Study on sgp-Regular and sgp-Normal Spaces Mahesh Bhat

Study on sgp-Regular and sgp-Normal Spaces

Mahesh Bhat

and Md. Hanif PAGE

1

Department of Mathematics,

S.D.M. College, Honnavar, Karnataka, INDIA.

2

Corresponding Author, Department of Mathematics, KLE Technological University, Hubli, Karnataka, INDIA.

email:[email protected], [email protected] (Received on: October 10, Accepted: October 14, 2017)

ABSTRACT

The purpose of this paper is to introduce and study new classes of spaces called sgp-regular and sgp-normal spaces in topological spaces using sgp-clsoed set and some of their properties are discussed.

2010 Mathematics Classification: 54A05, 54D10

Keywords: sgp-closed set, sgp-regular space, sgp-normal space.

1. INTRODUCTION

The study of generalized closed (in short, g-closed) sets initiated by Levine

in 1970 and defined a T

space to be one in which the closed sets and g-closed sets coincide. The notion has been studied extensively in recent years by many topologists. Maheshwari and Prasad

introduced and studied s-regular and s-normal spaces in topological spaces. Dorsett

defined and studied the concepts of semi normal spaces and semi-regular spaces respectively.

In 1990, Arya and Nour

obtained some characterization of s-normal spaces. Munshi

introduced g-regular and g-normal spaces using g-closed sets in topological spaces. Further Noiri and Popa

investigated the concepts introduced by Munshi

.

Recently Navalagi and Mahesh Bhat

introduced the notion of sgp-closed set utilizing pre closure operator. The notions of sgp-open sets, sgp-contonuity are introduced in

. In this paper we continue the study of sgp-closed sets, with introducing and characterizing sgp- regualr and sgp-normal spaces.

2. PRELIMINARIES

In this paper (X,τ ) and (Y,σ) (or simply X and Y) denote topological spaces on which

no separation axioms are assumed unless explicitly stated. If A is any subset of space X, then

Cl(A) and Int(A) denote the closure of A and the interior of A in X respectively.

We use following definitions in this paper;

Definition 2.1: A subset A of space X is called

(i) a semi-open set

if A ⊆ Cl(Int(A)) (ii) a semi-closed set

if Int(Cl(A)) ⊆ A

(iii) pre-open

, if A ⊆ Int (Cl (A))). The complements of these sets are their respective closed sets in the space X.

Definition 2.2

: The pre closure of a subset A of X is the intersection of all pre-closed sets containing A in X and is denoted by PCl(A).

Definition 2.3: A subset A of a space X is called (i) Generalized-closed (in short, g-closed) set

if Cl (A) ⊆ U and U is open in X. The complement of g-closed set is g-open set.

(ii) Semi generalized pre closed (briefly, sgp-closed) set

if PCl(A) ⊆ U, whenever A ⊆ U and U is semi-open in X.The complement of sgp-closed set is sgp-open set and the family of all sgp-open sets of X is denoted by SGPO(X).

Definition 2.4

: A space X is said to be

T

-space if every sgp-closed set is closed set in it.

Definition 2.5

: A function f: X → Y is sgp-irresolute if inverse image of sgp-closed set in Y is sgp-closed set in X.

Definition 2.6

: A function f: X → Y is sgp-clsoed function if the image of every closed set in X is sgp-closed set in Y.

3. sgp-REGULAR SPACES

In this section, we introduce sgp-regular spaces by using sgp-closed sets in topological spaces and discuss some properties.

Definition 3.1: A topological space X is said to be sgp-regular if for every sgp-closed set F and a point x ∉ F, there exist disjoint open sets U and V such that F ⊆ U and x ∈ V.

Theorem 3.2: Every sgp-regular space is regular.

Proof: Let X be a sgp- regular space. Let F be any closed set in X and a point x ∉ F. Then F is sgp-closed set in X and x ∉ F. Since X is sgp- regular space, there exist disjoint open sets U and V such that F ⊆ U and x∈V. Hence X is regular space.

The converse of the above theorem need not be true as seen from the following example.

Example 3.3: Let X = {a, b, c}, τ = {X,φ, {a}, {b, c}}. Then the space X is regular but not a

sgp-regular space.

X is regular: Consider the closed set {b, c} and a point ‘a’ such that a∉{b, c}. Then {b, c} and {a} are open sets such that {b, c} ⊂ {b, c}, a∈{a} and {b, c} ∩ {a} = φ. Similarly for the closed set {a} and a point ‘c’ such that c ∉ {a}. Then there exist open sets {a} and {b, c} such that {a} ⊂ {a}, c ∈ {b, c} and {a} ∩ {b, c} = φ . It follows that X is regular space.

X is not a sgp-regular: Consider the sgp-closed set {b} and a point ‘c’ such that c∉{b}. Then there exist open sets {b, c} and {a} such that {b} ⊂ {b, c}, c ∉ {a} and {b, c} ∩ {a} = φ.

Therefore (X, τ ) is not a sgp-regular space.

Theorem 3.4: If a space X is regular and

T

-space, then X is sgp-regular space.

Proof: Let X be a regular space. Let x ∈ X and A be a sgp-closed set in X such that x ∉ A.

Since X is

T

-space, A is a closed set in X. As X is a regular space, there exist disjoint open sets G and H such that A ⊆ G and x ∈ H. Hence X is sgp-regular space.

Theorem 3.5: The following statements are equivalent for a topological space X.

i) X is a sgp-regular space.

ii) For each x∈X and each sgp-open set A of x there exists an open set V containing x such that Cl(V) ⊆ A.

Proof: (i) ⇒ (ii): Let A be any sgp-open set of X. Then there exists a sgp-open set G such that x ∈ A ⊆ X. Since X-G is sgp-closed and x ∉ X-G, by hypothesis, there exist open sets U and V such that X-G ⊆ U, x ∈ V and U ∩ V = φ and so V ⊆ X-U. Now Cl(V) ⊆ Cl(X – U) = X – U and X – G ⊆ U implies X – U ⊆ G ⊆ A. Therefore Cl(V) ⊆ A.

(ii) ⇒ (i): Let F be a sgp-closed set and x ∉F. Then x ∈X – F and X – F is sgp-open and so X – F is a sgp-open set containing x. By hypothesis, there exists an open set V containing x such that x ∈V and Cl(V) ⊆ X – F, which implies F ⊆ X – Cl(V). Then X – Cl(V) is an open set containing F and V ∩ X – Cl(V) = φ . Therefore X is a sgp-regular.

Theorem 3.6: A space X is a sgp-regular if and only if for each sgp-closed set F of X and each x ∈ X – F, there exist open sets U and V of X such that x∈ U and F ⊆ V and Cl(U) ∩ Cl(V)

= φ.

Proof: Necessity: Let F be a sgp-closed set of X and x ∉ F. Then there exist open sets U

and V such that x ∈U

, F ⊆ V and U

∩ V =φ. This implies U

∩ Cl(V) = φ. Since X is sgp-regular, there exist open sets G and H of X such that x ∈G, Cl(V) ⊆ H and G ∩ H = φ. This implies Cl(G) ∩ H = φ . Now, put U = U

∩ G, then U and V are open sets of X such that x ∈ U, F ⊆ V and Cl(U) ∩ Cl(V) = φ .

Sufficiency: if for each sgp-closed set F of X and each point x ∈ X – F, there exist open sets G and H such that x ∈ G, F ⊆ H and Cl(G) ∩ Cl(H) = φ . This implies that x ∈ G, F ⊆ H and G

∩ H = φ . Hence X is sgp-regular.

Theorem 3.7: Every subspace of a sgp-regular space is sgp-regular space.

Proof: Let X be a sgp-regular space. Let Y be a subspace of X. Let p ∈ Y and F be a sgp- closed set in Y such that p ∉ F. Then there is a closed set and so sgp-closed set A of X with F

= Y∩ A and p ∉A. Therefore we have p ∈X, A is a sgp-closed set in X such that p ∉A. Since X is a sgp-regular, there exist open sets G and H such that p ∈G, A ⊆ H and G ∩ H = φ. Note that Y∩G and Y∩ H are open sets in Y. Also p ∈G and p ∈Y which implies p ∈ Y∩G and A ⊆ H. This implies Y ∩ A ⊆ Y ∩ G, F ⊆ Y ∩ H. Also (Y ∩ G) ∩ (Y ∩ H) = φ . Hence Y is a sgp- regular space.

Theorem 3.8: For a topological space X. Then the following conditions are equivalent.

(i) X is sgp-regular space.

(ii) For each point x ∈ X and for each sgp-open set U of X, there exists an open set V of X such that x ∈ V ⊆ Cl(V) ⊆ U.

(iii) For each point x ∈ X and for each sgp-closed set A not containing x, there exists an open set V of X such that Cl(V) ∩ A= φ .

Proof: (i) ⇔ (ii): Follows from the Theorem 3.5.

(ii) ⇒ (iii): Let x ∈ X and A be a sgp-closed set such that x ∉A. Then X- A is a sgp-open set such that x ∈ X- A. By hypothesis, there exists an open set V of x such that x ∈ V ⊆ Cl(V) ⊆ X- A. That is x ∈ V, V ⊆ Cl(V) and Cl(V) ⊆ X- A. So x ∈ V and Cl(V) ∩ A= φ .

(iii) ⇒ (ii): Let x ∈ X and U be a sgp-open set in X such that x∈X. Then X- U is sgp-closed set and x∉X- U. Then by hypothesis, there exists an open set V of x containing x such that Cl(V) ∩ X- U = φ . Therefore x ∈ V, Cl(V) ⊆ U. So x ∈ V ⊆ Cl(V) ⊆ U.

Theorem 3.9: If a function f: X → Y is bijective, sgp-irresolute and open from sgp-regular space X into a topological space Y then Y is sgp-regular space.

Proof: Let y ∈ Y and F be a sgp-closed set of Y with y ∉ F. Since f is sgp-irresolute, f

(F) is sgp-closed set in X. As f is bijective, let f (x) = y then x = f

(y) and x ∉ f

(F). Again since X is sgp-regular space, there exist open sets U and V such that x ∈ U and f

(F) ⊆ G and U ∩ V = φ . Since f is open and bijective, we have y ∈ f(U), F ⊆ f(V) and f(U) ∩ f(V) = f(U ∩ V) = f( φ ) = φ . Hence Y is sgp-regular space.

Theorem 3.10: If f: X → (Y, σ) is injective, sgp-closed function from a topological space (X, τ) into sgp-regular space Y. If X is

T

-space, then X is sgp-regular.

Proof: Let x ∈ X and F be a sgp-closed set in X such that x ∉ F. Since X is

T

-space, F is closed in X. Then f(F) is sgp-closed set with f(x) ∉ f(F) in Y as f is sgp-closed function. Again since Y is sgp-regular, then there exist disjoint open sets U and V such that f(x) ∈ U and f(F)

⊆ V. Therefore x ∈f

(U) and F ⊂ f

(V). Hence X is sgp-regular space.

4. sgp-NORMAL SPACES

In the section, we introduce the notions of sgp-normal spaces and obtain some of their properties.

Definition 4.4.1: A topological space X is said to be sgp-normal if for each pair of disjoint sgp-closed sets A and B, there exists a pair of disjoint open sets U and V in X such that A ⊆ U and B ⊆ V.

Theorem 4.2: Every sgp-normal space is normal, but not conversely.

Proof: Let X be a sgp-normal space. Let A and B be a pair of disjoint closed sets in X. Since every closed set is sgp-closed. Therefore A and B are sgp-closed sets in X. Again since X is sgp-normal, there exists a pair of disjoint open sets G and H such that A ⊆ G and B ⊆ H. Hence X is normal.

Example 4.3: Let X = {a, b, c}, τ = {X, φ , {a}, {b, c}}.Then the space X is normal but not a sgp-normal.

Theorem 4.4: If a space X is normal and

T

-space, then X is sgp-normal space.

Proof: Let A and B be two disjoint sgp-closed sets in X. Since X is

T

-space, A and B are disjoint closed sets in X. Again since X is normal, there exists a pair of disjoint open sets M and N such that A ⊆ M and B ⊆ N. Hence X is sgp-normal space.

Theorem 4.5: If f: X → Y is bijective open sgp-irresolute function from a sgp-normal space onto topological space Y, then Y is sgp-normal.

Proof: Let A and B be disjoint sgp-closed sets in Y. Then f

(A) and f

(B) are disjoint sgp- closed sets in X as f is sgp-irresolute. Since X is sgp-normal, there exist disjoint open sets G and H in X such that f

(A) ⊆ G and f

(B) ⊆ H. Again since f is bijective and open, f(G) and f(H) are disjoint open sets in Y such that A ⊆ f(G) and B ⊆ f(H). Hence Y is sgp-normal.

Theorem 4.6: A sgp-closed subspace of a sgp-normal space is sgp-normal.

Proof: Let X be a sgp-normal space. Let Y be a sgp-closed subspace of X. Let A and B be a

pair of disjoint sgp-closed sets in Y. Then A and B are disjoint sgp-closed sets in X. Since X

is sgp-normal, there exist disjoint open sets G and H in X such that A ⊆ G and B ⊆ H. Since

G and H are open in X, then Y ∩ G and Y ∩ H are open in Y. Also we have A ⊆ G, B ⊆ H

which implies Y ∩ A ⊆ Y ∩ G, Y ∩ B ⊆ Y ∩ H. So A ⊆ Y ∩ G, B ⊆ Y ∩ H and (Y ∩ G) ∩

(Y ∩ H) = Y ∩ (G ∩ H) = φ . Thus for each pair of disjoint sgp-closed sets A, B in Y, there

exist disjoint open sets Y ∩ G and Y ∩ H such that A ⊆ Y ∩ G and B ⊆ Y ∩ H. Hence Y is

sgp-normal.

Theorem 4.7: Let X be a topological space. The following conditions are equivalent.

(1) X is sgp-normal space.

(2) For each sgp-closed set A and each sgp-open set U such that A ⊆ U, there exists an open set V such that A ⊆ V ⊆ pCl(V) ⊆ U.

(3) For each pair A, B of distinct sgp-closed sets there exists an open set V such that A ⊆ V and pCl(V) ∩ B = φ .

(4) For each pair A, B of distinct sgp-closed sets there exists open set U and V such that A

⊆ U, B ⊆ V and pCl(U) ∩ pCl(V) = φ .

Proof: (1) ⇒ (2): Let A be a sgp-closed set and U be sgp-open set such that A ⊆U. Then A and X – U are disjoint sgp-closed sets in X. Since X is sgp-normal, there exist disjoint open sets V and W in X such that A ⊆ V and X – U ⊆ W. Now X – W ⊆ X – (X – U) that is X – W ⊆ U. Also V ∩ W = φ , which implies V ⊆ X – W implies that pCl(V) ⊆ pCl(X – W) that is pCl(V) ⊆ X – W. Therefore pCl(V) ⊆ X – W ⊆ U. So pCl(V) ⊆ U. Hence A ⊆ V ⊆ pCl(V)

⊆ U.

(2) ⇒ (3): Let A and B be a pair of disjoint sgp-closed sets in X. Now A ∩ B = φ, so A ⊆ X – B, where A is sgp-closed and X – B is sgp-open. Then by (2), there exists open set V such that A ⊆ V ⊆ pCl(V) ⊆ X – B. Now pCl(V) ⊆ X – B implies pCl(V) ∩ B = φ . Thus A ⊆ V and pCl(V) ∩ B = φ .

(3) ⇒ (4): Let A and B be a pair of disjoint sgp-closed sets in X. From (3), there exists an open set U such that A ⊆ U and pCl(U) ∩ B = φ. Now pCl(V) is α-closed so sgp-closed set and B is sgp-closed set. Therefore again from (3), there exists an open set V such that B ⊆ V and pCl(V) ∩ pCl(U) = φ.

(4) ⇒ (1): Let A and B be a pair of disjoint sgp-closed sets in X. Then form (4), there exists an open sets U and V in X such that A ⊂ U, B ⊂ V and pCl(U) ∩ pCl(V) = φ . So A ⊆ U, B ⊆ V and U ∩ V = φ. Hence X is sgp-normal space.

Theorem 4.8: A topological space X is sgp-normal if and only if for any disjoint sgp-closed sets A and B of X, there exist open sets U and V of X such that A ⊆ U, B ⊆ V and pCl(U) ∩ pCl(V) = φ .

Proof: The proof follows from the Theorem 4.7.

REFERENCES

1. S. P. Arya and T. M. Nour, Characterization of s-normal spaces, Indian J. Pure Appl.

Math., 21(8), 717-719 (1990).

2. Mahesh Bhat and Md. Hanif PAGE, Properties of sgp-closed and sgp-open Functions in Topological Spaces.(Communicated).

3. S. G. Crossely and S. K. Hilderbrand. Semi closure, Texas Jl. Sci., 22, 99-112 (1971).

4. C. Dorsett, Semi-normal spaces, Kyungpook Math. J., 25, 173-180 (1985).

5. N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math., Monthly, 70, 36-41 (1963).

6. N. Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19(2), 89-96 (1970).

7. S. N. Maheswari and R. Prasad, On s-regular spaces, Glasnik Mat. Ser. III, 10, 347-350 (1975).

8. S. N. Maheswari and R. Prasad, On s-normal spaces, Bull. Math. Soc. Sci. Math. R. S.

Roumanie, 22, 27-29 (1978).

9. A. S. Mashhour, M. E. Abd El-Monsef and Noiri. T., Strongly Compact spaces, Delta J.

Sci.8, 30-46 (1984).

10. B. M. Munshi, separation axioms, Acta Ciencia Indica 12, 140-144 (1986).

11. Govindappa Navalagi and Mahesh Bhat, on sgp-closed sets in Topological Spaces, Journal of Applied Mathematical Analysis and Applications, 3(1), 45-58 (2007).

12. T. Noiri and V. Popa, on g-regular spaces and some functions, Mem. Fac. Sci. Kochi Univ.

Math. 20, 67-74 (1999).

13. Md. Hanif PAGE, On θgs-Regular and θgs-Normal Spaces, International Journal of

Advances in Science and Technology, Vol.5, No.5, 149-155 (2012).