Some Propertıes of Rw-Locally Closed Sets In Topologıcal Spaces

In this paper, we introduce three weaker forms of locally closed sets called RW-LC sets, RW-LC* set and RW-LC** sets each of which is weaker than locally closed set and study some of their properties in topologivcal spaces.

Illustrations explain the paper; they do not show study data.

AI-generated illustration

↗ Enlarge

Hierarchy of Locally Closed Set GeneralizationsClose ×

AI-generated conceptual illustration (Abstract geometric shapes representing mathematical sets in a topological space). It explains the idea and does not show study data.

Open full-size image ↗

Introduction Kuratowski and Sierpinski

introduced the notion of locally closed sets in topological spaces. According to Bourbaki [6] , a subset of a topological space (X, τ) is locally closed in (X, τ) if it is the intersection of an open set and a closed set in (X, τ). Stone [2] has used the term FG for locally closed set. Ganster and Reilly [7] 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 α-locally closed, generalized α-locally closed, semi locally closed, semi generalized locally closed, regular generalized locally closed, strongly locally closed and w-locally closed sets and their continuous maps in topological space respectively. Recently as a generalization of closed sets rw-closed sets and continuous maps were introduced and studied by Benchalli et al. [12] . A subset A of a topological space (X, T) is said to be rw-closed set if Cl(A) U whenever A U and U is semi regular-open.

16. g-lc set if A=U F [3] where U is g-open and F is l g-closed in X. 17. l g-lc* set if A=U F [3] where U is g-open and F closed in X. 18. l g-lc** set if A=U F [3] where U is open and F is l g-closed in X. 19. αrω-lc set if A=U F [14] where U is αrω-open and F is αrω-closed in X. 20. αrω-lc* set if A=U F [14] where U is αrω-open and F is closed in X. 21. αrω-lc** set if A=U F [14] where U is open and F is αrω-closed in X.

2.1 Lemma: [14] (i) Every closed set is rw-closed set. (ii) Every w-closed set is rw-closed set.

(iii) Every -closed set is rw-closed set. (iv) Every -closed set is rw-closed set. (v) Every rw-closed set is rg-closed set.

Lemma:

The space (X, τ) is Trw-space if every rw-closed set is closed set.

rw-locally closed sets in topological spaces 3.1 Definition:

A Subset A of t.s (X, τ) is called rw-locally closed (briefly RW-LC) if A=U F where U is rw-open in (X, τ) and F is rw-closed in (X, τ). The set of all rw-locally closed sets of (X, τ) is denoted by RW-LC(X, τ). (ii) Let X={a, b, c, d} and Τ ={X, , {a}, {b}, {a, b},{a, b, c}} then {c} is RW-LC but not l gc-lc set in (X, τ).

Remark:

-g lc sets and RW-LC sets are independent of each other as seen from the following example 3.17 Example: i) Let X={a,b,c,d} and τ ={X, ,{a},{b},{a,b},{a,b,c}} then {b,d} is -g -lc but not RW-LC set in (X, τ).

(ii) Let X={a,b,c} and τ ={X, ,{a},{a,b},{a,c}} then {a,b} is RW-LC but not -g-lc set in (X, τ).

Definition:

A subset A of (X, τ) is called a RW-LC * set if there exists a rw-open set G and a closed F of (X, τ) s.t A=G F the collection of all RW-LC* sets of (X, τ) will be denoted by RW-LC*(X, τ).

Definition:

A subset B of (X, τ) is called a RW-LC ** set if there exists an open set G and rw closed set F of (X, τ) s.t B=G F the collection of all RW-LC** sets of (X, τ) will be denoted by RW-LC**(X, τ). (iii) The set {c} is RW-LC* set but not a w-lc* set in (X, τ). (iv) The set {c} is RW-LC** set but not a w-lc** set in (X, τ).

Theorem

Theorem

Example:

Let X={a,b,c,d} and τ ={ X, ,{a},{b},{a,b},{a,b,c}} (i) The set {a,b,d} is RW-LC set but not a RW-LC* set in (X, τ). (ii) The set {a,d} is rg-lc* set but not a RW-LC* set in (X, τ). (iii) The set {a,d} is rg-lc* set but not a RW-LC* set in (X, τ). (iv) The set {b,d} is rg-lc set but not a RW-LC* set in (X, τ). (v) The set {b,d} is rg-lc set but not a RW-LC** set in (X, τ). (vi) The set {b,d} is αrω-lc set but not a RW-LC set in (X, τ).

Example:

Let X={a,b,c,d} and τ ={ X, ,{a},{b},{a,d},{a,b,d}} (i) The set {b} is RW-LC* set but not a -lc* set in (X, τ). (ii) The set {d} is RW-LC** set but not a -lc** set in (X, τ).

Example:

Let X={a,b,c} and τ ={ X, ,{a},{c},{a,c},{b,c}} (i) The set {c} is RW-LC* set but not a c set in (X, τ). (ii) The set {c} is RW-LC** set but not a c set in (X, τ).

Example: i ) Let

X={a,b,c,d} and τ ={ X, ,{a},{b},{a,b}} then the set {c} is RW-LC set but not a RW-LC** set in (X, τ) ii) Let X={a,b,c} and τ ={ X, ,{a},{b,c}} then the set {c} is RW-LC set but not a αrω-lc set in (X, τ). (ii) Let X={a,b,c,d} and τ ={ X, ,{a},{b},{a,b}} then set {c} is RW-LC* set but not a RW-LC** set in (X, τ).

Remark:

From the above discussion and known results we have the following implications in the diagram.

Theorem:

If RWO(X, τ) = τ then (i) RW-LC(X, τ) = LC (X, τ). (ii) RW-LC(X, τ) = W-LC (X, τ). (iii)RW-LC(X, τ)  GLC (X, τ).

Proof: (i) For any space (X, τ), W.k.t LC(X, τ)  RW-LC(X, τ). Since RWO(X, τ) = τ, that is every rw-open set is open and every rw-closed set is closed in (X, τ), RW-LC(X, τ) LC (X, τ); hence RW-LC(X, τ) = LC (X, τ). (ii) For any space (X, τ), LC(X, τ) W-LC(X, τ) RW-LC(X, τ) From (i) it follows that RW-LC(X, τ) = W-LC(X, τ). (iii) For any space (X, τ), LC(X, τ) GLC (X, τ) from (i) RW-LC(X, τ) = LC (X, τ) and hence RW-LC(X, τ) GLC (X, τ).

Theorem:

If RWO(X, τ) = τ, then RW-LC*(X, τ) = RW-LC** (X, τ) = RW-LC(X, τ). Proof: For any space,(X, τ) LC (X, τ)  RW-LC*(X, τ)  RW-LC(X, τ) and LC (X, τ)  RW-LC**(X, τ)  RW-LC(X, τ) since RWO(X, τ) = τ. RW-LC(X, τ) =LC(X, τ) by theorem 3.39,it follows that LC(X, τ) =RW-LC*(X, τ) = RW-LC**(X, τ) = RW-LC(X, τ).

Remark:

The converse of the theorem 3.40 need not be true in general as seen from the following example.

Example:

Let X={a, b, c} with the topology τ= { X, , {a}, {b}, {a,b}} then RW-LC*(X, τ) = RW-LC**(X, τ) = RW-LC(X, τ) =P(X). However RWO(X, τ) ={ X, , {a}, {b}, {c}, {a,b} } τ.

Theorem:

If GO(X, τ) = τ, then GLC(X, τ)  RW-LC(X, τ) Proof: For any space (X, τ) w.k.t LC(X, τ)  GLC(X, τ) and LC (X, τ) RW-LC(X, τ).....(i) GO(X, τ) =T, that is every g-open set is open and every g-closed set is closed in (X, τ) and so GLC(X, τ) LC(X, τ) that is GLC(X, τ) =LC(X, τ)......(ii) from (i) and (ii) we have GLC(X, τ)  RW-LC(X, τ). From (i) and (ii) we have RW-LC(X, τ) = RW-LC*(X, τ).

Remark:

The converse of the theorem 3.44 need not be true in general as seen from the following example.

Example:

Consider X= {a,b,c,d} and τ = { X, ,{a,b},{c,d}},then RW-LC(X, τ) = RW-LC*(X, τ) =P(X). But RWC(X, τ) =P(X) and LC(X, τ) = { X, ,{a,b},{c,d}} That is RWC(X, τ) LC(X, τ).

Theorem:

For a subset A of (X, τ) if A RW-LC(X, τ) then A = U (rw-cl(A)) for some open set U.

1. Pushpalatha A. Studies on generalizations of mappings in topological spaces, Ph.D., thesis, Bharathiar University, Coimbatore, 2000.
2. Kuratowski C, Sierpinski W. Sur les differences deux ensembles fermes, Tohoku Math. Jl. 1921; 20:22-25.
3. Perk HP, Perk JK. on locally c-continuous functions,solitions and fractals 2004; 19:995-1002.
4. Arockiarani I, Balachandran K, Ganster M. Regular generalized locally closed sets and rgl-continuous functions, Indian J Pure Appl Math. 1997; 28:661-669.
5. Balachandran K, sundaram P, Maki H. g-lc and glc-continuous function, indian j pure appp Math. 1966; 27:235-244.
6. Ganster M, Reilly IL. locally closed sets ad lc-contınuous function internret, j math sci. 1989; 12:417-424.
7. Sheik john M. a study on generalization of closed sets on continuous maps in topological & bitopological spaces, Ph.D thesis, Bharathiar university, Coimbatore, 2002.
8. Stone M. Application of the theorey of boolean rings to general topology, trans, amer, math, soc, al, 1937, 374-481.
9. Bourkbaki N. General topology, part I adison-wesley, reading mass., 1966.
10. Wali RS, Mandalgeri PS. On α Regular ω-closed sets in topological spaces, Int. J ofMath Archive. 2014; 5(10):68-76.
11. Wali RS, Mandalgeri PS. On α Regular ω-open sets in topological spaces, J of comp & Math Sci. 2014; 5(6):490-499.
12. Benchalli SS, Wali RS. on rw-Closed sets is Topological Spaces, Bull, Malays, Math, sci, soc. 2007; 30:99-110.
13. Gnanambal Y. On generalized pre regular closed sets in topological spaces, Indian J Pure Appl Math. 1997; 28(3):351-360.
14. Gnanambal Y. On generalized pre-regular Closed sets and generalized of locally closed sets in Topological Spaces, Ph.D thesis, Bharathiar university, Coimbatore, 1998.