  cl(A)) 
(W  cl(B)) = V  W  (cl(A)  cl(B)). Here V  W is -open by assumption. Thus A  B  LC*(X,).
Remark 3.25
The assumption that A and B are separated cannot be removed from the Theorem 3.24.
Example 3.26
Let X = {a, b, c} and  = {, {a, b}, X}. Then {, {a}, {b}, {c}, {a, b}, X}  LC*(X,) and is closed under finite
unions. Let A = {a}LC*(X,) and B={c} LC*(X,) but {a, c} -LC*(X,). Since they are not separated,{a}
 cl({c}) = {a}  {b, c} =  but cl{a}  {c} = X  {c} = {b, c} is not equal to .
Definition 3.27
A topological space (X,) is called -submaximal if every dense set in it is -open.
Theorem 3.28
If (X,) is submaximal then it is -submaximal but converse need not be true in general.
Example 3.29
Let X = {a, b, c} and  = {, {a}, {a, c}, X}. Then Topological space (X,) is -submaximal but not submaximal.
Theorem 3.30
A topological space (X,) -submaximal if and only if P(X) = LC*(X,).
Proof
Necessity: Let AP(X) and U = A  (X-cl(A)).Then it follows cl(U)=X. Since (X,) is -sub maximal, U is open, so ALC*(X,) from the Theorem 3.16. Hence P(X) = LC*(X,).
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Sufficiency: Let A be dense sub set of (X,).Then by assumption and Theorem 3.16(iv) that A  (X-cl(A)) = A holds,
ALC*(X,) and A is -open. Hence (X,) -sub maximal.
4. -Locally Continuous Maps
In this section, we define LC-continuous maps. Also we define LC*-continuous maps, LC**-continuous
maps which are weaker than LC-continuous and stronger than LC-continuous maps and study their relations with
existing ones. We also define LC-irresolute maps.
Definition 4.1
A function f: (X, )  (Y, ) is called LC-continuous (resp. LC*-continuous, LC**-continuous) if f-1(G) 
LC(X,) (resp. f-1(G)  LC*(X,), f-1(G)  LC**(X,)) for each open set G of (Y, ).
Theorem 4.2
If f: (X,)  (Y,) is LC-continuous then f is LC-continuous (resp.LC*-continuous and LC**-continuous).
The converse of the above theorem need not be true as seen from the following example.
Example 4.3
Let X = Y = {a, b, c},  = {, {a}, X} and  = {, {a}, {b},
{a, b}, Y}. Then the identity map f: (X,)  (Y,), f is
LC-continuous,LC*-continuous and LC**-continuous but not LC-continuous, since for the open set A = {a, b} 
(Y, ), f-1({a, b}) = {a, b}  LC(X, ).
Theorem 4.4
Every LC-continuous function is LC -continuous.
Following example shows that converse need not be true.
Example 4.5
Let X = {a, b, c} = Y,  = {, {a, b}, X} and  = {, {b}, {b, c}, Y}. Then the identity map f: (X,)  (Y,) is LC
-continuous but not lc-continuous, since for the open set {b} in (Y,), f-1({b}) = {b} is not lc- set in (X,).
Remark 4.6
Composition of two LC -continuous (resp. LC*-continuous, LC**-continuous) maps need not be LC continuous (resp. LC*-continuous, LC**-continuous) as seen from the following example.
Example 4.7
Let X = Y = Z = {a, b, c},  = {, {a}, {b, c}, X} and
(X,)  (Y,) by
 = {, {a, b}, Y},  = {, {a} {a, b}, Z}. Define a map f:
f (a) = c, f (b) = a f(c) = b and g: (Y,)  (Z,) be the identity map. Then both f and g are LC -
continuous (LC*-continuous, LC**-continuous) but the composition gof is not LC - continuous (resp. LC*continuous, LC**- continuous), since for the open set A = {a} in (Z, ), (gof)-1({a}) = f–1(g-1{a}) = f-1{a} = {b}LC
(X,) (resp. {a} LC*(X,), {b}  LC**(X,)).
Theorem 4.8
If f: (X,)  (Y,) is LC*-continuous or LC**-continuous then f is LC -continuous.
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P. G. Patil
Proof
Let f: (X, )  (Y, ) is LC*-continuous or LC**-continuous. By Remark 5.2.16(ii), f is LC -continuous.
The converse of the above theorem need not be true as seen from the following examples.
Example 4.9
Let X = {a, b, c} = Y,  = {, {a, b}, X} and  = {, {a},
{a, c}, X}. Let f: (X,)  (Y,) be the identity map. Then f
is LC -continuous and LC**-continuous but not LC*-continuous. For the open set A = {a, c}  (Y,), f-1({a, c}) = {a,
c}  LC*(X, ) but {a, c}  LC (X, ) and {a, c}  LC**(X, ).
Example 4.10
Let X = {a, b, c, d, e}, Y = {a, b, c, d},  = {, {a}, {a, b}, X} and  = {, {a}, {a, b}, {a, b, c}, X}. Define a map f:
(X,)  (Y,) by f(a) = c, f(b) = b, f(c) = a, f(d) = f(e) = d. Then f is LC -continuous, LC*-continuous but not LC**continuous. For the open set A = {a, b}  (Y,), f-1({a, b}) = {b, c}  LC **(X, ).
Theorem 4.11
Any map defined on a door space is LC –continuous.
Proof
Let f: (X,)  (Y,) be a map, where (X,) be a door-space and (Y,) be any topological space. Let A   (resp. A 
LC (Y,)). Then by the assumption on (X,), f-1(A) is either open or closed. In both cases f-1(A)  LC (X,) and
therefore f is LC –continuous.
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