m ˆ m m m ˆ ˆ ˆ m m ˆ ˆ m m ˆ ˆ m m m ˆ ˆ ˆ m m ˆ ˆ On -T , -T , -T , -R , -R Spaces m m m m m ˆ ˆ ˆ ˆ ˆ

On µ ˆ -T

0

, µ ˆ -T

1

, µ ˆ -T

2

, µ ˆ -R

0

, µ ˆ -R

1

Spaces

S. PIOUS MISSIER¹ and E. SUCILA²

1Department of Mathematics,

V.O. Chidambaram College, Tuticorin, INDIA.

2Department of Mathematics,

G. Venkataswamy Naidu College, Kovilpatti, INDIA.

(Received on: June 5, 2013) ABSTRACT

M.K.R.S. Veerakumar introduced µ-closed sets in topological space. Using µ-closed sets recently S.Pious Missier and E.Sucila have defined and studied the notions of

µ ˆ

-closed and

µ ˆ

-opensets in topological spaces. In this paper, we introduce and investigate

µ ˆ

-Ti, i = 0,1,2 and

µ ˆ

-Ri, i = 1,2 using these

µ ˆ

-opensets in topological spaces.

Keywords:

µ ˆ

-T_i (i = 0,1,2) spaces,

µ ˆ

-R_i(i = 1,2) spaces, semi symmetric, always

µ ˆ

-open map,

µ ˆ

-irresolute map.

2000 Mathematics Subject code Classification: 54B40, 54C05, 54D05.

1. INTRODUCTION

In 1963, the concept of semiopen sets in topology was introduced by N.

Levine³. In⁴, N. Levine generalized the concept of closed set to generalized closed set. The concept of semi-Ti, i = 0,1,2 spaces has been defined by S.N. Maheshwari and R.Prasad in⁵, which is weaker than Ti, i = 0,1,2 spaces. The notion of µ-closed sets was introduced by M.K.R.S. Veerakumar¹¹. Recently the authors introduced⁸

µ ˆ

-closed

sets in topological spaces using µ-closed set.

In this paper, we introduce and investigate

µ ˆ

-Ti, i = 1,2,3 and

µ ˆ

-Ri, i = 1,2 using

µ ˆ

- open sets in topology. Further, we study their basic properties and preservation theorems of these new spaces.

2. PRELIMINARIES

Throughout this paper, by (X, τ), (Y, σ) (or simply X and Y), we always mean topological spaces on which no separations

axioms are assumed unless explicity stated.

Let A be a subset of X. The interior and closure of A in X are denoted by int(A) and cl(A) respectively. The following definitions and results are useful in the sequel.

Definition: 2.1

A subset A of a topological space (X,τ) is called:

1. semiopen³ if A ⊆ cl(int(A)) 2. α-open⁷ if A ⊆ int(cl(int(A)))

The complement of semi-open (resp.

α-open) set is called semi closed (resp. α- closed). The intersection of all semiclosed (resp. α-closed) sets containing a subset A of X is called semiclosure (resp. α-closure) of A is denoted by scl(A) (resp. αcl(A)).

Definition: 2.2

A subset A of a topological space (X, τ) is called

1. gα*-closed⁶ if αcl(A) ⊆ int(U) whenever A ⊆ U and U is α-open in (X, τ). The complement of gα*-closed set is called gα*-open.

2. µ-closed set [11] if cl(A) ⊆ U whenever A ⊆ U and U is gα*-open in (X, τ).

The complement of µ-closed set is called µ-openset.

3.

µ ˆ

-closed⁸ if scl(A) ⊆ U whenever A ⊆ U and U is µ-open in (X, τ). The complement of

µ ˆ

-closed set is called

µ ˆ

-openset.

The class of

µ ˆ

-closed subsets of X is denoted by

µ ˆ

c(X, τ) and the class of

µ ˆ

- open subsets of X is denoted by

µ ˆ

o(X, τ)⁸.

The intersection of all

µ ˆ

-closed sets

containing a subset A of X is called

µ ˆ

-closure of A is denoted by

µ ˆ

cl(A)⁹. Definition : 2.3

A space X is said to be : 1. semi-T0

5 if for each pair of distinct points in X, there is an semi-openset containing one of the points but not the other.

2. semi-T1

5 if for each pair of distinct points x and y of X, there exists semi- open sets U and V containing x and y respectively such that y ∉ U and x ∉ V.

3. semi-T2

5 if for each pair of distinct points x and y of X, there exist disjoint semi-open sets U and V containing x and y respectively.

4. semi-R0

2 if for each semi-openset G in X and x ∈ G such that scl({x}) ⊂ G.

5. semi-R1

2 if for x, y ∈ X with scl({x}) ≠ scl({y}), there exist disjoint semi-open sets U and V such that scl({x}) ⊂ U and scl({y}) ⊂ V.

Definition: 2.4

A mapping f : X → Y is said to be always α-open¹ if the image of every α- openset of X is an α-openset in Y.

Definition: 2.5

A mapping f : X → Y is called an

µ ˆ

-continuous⁹ if the inverse image of each openset in Y is an

µ ˆ

-openset in X.

Definition: 2.6

A mapping f : X → Y is called

µ ˆ

- irresolute [10] if the inverse image of each

µ ˆ

-openset in Y is an

µ ˆ

-openset in X.

3. ON

µ ˆ

-T0 SPACES

We introduce the following definition.

Definition 3.1

A topological space (X, τ) is said to be

µ ˆ

-T0, if for each pair of distinct points x, y of X, there exists an

µ ˆ

-openset containing one point but not the other. Clearly every semi-T0 space is

µ ˆ

-T0, since every semiopen set in X is

µ ˆ

-open in X.

However the converse is not true in general as shown by the Example 3.2.

Example: 3.2

Let X = {a, b, c} and τ = {X, ϕ, {a}, {b, c}}. Here X is

µ ˆ

-T0 but not semi– T0, since there is no semiopen set containing b but not containing c.

The following Theorem 3.3 characterizes

µ ˆ

-T0 spaces.

Theorem: 3.3

A space X is an

µ ˆ

-T0 space if and only if

µ ˆ

-closures of distinct points are distinct.

Proof:

Let x, y ∈ X with x ≠ y and X be an

µ ˆ

-T0 space. We shall show that

µ ˆ

_{cl({x}) ≠}

µ ˆ

cl({y}). Since X is

µ ˆ

-T0 space, there exists an

µ ˆ

-open set G such that x ∈ G and y ∉ G. Also x ∉ X – G and y ∈ X – G where X – G is

µ ˆ

-closed set in X. Since

µ ˆ

cl({y}) is the intersection of all

µ ˆ

-closed sets which

contain y, y ∈

µ ˆ

cl({y}) but x ∉

µ ˆ

cl({y}) as x ∉ X – G. Therefore

µ ˆ

_{cl({x}) ≠}

µ ˆ

cl({y}).

Conversely, suppose that for any pair of distinct points x, y ∈ X,

µ ˆ

cl({x}) ≠

µ ˆ

cl({y}). Then there exists atleast one point

z ∈ X such that z ∈

µ ˆ

cl({x}) but z ∉

µ ˆ

cl({y}). We claim that x ∉

µ ˆ

cl({y}).

If x ∈

µ ˆ

cl({y}), then

µ ˆ

cl({x}) ⊂

µ ˆ

cl({y}). So z ∈

µ ˆ

cl({y}), which is a contradiction. Hence x ∉

µ ˆ

cl({y}). Now x ∉

µ ˆ

cl({y}) ⇒ x ∈ X-

µ ˆ

cl({y}), which is an

µ ˆ

-openset in X containing x but not y.

Hence x is an

µ ˆ

-T0 space.

Definition: 3.4

A mapping f : X → Y is said to be

always

µ ˆ

-open if the image of every

µ ˆ

-openset of X is

µ ˆ

-open in Y.

Proposition: 3.5

The property of a space being

µ ˆ

-T0

is preserved under one-one, onto and always

µ ˆ

-open mapping.

Proof:

Let X be a

µ ˆ

-T0 space and Y be a topological space. Let f : X → Y be a one- one, onto, always

µ ˆ

-open mapping from X to Y. Let u, v ∈ Y with u ≠ v. Since f is one- one onto, there exists distinct points x, y ∈ X such that f(x) = u, f(y) = v. Since X is an

µ ˆ

-T0 space, there exists

µ ˆ

-openset G in X such that x ∈ G and y ∉ G. Since f is always

µ ˆ

-open, f(G) is an

µ ˆ

-openset in

Y containing f(x) = u but not containing f(y)

= v. Thus there exists an

µ ˆ

-open set f(G) in Y such that u ∈ f(G) but v ∉ f(G) and hence Y is an

µ ˆ

-T0 space.

4. ON

µ ˆ

-T1 SPACES

In this section, we define

µ ˆ

-T1

spaces and discuss its properties.

Definition: 4.1

A space X is said to be

µ ˆ

-T1, if for each pair of distinct points x, y of X, there exists a pair of

µ ˆ

-opensets, one containing x but not y and the other containing y but not x.

It is easy to verify the following.

(i) Every semi–T1 space is an

µ ˆ

-T1 space.

(ii) Every

µ ˆ

-T1 space is an

µ ˆ

-T0 space.

The converses of the above two statements are not true as shown by the examples.

Example: 4.2

Let X = {a, b, c} and τ = {X, ϕ, {a}, {b, c}}. Then X is an

µ ˆ

-T1 space but not semi-T1 space, since there is no semi openset containing b but not containing c.

Example: 4.3

Let X = {a, b, c} and τ = {X, ϕ, {a}, {a, c}}. Then, X is an

µ ˆ

-T0 space but not an

µ ˆ

-T1, since for distinct point a and b of X there is no pair of

µ ˆ

-opensets containing one point but not the other.

Theorem: 4.4

A space X is an

µ ˆ

-T1 space if {x} is

µ ˆ

-closed in X for every x ∈ X.

Proof :

Let x, y be two distinct points of X such that {x} and {y} are

µ ˆ

-closed in X.

Then X – {x} and X – {y} are

µ ˆ

-open in X such that y ∈ X – {x} but x ∉ X – {x} and x

∈ X–{y} but y ∉ X – {y}. Hence, X is an

µ ˆ

-T1 space.

Theorem: 4.5

If {x} is

µ ˆ

-closed for each x in X and scl({x}) is µ-closed for each x in X then a space X is semi-symmetric.

Proof:

Suppose x ∈ scl({y}) and y ∉ scl({x}). Since y ∈ X – scl({x}) and scl({x}) is µ-closed implies X-scl({x}) is µ-open also {y} is

µ ˆ

-closed by definition of

µ ˆ

-closed, scl({y}) ⊂ X – scl({x}). Thus x ∈ X – scl({x}), a contradiction.

Next, we have the following invariant properties.

Theorem: 4.6

Let f : X → Y be an

µ ˆ

-irresolute, injective map. If Y is

µ ˆ

-T1, then X is

µ ˆ

-T1. Proof:

Assume that Y is

µ ˆ

-T1. Let x, y ∈ X be such that x ≠ y. Then there exists a pair of

µ ˆ

-opensets U, V in Y such that f(x)∈U,

f(y)∈V and f(x)∉V, f(y)∉U. Then x ∈ f^-1(U), y ∈ f^-1(V), x ∉ f^-1(V) and y ∉ f^-

1(U). Since f is

µ ˆ

-irresolute, X is

µ ˆ

-T1.

5. ON

µ ˆ

-T2 SPACES Definition: 5.1

A space X is said to be

µ ˆ

-T2, if for each pair of distinct points x, y of X, there exist disjoint

µ ˆ

-opensets U and V such that x ∈ U and y ∈ V.

Clearly the following holds.

Every semi-T2 space is an

µ ˆ

-T2

space. However the converse is not true in general as shown by the example.

Example: 5.2

The space defined in Example 4.2 is an

µ ˆ

-T2 space but not an semi-T2 space, since the semi-openset {b, c} containing both b and c of X.

Also every

µ ˆ

-T2 space is

µ ˆ

-T1

space.

We have the following invariant properties.

Theorem: 5.3

Let f : X → Y be an

µ ˆ

-irresolute and injective. If Y is

µ ˆ

-T2, then X is

µ ˆ

-T2. Proof :

Similar to Theorem 4.6.

The following theorem 5.4 gives characterizations of

µ ˆ

-T2 spaces.

Theorem: 5.4

In a space X, the following statements are equivalent.

(a) X is

µ ˆ

-T2.

(b) For each y ≠ x ∈ X, there exists an

µ ˆ

-

openset U such that x ∈ U and y ∉

µ ˆ

cl(U).

(c) For each x ∈ X , ∩ {

µ ˆ

cl(U) / U is

µ ˆ

- open in X and x ∈ U} = {x}.

(d) The diagonal ∆ = {(x, x)/ x ∈ X} is

µ ˆ

- closed in X × X.

Proof:

(a) ⇒ (b) : Assume that the space X is

µ ˆ

- T2. Let x, y ∈ X such that y ≠ x. Then there are disjoint

µ ˆ

-opensets U and V in X such that x ∈ U and y ∈ V. Clearly X – V is

µ ˆ

-closed such that

µ ˆ

cl(U) ⊂ X – V, y ∉ X – V and therefore y ∉

µ ˆ

cl(U).

(b) ⇒ (c) : Assume that for each y ≠ x, there exists an

µ ˆ

-openset U such that x ∈ U and y ∉

µ ˆ

cl(U). So, y ∉∩ {

µ ˆ

cl(U) / U is

µ ˆ

-open in X and x ∈ U} = {x}.

(c) ⇒ (d) : We claim that X - ∆ is

µ ˆ

-open in X × X. Let (x, y) ∉ ∆. Then y ≠ x.

Since ∩ {

µ ˆ

cl(U)/ U is

µ ˆ

-open in X and x ∈U} = {x}, there is some

µ ˆ

- openset U in X with x ∈ U and y ∉

µ ˆ

cl(U)). Since U ∩ (X -

µ ˆ

cl(U)) = ϕ, U × (X -

µ ˆ

cl(U)) is an

µ ˆ

- open set such that (x, y) ∈ U × (X -

µ ˆ

cl(U)) ⊂ X – ∆. Hence ∆ is

µ ˆ

-closed in X × X.

(d) ⇒ (a) : If y ≠ x, then (x, y) ∉ ∆ and thus there exist

µ ˆ

-open sets U and V such that (x, y) ∈ U × V and (U × V) ∩∆ = ϕ. Thus, for the

µ ˆ

-open sets U and V, we have x ∈ U, y ∈ V and U ∩ V = ϕ. Hence X is

µ ˆ

-T2.

Remark: 5.5

From the above Propositions and Examples, we have the following diagram.

Here A → B represents A implies B but not conversely.

In 1975, S. N. Maheshwari and R. Prasad⁵ have shown that the following implications hold.

T2 semi-T2

T1 semi-T1

T0 semi-T0

Now the authors have shown that the following implications hold.

T2 semi-T2

µ ˆ

-T2

T1 semi-T1

µ ˆ

-T1

T0 semi-T0

µ ˆ

-T0

6. ON

µ ˆ

-R0 SPACES AND

µ ˆ

-R1 SPACES

In this section, we define

µ ˆ

-R0

spaces and

µ ˆ

-R1 spaces and study some of the properties.

Definition: 6.1

A space X is called an

(i)

µ ˆ

-R0 space if

µ ˆ

cl({x}) ⊂ U whenever U is

µ ˆ

-open set and x ∈ U.

(ii)

µ ˆ

-R1 space if for x, y ∈ X with

µ ˆ

cl({x}) ≠

µ ˆ

cl({y}), then there exist disjoint

µ ˆ

-open sets U and V such that

µ ˆ

cl({x}) ⊂ U and

µ ˆ

cl({y}) ⊂ V.

Remark: 6.2

Every

µ ˆ

-R1 space is an

µ ˆ

-R0

space.

Proposition: 6.3

A topological space (X, τ) is

µ ˆ

-R0

then for any F ∈

µ ˆ

c(X, τ), x ∉ F implies F ⊂ U and x ∉ U for some U ∈

µ ˆ

_{o(X, τ).}

Proof:

Suppose that F ∈

µ ˆ

c(X, τ) and x ∉ F. Then x ∈ X – F also. X – F is

µ ˆ

-open.

Since (X, τ) is

µ ˆ

-R0,

µ ˆ

cl({x}) ⊂ X – F. Set U = X -

µ ˆ

cl({x}) then U ∈

µ ˆ

o(X, τ), F ⊂ U and x ∉ U.

Proposition: 6.4

A topological space (X, τ) is

µ ˆ

-R0

then for any nonempty set A and G ∈

µ ˆ

o(X, τ) such that A ∩ G ≠ ϕ, there

exists F ∈

µ ˆ

c(X, τ) such that A ∩ F ≠ϕ and F ⊂ G.

Proof:

Let A be a nonempty set of X and G

∈

µ ˆ

o(X, τ) such that A ∩ G ≠ ϕ. There exists x ∈ A ∩ G. Since x ∈ G ∈

µ ˆ

o(X, τ),

µ ˆ

cl({x}) ⊂ G. Set F =

µ ˆ

cl({x}), then F ∈

µ ˆ

c(X, τ), F ⊂ G and A ∩ F ≠ ϕ.

Proposition: 6.5

A topological space (X, τ) is

µ ˆ

-R0

space if and only if for any x and y in X,

µ ˆ

cl({x}) ≠

µ ˆ

cl({y}) implies

µ ˆ

cl({x}) ∩

µ ˆ

cl({y}) = ϕ.

Proof:

Necessity: Suppose that (X, τ) is

µ ˆ

-R0 and x, y ∈ X such that

µ ˆ

cl({x}) ≠

µ ˆ

cl({y}).

Then, there exists z ∈

µ ˆ

cl({x}) such that z

∉

µ ˆ

cl({y}) (or z ∈

µ ˆ

cl({y}) such that z ∉

µ ˆ

cl({x}). There exists V ∈

µ ˆ

o(X, τ) such that y ∉ V and z ∈ V, hence x ∈ V. Therefore we have x ∉

µ ˆ

cl({y}). Thus x ∈ X -

µ ˆ

cl({y}) ∈

µ ˆ

o(X, τ), sinceX is

µ ˆ

-R0,

µ ˆ

cl({x}) ⊂ X -

µ ˆ

cl({y}) and

µ ˆ

cl({x}) ∩

µ ˆ

cl({y}) = ϕ. The proof for otherwise is similar.

Sufficiency: Let V ∈

µ ˆ

o(X, τ) and let x ∈ V. Now we will show that

µ ˆ

cl({x}) ⊂ V.

Let y ∉ V(ie) y ∈ X – V. Then x ≠ y and x

∉

µ ˆ

cl({y}). This shows that

µ ˆ

cl({x}) ≠

µ ˆ

cl({y}). By assumption

µ ˆ

cl({x}) ∩

µ ˆ

cl({y}) = ϕ. Hence y ∉

µ ˆ

cl({x}). Therefore

µ ˆ

cl({x}) ⊂ V.

Theorem : 6.6

The following properties are equivalent for a space X.

(a) X is an

µ ˆ

- R0 space.

(b) x ∈

µ ˆ

cl({y}) if and only if y ∈

µ ˆ

cl({x}) for points x and y in X.

Proof :

(a) ⇒ (b). Suppose X is

µ ˆ

-R0. Let x ∈

µ ˆ

cl({y}) and U be any

µ ˆ

-openset such that y ∈ U. By hypothesis, x ∈ U.

Therefore, every

µ ˆ

-openset containing y contains x. Hence y ∈

µ ˆ

cl({x}).

(b) ⇒ (a). Let V be an

µ ˆ

-openset and x ∈ V. If y ∉ V, then x ∉

µ ˆ

cl({y})and hence y ∉

µ ˆ

cl({x}). This implies that

µ ˆ

cl({x}) ⊂ V. Hence X is

µ ˆ

-R0. Theorem: 6.7

If a space X is

µ ˆ

-R1 and

µ ˆ

-T0 then it is

µ ˆ

-T2.

Proof :

Let x, y be any two distinct points in X. By Theorem 3.3, X is

µ ˆ

-T0 which implies that

µ ˆ

-closures of distinct points are distinct. Since X is

µ ˆ

-R1, there exist disjoint

µ ˆ

-open sets U and V such that

µ ˆ

cl({x}) ⊂ U and

µ ˆ

cl({y}) ⊂ V and so x ∈ U and y ∈ V. Hence X is an

µ ˆ

-T2 space.

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µ ˆ

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µ ˆ

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