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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-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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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.

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(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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in (X,) and A LC*(Z, /Z) then ALC(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 AP(X) and U = A (X-cl(A)).Then it follows cl(U)=X. Since (X,) is -sub maximal, U is open, so ALC*(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, ALC*(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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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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