Relations of Pre Generalized Regular Weakly Locally Closed Sets in Topological Spaces

In this paper pgrw-locally closed set, pgrw-locally closed*-set and pgrw-locally closed**-set are introduced. A subset A of a topological space (X,t) is called pgrw-locally closed (pgrw-lc) if A=GÇF where G is a pgrw-open set and F is a pgrw-closed set in (X,t). A subset A of a topological space (X,t) is a pgrw-lc* set if there exist a pgrw-open set G and a closed set F in X such that A= GÇF. A subset A of a topological space (X,t) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that A=GÇF. The results regarding pgrw-locally closed sets, pgrw-locally closed* sets, pgrw-locally closed** sets, pgrw-lc-continuous maps and pgrw-lc-irresolute maps and some of the properties of these sets and their relation with other lc-sets are established.


Introduction
According to Bourbakia 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. Gangster and Reilly used locally closed sets to define LC-Continuity and LCirresoluteness. Balachandran, Sundaram and Maki introduced the concept of generalized locally closed sets in topological spaces and investigated some of their properties.

Definition:
Let (X, τ) be a topological space and A⊆X. The intersection of all closed (resp pre-closed, αclosed and semi-pre-closed) subsets of space X containing A is called the closure (resp pre-closure, α-closure and Semi-preclosure) of A and denoted by cl(A) (resppcl(A), αcl(A), spcl(A)).

3.
pgrw-locally-closed sets 3.1 Definition: A subset A of a topological space (X,) is pgrw-locally closed (pgrw-lc) if Ą=G∩F where G is a pgrw-open set and F is a pgrw-closed set in (X,).
The set of all pgrw-locally closed subsets of (X,) is given by PGRWLC(X,).

Theorem: subset A of X is pgrw-lc if and only if its complement A c is the union of a pgrw-open set and a pgrw-closed set.
Proof: A is a pgrw-lc set in (X,).
A=G∩F where G is a pgrw-open set and F is a pgrw-closed set.
 A c =(G∩F) c = G c ∪F c where G c is a pgrw-closed set and F c is a pgrw-open set.
Conversely,A is a subset f (X,) such that A c =G∪F where G is a pgrw-open set and F is a pgrw-closed set.
 (A c ) c =(G∪F) c  A= G c ∩F c = F c ∩G c where F c is a pgrw-open set and G c is a pgrw-closed set.
 A is a pgrw-lc set.

Theorem:
i) Every pgrw-open set in X is pgrw-lc.
ii) Every pgrw-closed set in X is pgrw-lc Proof: i) A is a pgrw-open set in X.
 A=SA∩X where Ą is pgrw-open and X is pgrw-closed.
The converse statements are not true.

Corollary: In X
Everyopen set is pgrw-lc.
i) every closed set is pgrw-lc.

Proof: i) A is open in X.
 Aispgrw-open in X.
 A is pgrw-lc in X.
ii) Ais closed in X.
The converse statements are not true.
3.9 Theorem: Every locally closed set in X is pgrw-lc.
Proof: A is a locally closed subset of X.
 A = G∩H, G is an open set and H is a closed set.
The converse statement is not true.
 A=G∩F, G is pgrw-open and F is pgrw-closed in X.
The other statements may be proved similarly.
The converse statements are not true.

Theorem:
In X every pgrw-locally closed set is i) gp-lc ii) gpr-lc iii) gsp-lciv) gspr-lc Proof: i) A is a pgrw-lc set in X.
 A=G∩H, G is pgrw-open and H is pgrw-closed.

 A=G∩H, G is gp-open and H is gp-closed.
 A is a gp-lc set in (X,).
The other statements may be proved similarly. The set of all pgrw-lc* subsets of (X,) is denoted by PGRWLC*(X,).

Theorem: Every lc-set of X is a pgrw-lc*-set .
Proof: A is alc-set in X.
 A=G∩C, G is open and C is closed in X.

 A=G∩C, G is pgrw-open and C is closed in X.
 A is a pgrw-lc*-set in X.
The converse statement is not true.

Theorem:
Every pgrw-lc*-set of X is a pgrw-lc set.

Proof:
A is a pgrw-lc*-set in X.

A=G∩C where G is pgrw-open & C is pgrw-closed in X.
 A is a pgrw-lc-set in X.

Theorem: A subset A of X is pgrw-lc* iff A= G∩cl(A) for some pgrw-open set G.
Proof:A is a pgrwlc*-set in X.
 A=G∩F for a pgrw-open set G and a closed set F in X.
 AG and AF, a closed set. Therefore cl(V)-V is pgrw-closed.

Definition: Ą subset Ą of (X,) is a pgrw-lc**-set if there exists an open set G and a pgrw-closed set F such that Ą=G∩F.
The set of all pgrw-lc**-sets of (X,) is denoted by PGRWLC**(X,).

Theorem: Every lc-set of X is a pgrw-lc**-set.
Proof: Ą is alc-set X.

 A=G∩F where G is open and F is pgrw-closed in X.
 A is a pgrw-lc**-set in X.
The converse statement is not true.

 A=G∩F where G is pgrw-open and F is pgrw-closed.
 A is a pgrw-lc-set.
The converse statement is not true.
 A = P∩F where P is a pgrw-open set and F is a closed set in X and B is closed.

ii) APGRWLC**(X, τ) and B is open in X.
A=P∩F where P is an open set and F is a pgrw-closed set in X and B is open.
Similarly (ii) may be proved.
ii) If f is lc-continuous, then f is pgrw-lc-continuous.
iii) If f is regular-lc-continuous, then f is pgrw-lc-continuous. iv) If f is #rg-lc-continuous, then f is pgrw-lc-continuous. v) If f is -lc-continuous, then f is pgrw-lc-continuous. Proof: i) A map f is lc-continuous.
Similarly, the other statements may be proved.
The converse statements are not true.

Example
not a lc-set. Therefore f is not lc-continuous. ii) not a lδc-set. Therefore f is not lδc-continuous. iii) not a regular-lc-set. Therefore f is not regular-lc-continuous. iv) not a α-lc-set.Therefore, f is not α-lc-continuous.

Example:
Consider the spaces in 6.5, #rg-closed sets in X are X,ɸ, ii) gpr-lc-continuous.
iii) gsp-lc-continuous iv) gspr-lc-continuous Similarly the other statements may be proved 6.8 Theorem: If X is a door space, then every map i is i. pgrw-lc-continuous.
ii. pgrw-lc*-continuous iii. pgrw-lc**-continuous Proof : i) X is a door space and f is a map. ⇒Aєf -1 (A) is a pgrw-lc set in X.
Similarly the other statements may be proved.
Proof: X is a pgrw-sub-maximal space.
Similarly the other statements may be proved.
The converse statement is not true.