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International Journal of Mathematical Archive-3(12), 2012, 4959-4971
Available online through www.ijma.info ISSN 2229 – 5046
PAIRWISE Q* SEPARATION AXIOMS IN BITOPOLOGICAL SPACES
P. Padma1*, S. Udayakumar2 and K. Chandrasekhararao3
1
Department of Mathematics, PRIST University, Kumbakonam, India
Department of Mathematics, A.V.V.M Sri Puspam college, Poondi, India
3
Department of Mathematics, SASTRA University, kumbakonam , india
2
(Received on: 17-11-12; Revised & Accepted on: 28-12-12)
ABSTRACT
The aim of this paper is to introduce the concepts of pairwise Q* Ti ( i = 0, 1, 2, 3, 4 ) and pairwise Q* normal,
pairwise Q* regular, pairwise Q* US, pairwise Q* KC space, pairwise urysohn space, pairwise Q* 𝑇𝑇7⁄8 , pairwise Q*
𝑇𝑇1 1⁄2 , pairwise Q* 𝑇𝑇1 2⁄ 3 , pairwise Q* 𝑇𝑇2 1⁄2 space, pairwise Q* 𝑇𝑇4 1⁄2 space, pairwise Q* 𝑇𝑇5 1⁄2 space and study its
general properties.
Keywords: pairwise Q* T0, pairwise Q* T1, pairwise Q* T2, pairwise Q* T3, pairwise Q* T4, pairwise Q* normal,
pairwise Q* regular, pairwise Q* US, pairwise Q* KC space, pairwise urysohn space, pairwise Q* 𝑇𝑇7⁄8 , pairwise Q*
𝑇𝑇1 1⁄2 , pairwise Q* 𝑇𝑇1 2⁄ 3 , pairwise Q*𝑇𝑇2 1⁄2 space, pairwise Q* 𝑇𝑇4 1⁄2 space, pairwise Q* 𝑇𝑇5 1⁄2 space.
AMS Subject Classification: 54E55.
1. INTRODUCTION
The separation axioms of topological spaces are usually denoted with the letter T after the German “Trennung” which
means separation. Most separation axioms are defined interms of generalized closed sets and their definitions are
deceptively simple .The first separation axiom between T0 and T1was introduced by J. W. T. Youngs [22].
The separation axioms that were studied together in this way were the axioms for Hausdorffspaces, regular spaces and
normal spaces. Separation axioms and closed sets in topological spaces have been very useful in the study of certain
objects in digital topology [11, 12]. Ivan really discussed pairwise T0, T1, T2, T3, T4 - spaces in [7].
Levine introduced the concept of generalized closed sets in topological spaces and he initiated the notion of T1/2- spaces
in unital topology which is properly placed between T0 - space and T1-spaces by defining T1/2 - space in which every
generalized closed set is closed .
The notion of R0 topological spaces introduced by Shanin in 1943. Later, A. S. Davis [9] rediscovered it and studied
some properties of this weak separation axiom. Several topologists further investigated properties ofR0 topological
spaces and many interesting results have been obtained in various contexts. In the same paper, A. S. Davis
alsointroduced the notion of R1 topological space which is independent of bothT0 and T1 but strictly weaker than
T2.Bitopological forms of these concepts have appeared in the definitions of pairwise R0 and pairwise R1 spaces given
by Mrsevic [13].
Meanwhile in 1963, J.C. Kelly studied quasi metrics and showed that a quasi metric p on X ≠φ gives rise in a natural
way to another quasi metric p* called conjugate of p, defined by p*(x, y) = p (y, x) for all x, y∈ X. These results
culminate in a bitopological space (X, p, p*) called a bi-quasi metric space as a natural structure.
In the year 2010, K. ChandrasekharaRao and P. Padma [5] introduced and studied the concept of lower separation
axioms in bitopological spaces.
The notion of Q* - closed sets in a topological space was introduced by Murugalingam and Lalitha [14, 15] in 2010.
Recently P. Padma and S. Udayakumar [17, 18, 19] introduced and studied the concepts of (τ1, τ2)* - Q* closed sets
and (τ1, τ2)* - Q* continuous maps in bitopological spaces.
Corresponding author: P. Padma1*
Department of Mathematics, PRIST University, Kumbakonam, India
1
International Journal of Mathematical Archive- 3 (12), Dec. – 2012
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P. Padma* & S. Udayakumar/ Pairwise Q* Separation Axioms in Bitopological spaces /IJMA- 3(12), Dec.-2012.
The main focus of my paper is to introduce new separation axioms called pairwise Q* Ti spaces ( i = 0 , 1 , 2 , 3, 4 ),
pairwise Q* normal, pairwise Q* regular, pairwise Q* US, pairwise Q* KC space, pairwise urysohn space, pairwise Q*
𝑇𝑇7⁄8 , pairwise Q* 𝑇𝑇1 1⁄2 , pairwise Q* 𝑇𝑇1 2⁄ 3 , pairwise Q* 𝑇𝑇2 1⁄2 space, pairwise Q* 𝑇𝑇4 1⁄2 space and pairwise Q* 𝑇𝑇5 1⁄2
space in bitoplogical spaces.
2. PRELIMINARIES
Throughout this paper X and Y always represent nonempty bitopological spaces (X, τ1, τ2) and (Y, σ1, σ2). For a subset
A of X , τi - cl (A), τi - Q*cl (A) ( resp. τi - int (A) .τi - Q*int ( A ) ) represents closure of A and Q*closure of A(resp.
interior of A , Q* - interior of A) with respect to the topology τi . Now we shall require the following known definitions
are prerequisites.
Definition 2.1: A subset A of a bitopological spaces (X, τ1, τ2) is called
i) τ1τ2- Q* closed if τ1- int( A ) = φ and A is τ2 - closed.
ii) τ1τ2- Q* open if X – A is τ1τ2- Q* closed in X.
Definition 2.2: Let (X, τ1, τ2) be a bitopological spaces. Let A ⊂X. The intersection of all τ1τ2- Q* closed sets of X
containing a subset A of X is called τ1τ2-Q* closureof A and is denoted by τ1τ2- Q*cl (A).
Definition 2.3: Let (X, τ1, τ2) be a bitopological spaces. Let A ⊂X. The union of all τ1τ2- Q* open sets contained in a
subset A of X is called τ1τ2- Q* interior of A and is denoted by τ1τ2- Q* int (A).
Definition 2.4 [15]: A space X is called τ1τ2- Q* - Tspace, if every τ1τ2- Q* - closed set is τ2 - closed.
3. PAIRWISE Q* SEPARATION AXIOMS
In this section we study the new type of separation axioms by using τ1τ2- Q* closed sets. The family of all τ1τ2 - Q*open
[resp.τ1τ2 - Q*closed ] sets in X will be denoted by τ1τ2 - Q* O (X) [ respectively τ1τ2 - Q*cl (X)].
Definition 3.1: A bitopological space (X, τ1, τ2) is pairwise Q*-R0 if foreach τi - Q* open set G, x ∈ G implies τj - Q*
cl ({x}) ⊂ G, where i, j = 1, 2 and i ≠ j.
Example 3.1: Let X = {a, b, c}, τ1 = {φ, X, {a}, {a, c}} and τ2 = {φ, X, {b, c}, {b}}. Clearly the space (X, τ1, τ2) is
pairwise Q* R0.
Theorem 3.1: In a bitopological space (X, τ1, τ2) the following statements are equivalent:
i)
(X, τ1, τ2) is pairwise Q*- R0.
ii)
For any τi - Q* closed set F and a point x ∉ F, there exists a U ∈ Q* O(X,τ2 ) such that x ∉ U and F ⊂ U for i, j =
1, 2 and i ≠ j.
iii) For any τi - Q* closed set F and a point x ∉ F, τj - Q* cl ({x}) ∩ F =φ, for j = 1, 2 and i ≠ j.
Proof: i) ⇒ii): Let F be a τj - Q* closed set F and a point x ∉ F. Then by i)
τj - Q* cl ( { x } )⊂X – F, where i , j = 1, 2 and i ≠ j.
Let U = X - τj - Q* cl ({x}) then U ∈ Q* O(X,τ2 ) and also F ⊂U and x ∉ U.
ii) ⇒iii): Let F be a τj - Q* closed set F and a point x ∉ F. Suppose the given conditions hold.
Since U∈ Q* O( X ,τ2 ), U ∩τj - Q* cl ({x}) = φ.
Then F ∩ τj - Q* cl ( { x } ) = φ, where i, j = 1 , 2 and i ≠ j.
iii) ⇒i) : Let G ∈ Q* O(X,τi) and x ∈ G. Now X - G isτj - Q*closed and x ∉X – G. By iii), τj - Q* cl ({x})∩ (X – G) =
φ and hence τj - Q* cl ({ x })⊂ Gfori , j = 1, 2and i ≠j.
Therefore, the space (X, τ1, τ2) is pairwise Q* - R0.
Definition 3.2: A space (X, τ1, τ2) is said to be pairwise Q* - R1 if for each x, y∈ X, τi - Q* cl ({x}) ≠ τj - Q* cl ({y}),
there exist disjoint sets U ∈ Q* O( X ,τj ) and V ∈ Q* O (X,τi) such that τj - Q* cl ({x}) ⊂ U and τj - Q* cl ({y}) ⊂ V
where i , j = 1 , 2 and i ≠ j.
Theorem 3.2: If (X, τ1, τ2) is pairwise Q* - R1, then it is pairwise Q* - R0.
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Proof: Suppose that (X, τ1, τ2) is pairwise Q* - R1. Let U be aτi - Q* open set and x ∈ U. If y ∉ U, then y ∈ X - U and
x ∉τi - Q* cl ({y}).
Therefore, for each point y ∈ X – U, τj - Q* cl ({x}) ≠ τi - Q* cl ({y}).
Since (X, τ1, τ2) is pairwise Q*- R1, there exist a τi - Q* open set Uy and a τj - Q* openset Vy such that τj - Q* cl ({x}) ⊂
Uy, τi - Q* cl ({y}) ⊂Vy and Uy∩Vy= φ, where i, j = 1, 2 and i ≠ j.
Let A = ∪{Vy /y ∈X – U}, then X – U⊂A, x∉A and A isτj - Q*open set. Therefore,τj - Q*- cl ({x}) ⊂ X - A ⊂ U.
Hence (X, τ1, τ2) is pairwise Q* -R0.
Remark 3.1: The converse of theorem need not be true in general. The space (X, τ1, τ2) [in Example 3.1] is pairwise
Q* -R0 but not pairwise Q*-R1.
Definition 3.3: A bitopologicalspace X is called pairwise Q* T0 if for any pair of distinct points x, y of X, there exists
a set which is either τi -Q* open or τj -Q* open containing one of the points but not the other, where i, j = 1, 2 and i ≠ j.
Theorem 3.3: A bitopologicalspace X is called pairwise Q* T0 if either (X, τ1) or (X, τ2) is Q* T0.
Proof: The proof is obvious.
Theorem 3.4 - The product of an arbitrary family of pairwise Q* T0 space is pairwise Q* T0.
Proof: Let (X, τ1, τ2) =∏𝛼𝛼 𝜖𝜖 ∆(𝑋𝑋𝛼𝛼 , 𝜏𝜏1𝛼𝛼 , 𝜏𝜏 2𝛼𝛼 ), where τ1andτ2 are the product topologies on X generated by 𝜏𝜏1𝛼𝛼 and 𝜏𝜏2𝛼𝛼
respectively and X=∏𝛼𝛼 𝜖𝜖 ∆ 𝑋𝑋𝛼𝛼 . Let x = (xα) and y = (yα) be two distinct points of X.
Hence x α≠ yα for some α∈∆ .
But (𝑋𝑋𝜆𝜆 , 𝜏𝜏1𝜆𝜆 , 𝜏𝜏 2𝜆𝜆 ) is pairwise Q* T0, there exist either a 𝜏𝜏1𝜆𝜆 - Q* open set Uα containing xλ but not y λ or a 𝜏𝜏2𝜆𝜆 - Q* open
set Vα containing yλ but not x λ.
Define U = ∏ 𝜆𝜆≠ 𝛼𝛼 (𝑋𝑋𝜆𝜆 × 𝑈𝑈𝛼𝛼 ) and V= ∏ 𝜆𝜆≠ 𝛼𝛼 (𝑌𝑌𝜆𝜆 × 𝑉𝑉𝛼𝛼 ). Then U is 𝜏𝜏1 - Q* open and V is 𝜏𝜏2 - Q* open. Also, U contains
x but not y.
Theorem 3.5: Let f: X→Y be a bijection, pairwise Q* continuous and Y is pairwise Q* T0space, then X is a pairwise
Q* T0 space.
Proof: Let f: X→Y be a bijection, pairwise Q* continuous map and Y is a pairwise Q* T0 space. To prove that X is a
pairwise Q* T0 space.
Let x1, x 2 ∈X with x 1≠ x 2.
Since f is a bijection, there exists y1,y2∈Y with y1≠y2 such that f(x1)= y1 and f(x2) = y2 .
⇒ x 1 = f – 1(y1) and x 2 = f – 1 (y2).
Since Y is pairwise Q* T0space, there exists a set which is either τi - Q* open or τj - Q* open set M in X such that
y1∈M and y2 ∉ M.
Since f is pairwise Q* continuous map, f – 1(M) is a pairwise Q* open set in Y. Now we have
y1 ∈ M⇒ f – 1(y1) ∈ f – 1(M) ⇒ x1∈ f – 1(M) and x2 ∉ f – 1(M).
Hence for any two distinct points y1, y2 in Y, there exists pairwise Q* open set f – 1(M) in Y such that x1∈ f – 1(M) and
x2 ∉ f – 1(M).
Hence Y is a pairwise Q* T0 space.
Definition 3.4: A bitopological space X is called pairwise Q* T1 if for every distinct points x, y of X, there is a τi Q*open set U and a τ j - Q* open set V such that x ∈ U , y ∉ U and y ∈ V x ∉ V, where i, j = 1, 2 and i ≠ j.
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Example 3.2: Let X = {a, b, c}, τ1 = {φ, X, {a}, {a, c}} and τ2 = {φ, X, {b, c},{b}}. Then τ1 - Q* open sets are X, {a},
{a, c} and τ2 - Q* open sets are {b, c}, {b}, X.
Let a, b ∈ X. Then there is a τ 1 - Q* open set U = {a, c}& τ 2 - Q* open set V = {b, c} such that a ∈ U, b ∉ U and b∈
V, a∉ V. Let a, b ∈ X. Then there is a τ 1 - Q* open set U = {a} & τ 2 - Q* open set V = {b} such that a ∈ U, b ∉ U and
b ∈ V, a ∉ V.
Therefore, X is called pairwise Q* T1.
Corollary 3.1: A bitopologicalspace X is pairwise Q* T1 iff if it is pairwise Q* T0and pairwise Q* R0.
Theorem 3.6: Every pairwise Q* T1 space is pairwise T1.
Proof: Let X be a Q* T1 space. To prove that X is pairwise T1.
Since X is Q* T1, there exists a τ1 - Q*open set U and there exists a τ2 - Q* open set V such that x∈U, y ∉ U and y ∈ V,
x ∉ V.
Since every τ 1 - Q* open set is an τ 1 -open set and τ2 - Q* open set is an τ2 - open set we have U is τ 1 - open and V is τ2
- open.
⇒ U ∈τ 1 and V∈τ 2 such that x ∈U, y ∉ U and y ∈ V, x ∉ V.
Therefore, X is pairwise T1.
Remark 3.2: The converse of the above theorem is not true in general
ie) Pairwise T1 space is not pairwise Q* T1 space.
Theorem 3.6: The product of an arbitrary family of pairwise Q* T1 space is pairwise Q* T1.
Proof: Similar to the proof of the theorem 3.6.
Theorem 3.7: A bitopologicalspace X is called pairwise Q* T1 if either (X, τ1) or (X, τ2) is Q* T1.
Proof: Let (X, τ1, τ2) be pairwise Q* T1 space. Let x, y be two distinct points of X, then there exists a τ1 - Q* open set
U such that x ∈ U, y ∉ U. Thus, (X, τ1) is Q* T1.
Similarly, (X, τ2) is Q* T1.
Converse is obvious.
Definition 3.5: A bitopological space X is called pairwise Q* T2 or pairwise Q* Hausdorff if given distinct points x,
y of X, there is a τ i- Q*open set U and a τj- Q* open set V such that x∈U, y∈V, U ∩ V = φ where i, j = 1, 2 and i ≠ j.
Example 3.3: Let X = {a, b, c}, τ 1 = {φ, X, {a}, {a, b}} and τ2 = {φ, X, {b , c}, {b}}. Then τ1 - Q* open sets are X,
{a}, {a, b} and τ2 - Q* open sets are {b, c}, {b}, X. Let a, b ∈ X. Then there is a τ1 - Q* open set U = {a} & τ2 - Q*
open set V = {b} such that a ∈ U, b∈V and U ∩ V = φ . Let a, b ∈ X. Then there is a τ1 -Q* open set U = {a} & τ2 -Q*
open set V = {b, c} such that a ∈ U, b∈V and U ∩ V = φ .
Therefore, X is called pairwise Q* T2.
Corollary 3.2: A bitopological space X is pairwise Q* T2iff if it is pairwise Q* T1and pairwise Q* R1 .
Theorem 3.8: Every pairwise Q* T2 space is pairwise T1 space.
Proof: Let X is pairwise Q* T2 space. Let x ≠ y in X.
Since X is pairwise Q* T2 , there exists U ∈τ 1 and V ∈τ 2 such that x ∈ U and y ∈ V with U ∩ V = φ.
⇒there exists U ∈τ 1 and V∈τ 2 such that x ∈ U but y ∉ U and y ∈ V but x ∉ V.
⇒ X is pairwise T1 space.
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Theorem 3.9: Every pairwise Q* T2 space is pairwise T2 space.
Proof: Let X is pairwise Q* T2 space. To prove that X is pairwise T2 space.
Since X is pairwise Q* T2 space there exists a a τ1 - Q*open set U and a τ 2 - Q* open set V such that x ∈ U, y∈V,
U∩V=φ.
Since every τ1 - Q*open set is τ1 - open and τ2 - Q*open set is τ2 - open, we have U is τ1 - open and V is τ 2 - open.
⇒ U ∈τ 1 and V ∈τ 2 such that x ∈ U and y ∈V, U ∩ V =φ.
Therefore, X is pairwise T2 space.
Theorem 3.10: Every pairwise Q* T2 space is pairwise Q* T1 space.
Proof: Let X is pairwise Q* T2 space. Since X is pairwise Q* T2 space, there exists τ1 - Q*open set U and a τ2 - Q*
open set V such that x∈U, y ∈ V, U∩V= φ .
⇒ there exists U ∈τ 1 and V ∈τ 2 such that x ∈ U, but y ∉ U and y ∈ V but x ∉ V .
⇒ X is pairwise Q* T1space.
Remark 3.3: Every pairwise Q* - T1 space is pairwise Q* - T0.
Theorem 3.11: If a space (X, τ1, τ2) is pairwise Q* - T2, then it is pairwise Q* -R1.
Proof: Let (X, τ1, τ2) be pairwise Q* -T2.
Then for any two distinct points x, y of X, their exist a τi - Q* open set U and a τ j - Q* open set V such that x ∈U, y ∈
V and U ∩ V = φ where i , j = 1 , 2 and i ≠ j.
If (X, τ1, τ2) ispairwise Q* -T1, then {x} = τj - Q* cl({x}) and {y} = τi- Q* cl ({y}) and thus τi - Q* cl({x}) ≠ τj- Q*
cl({y}), where i, j = 1, 2 and i ≠ j.
Thus for any distinct pair ofpoints x, y of X such that τi - Q* cl({x}) ≠ τj - Q* cl({y}) where i, j = 1, 2 and i ≠ j, there
exist a τi - Q* open set U andτj - Q* open set V such that x∈V, y∈U and U ∩ V = φ where i, j = 1, 2 and i≠ j. Hence
(X, τ1, τ2) is pairwise Q* - R1.
Remark 3.4: The converse of the above is not true in general ie) pairwise Q* T1 space is not pairwise Q* T2 space.
Remark 3.5: If a bitopologicalspace X pairwise Q* Ti, then it is pairwise Q* T i – 1. i = 1; 2.
Definition 3.5: Let X be a bitopologicalspace. Then τ 1 is Q* regular w.r.to τ 2 if for each point x in X and each τ 1 - Q*
closed set P such that x ∉ P there is a τ1 - Q*open set U and a τ2 - Q* open set V disjoint from U such that x∈U and P
⊂V .X is pairwise Q* regular if τ1 is Q* regular w.r.to τ2 and τ2 is Q* regular w.r.to τ1.
Theorem 3.12: Every pairwise Q* regular space is pairwise regular space.
Proof: Let X be a pairwise Q* regular space. To show that X is pairwise regular. Let x ∈ X.
Since X is pairwise Q* regular there exists τ 1 - Q* closed set F such that x ∉F there is a τ 1 - Q*open set U and a τ2 Q* open set V disjoint from U such that x ∈ U and F ⊂ V. Since every τ 1 - Q* closed set is τ 1 - closed set, we have F is
a τ 1 - closed set in X such that x ∉ F .
Thus, for each point x in X and each τ 1 - closed set F such that x ∉ F there is a τ 1 - open set U and a τ 2 - open set V
disjoint from U such that x ∈ U and F ⊂ V.
Hence X is a pairwise regular.
Definition 3.6: X is pairwise Q* T3 if it is pairwise Q* regular and pairwise Q* T1.
Remark 3.6: Pairwise Q* T3 ⇒ Pairwise Q* T2.
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Theorem 3.13: Every pairwise Q* -T0, pairwise Q* regular space is pairwise Q* T1 and hence pairwise Q* T3.
Definition3.7: A bitopological space X is called pairwise Q* Urysohn, if for any two points xand y of X such that x ≠
y , there exists a τ i - Q*open set U and a τ j - Q* open set V such that x ∈ U, y ∈ V, τ j - Q* cl ( U ) ∩ τ i - Q* cl ( V ) =
φ, where i, j = 1, 2 and i ≠ j.
Remark 3.7: Obviously, Pairwise Q* T3 ⇒ Pairwise Q* Urysohn ⇒ Pairwise Q* T2.
Definition 3.8: X is said to be pairwise Q* normal if for each τ i - Q* closed set A and τ j - Q* closed set B with
A ∩ B=φ , there exists a τ i - Q*open set V ⊃ B and there exists a τ j - Q*open set U ⊃ A such that U ∩ V = φ, where
i, j = 1, 2 and i ≠ j.
Example 3.4: Let X = {a, b, c}, τ1 = {φ, X, {c}} and τ2 = {φ, X, {a, b}}. Then τ 1 - Q* closed sets are φ , {a, b} and τ2
- Q* closed sets are φ, {c}.
Take F = {a, b}, G = {c}. Then F is τ 1 - Q* closed set and G is τ2 - Q* closed set. Also F ∩ G = φ.
Then there exists a τ 2 -Q*open set U= {a, b} and τ1- Q*open set V ={c} such that F ⊆ U and G ⊆ V such that U∩V = φ
.
Therefore, X is pairwise Q* normal.
Theorem 3.13: Every pairwise Q* normal space is pairwise normal.
Proof: Let X be a pairwise Q* normal. To show that X is pairwise normal. Let A be τ 1 - Q* closed and B be τ 2 - Q*
closed with A ∩ B = φ .
Since X is pairwise Q* closed, there exists a τ 1 - Q*open set V ⊃ B and there exists a τ 2 - Q*open set U ⊃ A such that
U∩V=φ.
⇒ there exist a τ 1 - open set V ⊃ B and there exists a τ 2 - open set U ⊃ A such that U ∩ V = φ .
⇒ X is pairwise normal.
Definition 3.9: A pairwise Q* normal, pairwise Q* T1 space is called pairwise Q* 𝑻𝑻𝟒𝟒 space.
Definition 3.10: A bitopological space X is said to be a pairwise Q* completely normal provided that whenever A
and B are subsets of X such that τi - Q* cl (A) ∩ B = φ and A ∩ τ j - Q* cl (B) = φ there exists a τj - Q* open set U and
a τi - Q* open set V such that A ⊂ U, B ⊂ V, U ∩ V = φ, where i, j = 1, 2 and i ≠ j.
Definition 3.11: A pairwise Q* T1 space, pairwise Q* completely normal bitopological space is called pairwise Q* 𝑻𝑻𝟓𝟓
space.
Theorem 3.14: Every pairwise Q* completely normal space is pairwise Q* normal.
Proof: Let X be a pairwise Q* completely normal bitopological space.
Let A be a τi - Q* closed set and B be a τj - Q* closed set such that A ∩ B =φ. Then
τi - Q* cl (A) ∩ B =A ∩ B =φ and A ∩ τ j - Q* cl (B) = A ∩ B = φ.
By complete Q* normality, there exists a τj - Q* open set U and a τi - Q* open set V such that A⊂U, B⊂V, U ∩ V =φ.
Hence X is pairwise Q* normal.
Theorem 3.15: Every pairwise Q* completely normal space is pairwise completely normal.
Proof: The proof is obvious.
Theorem 3.16: If a bitopological space is pairwise Q* completely normal then every subspace is pairwise Q* normal.
Proof: Let (X, τ1, τ2) be pairwise Q* completely normal and (Y, τ1y, τ2y) be a subspace. Let F1 and F2 be disjoint Q*
closed in τ1y and τ2y respectively. F1y is τ1 - Q* closed
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⇒ F = τ1y - Q* cl (F1).
Then F1∩τ2 - Q* cl (F2) = τ1y - Q* cl (F1) ∩τ2 - Q* cl (F2)
= (Y∩τ1 - Q* cl (F) )∩τ2 - Q* cl (F2)
= τ2y - Q* cl (F2) ∩τ1y - Q* cl (F1)
= F1 ∩F2
=φ.
Similarly we can show that
τ1 - Q* cl (F1) ∩ F2 =φ .
Thus F1, F2 is a Q* separated pair of X. By pairwise Q* complete normality, there exists disjoint sets G1 ∈τ1 and G2 ∈τ2
such that, F2 ⊂ G1 F1 ⊂ G2.
Then F2 ⊂ Y ∩ G1, F1⊂ Y ∩ G2, (Y ∩ G1) ∩ (Y ∩ G2) = φ and Y ∩ G1∈τ1y, Y ∩ G2 ∈τ2y .
Hence (Y,τ1y, τ2y ) is pairwise Q* normal.
Definition 3.12: A subset A of a space (X, τ1, τ2) is said to be bi - Q* open if it is both τi - Q* open and τj - Q* open,
where i, j = 1, 2 and i ≠ j.
Definition 3.13: A bitopological space X is called τi τj - Q* Tb- space if every τi τj - Q* closed set is τi τj - closed, where
i, j = 1, 2 and i ≠ j.
Example 3.5: Let X = {a, b, c}, τ1= {φ, X, {a}} and τ2 = {φ, X, {a}, {b, c}}. Clearly φ, {b, c} are τ i τ j - Q* closed set.
Therefore, X is τi τ j - Q* - Tb space.
Definition 3.14: A bitopological space X is called pairwise Q*- compact if every proper, τj - Q* closed set is τi Q*compact and every proper τi - Q* closed set is τj - Q* - compact, where i, j = 1, 2 and i ≠ j.
Theorem 3.17: Every pairwise Q* closed, pairwise Q* continuous image of a pairwise Q* normal space is pairwise Q*
normal.
𝑜𝑜𝑜𝑜 𝑡𝑡𝑡𝑡
Proof: Let (X, τ1, τ2) be a pairwise Q* normal space. Let f: (X, τ1, τ2) �⎯⎯� (Y, 𝜏𝜏1∗ , 𝜏𝜏2∗ ) be a pairwise Q* closed ,
pairwise Q* continuous mapping. Let A and B be two disjoint subsets of Y, where A is 𝜏𝜏1∗ - Q* closed and B is 𝜏𝜏2∗ - Q*
closed. Then f – 1 (A) is τ1 - Q* closed and f – 1 (B) is τ2 - Q* closed.
Also, A ∩ B = φ ⇒ f – 1 (A∩ B) = f – 1 (φ) = φ.
Since X is pairwise Q* normal, there exists disjoint sets GA and GB such that f – 1 ( A ) ⊂ GA , f – 1 ( B ) ⊂ GB , where GA
is τ2 - Q* open and GB is τ 1 - Q* open.
Let 𝐺𝐺𝐴𝐴∗ = {y: f – 1 (y) ⊂ GA} and 𝐺𝐺𝐵𝐵∗ = {y: f – 1 (y) ⊂ GB}. Then 𝐺𝐺𝐴𝐴∗ ∩ 𝐺𝐺𝐵𝐵∗ = φ , A ⊂ 𝐺𝐺𝐴𝐴∗ , B ⊂ 𝐺𝐺𝐵𝐵∗ and since
𝐺𝐺𝐴𝐴∗ = Y – f (X – GA)
𝐺𝐺𝐵𝐵∗ = Y – f (X – GB).
Here 𝐺𝐺𝐴𝐴∗ is 𝜏𝜏2∗ - Q* open and 𝐺𝐺𝐵𝐵∗ is 𝜏𝜏1∗ - Q* open.
Hence (Y, 𝜏𝜏1∗ , 𝜏𝜏2∗ ) is pairwise Q* normal.
Definition 3.14: A bitopological space X is called pairwise Q* - KC space if every τi - Q* compact set is τj - Q*
closed and τj - Q* compact set is τi - Q* closed, where i, j = 1, 2 and i ≠ j.
Definition 3.15: A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝒃𝒃 space if its τ iτ j - Tb space and τjτ i - Q* Tb,
where i, j = 1, 2 and i ≠ j.
Definition 3.16: A bitopological space X is called τiτj - Q* Td - space if every τiτ j - Q* closedset is τiτj-semi closed,
where i, j = 1, 2 and i ≠ j.
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Example 3.6: Let X = {a, b, c}, τ1= {φ, X, {a}, {b, c}} and τ2 = {φ, X, {a, c}}. Then, X is τ iτ j- Q* - Td space. Since
X, φ, {b, c}, {c}, {a}, {b} are τiτj-semi closed sets and φ, {b} are τ I τ j- Q* closed sets.
Definition 3.17: A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝒅𝒅 space if its τ iτ j - Q* Td and τjτ i - Q* Td space,
where i, j = 1, 2 and i ≠ j.
Proposition 3.1: If (X, τ1, τ2) is τi τj - Q* -Tbspacethen it is an τi τj - Q* Td - space.
Proof: Let (X, τ1, τ2) be a pairwise Q* - Tb space.
Claim: τ i τ j - Q* - Tbspace then it is an τ i τ j - Q* Td - space. ie) To prove every τ iτj - Q* closedset is τiτj- semi closed .
Let A be τ iτj - Q* closed in X.
⇒ A is τ i τ j - closed. [Since X is τ i τ j - Q* - Tb space]
Since every τ iτj - closed set is τ iτj - semi closed, we have A is τ iτj -semi closed.
Thus, every τ iτj - Q* closed set is τiτj- semi closed.
Therefore, (X, τ1, τ2) is an τi τj - Q* Td - space.
Remark 3.8: The converse of the above theorem is not true in general. The following example supports our claim.
Example 3.7: In example 3.6, {c} τiτj- semi closed but not τiτj- closed
Definition 3.18: A bitopological space X is said to be a pairwise Q* US if for every sequence {xn} in X such that
( 𝜏𝜏 𝑖𝑖 )𝑄𝑄 ∗
� 𝜏𝜏 𝑗𝑗 � ∗
𝑄𝑄
xn�⎯⎯⎯� x and x n�⎯⎯⎯� y, it follows that x = y. Here ( 𝜏𝜏𝑖𝑖 )𝑄𝑄 ∗ [� 𝜏𝜏𝑗𝑗 �𝑄𝑄 ∗ ] is the family of all τi- Q* open [τ j- Q* open] set in
X.
Remark 3.9: pairwise Q* - T2 ⇒ pairwise Q* - KC ⇒ pairwise Q* - US ⇒ pairwise Q* - T1 .
Theorem 3.20: Every bi - Q* closed subspaceof a pairwise Q* normal space is pairwise Q* normal.
Proof: Let (Y, τ1y, τ2y) be a bi - Q* closed subspace of a pairwise Q* normal space (X, τ1, τ2). Let A be a τi y - Q*
closed set and B be a τj y - Q* closed set disjoint from A.
Since the space Y is bi - Q* closed, A is τi - Q* closed and τj - Q* closed, where i, j = 1, 2 and i ≠ j.
By pairwise Q* normality of (X, τ1, τ2), there exists a τj - Q* open set U and a τi - Q* open set V such that
A ⊂ U, B ⊂ V, U ∩ V = φ .
Thus,
A = A ∩ Y ⊂ Y∩ U
B = B ∩ Y ⊂ V ∩ Y.
⇒ (U ∩ Y) ∩ (V ∩ Y) = φ .
Also, U ∩ Y is τ j y - Q* open and V ∩ Y is τ i y - Q* open.
Thus, there exists a τi y - Q* open set V ∩ Y and a τj y - Q* open set U ∩ Y such that
A ⊂ ( U ∩ Y ), B ⊂ ( V ∩ Y ), ( U ∩ Y ) ∩ ( V ∩ Y ) = φ.
Hence (Y, τ1y, τ2y) is pairwise Q* normal.
Remark 3.10: The following implications are obvious.
pairwise Q* - R1 ⇒
pairwise Q* - R0 .
pairwise Q* - T5 ⇒
pairwise T5
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⇓
pairwise Q* - T4 ⇒
⇓
pairwise T4
⇓
pairwise Q* - T3 ⇒
⇓
pairwise T3
⇓
pairwise Q* - Urysohn⇒
⇓
pairwise Urysohn
⇓
pairwise Q* - T2 ⇒
⇓
pairwise T2
⇓
pairwise Q* - KC ⇒
⇓
pairwise KC
⇓
pairwise Q* - us ⇒
⇓
pairwise - us
⇓
pairwise Q* - T1 ⇒
⇓
pairwise T1
⇓
pairwise Q* - T0 ⇒
⇓
pairwise T0
4. LOWER Q* SEPARTION AXIOMS
In this section we study the new type of lower separation axioms called pairwise Q* 𝑇𝑇7⁄8 space, pairwise Q*
𝑇𝑇1 1⁄2 space pairwise Q* 𝑇𝑇1 2⁄3 spacepairwise Q* 𝑇𝑇2 1⁄2 space, pairwise Q* 𝑇𝑇4 1⁄2 space and pairwise Q* 𝑇𝑇5 1⁄2 space in
bitoplogical spaces.
Definition 4.1: A bitopological space X is said to be a τiτj- Q* 𝑻𝑻𝟕𝟕⁄𝟖𝟖 space if every τiτj - Q* closed set is τj - closed or τi
- Q* open, where i, j = 1, 2 and i ≠j.
Definition 4.1: A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝟕𝟕⁄𝟖𝟖 space if its τiτj- Q* closed and τjτ i - Q*
closed.
Theorem 4.1: Every pairwise Q* 𝑇𝑇7⁄8 space is pairwise Q* 𝑇𝑇1⁄2 space.
Proof: The proof is obvious.
Definition 4.1: A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝟏𝟏 𝟐𝟐⁄𝟑𝟑 space if every τi - Q* compact set is τj - Q*
closed , where i, j = 1, 2 and i ≠j.
Theorem 4.2: Pairwise Q* T2⇒pairwise Q* 𝑇𝑇1 2⁄3 .
Proof: Suppose that X is pairwise Q* T2. Let A be a τi - Q* compact set in X. Let p be any point in X – A. Take any x
∈A. But X is pairwise Q* Hausdorff.
Accordingly, there exists an τi - Q*open neighborhood Ux of x and τj- Q* open neighborhood Vx of p such that
Ux∩Vx= φ, where i, j = 1, 2 and i ≠j. The collection {Ux: X ∈ A} forms τ1 - Q* open covering of A.
But A is τi - Q* compact.
Hence there exists a finite τi - Q* open sub cover 𝑈𝑈𝑥𝑥 1 , 𝑈𝑈𝑥𝑥 2 , . . . , 𝑈𝑈𝑥𝑥 𝑛𝑛 for A.
The corresponding τj - Q* open sub cover 𝑉𝑉𝑥𝑥 1 , 𝑉𝑉𝑥𝑥 2 , . . . , 𝑉𝑉𝑥𝑥 𝑛𝑛 and 𝑈𝑈𝑥𝑥 𝑖𝑖 ∩𝑉𝑉𝑥𝑥 𝑖𝑖 = φ for i = 1 , 2 , … , n.
Hence W = ⋂𝑛𝑛𝑖𝑖=1 𝑉𝑉𝑥𝑥 𝑖𝑖 is a τj - Q* open neighborhood of pand W ∩A =φ.
That is, p ∈ W ∈ X – A.
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Consequently, X – A is τj - Q* open. But, then A is τj - Q* closed.
Hence X is pairwise Q*𝑇𝑇1 2⁄3 .
Proposition 4.1: Every pairwise Q* T2 space is pairwise Q* 𝑇𝑇1 1⁄2 .
Proof: Suppose that X is pairwise Q* T2 space. Let x ≠ y in X. Then there exists a τi - Q* open set U and τj - Q* open
set V such that U ∩ V = φ and x∈ U, y ∈ V.
⇒ X is pairwise Q* 𝑇𝑇1 1⁄2 .
Definition 4.2 - A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝟏𝟏 𝟏𝟏⁄𝟐𝟐 space if x and y aredistinct points X , there
exists adistinct τi- Q*open set U and a τj- Q* open set V such that x∈ U, y∈ V or x ∈V, y∈ U, where i, j = 1,2 and i ≠ j.
Theorem 4.1- The property of being a pairwise Q*𝑇𝑇1 1⁄2 space is hereditary.
Proof: Let X be a pairwise Q* 𝑇𝑇1 1⁄2 space. Let Y be a subspace of X, Let x, y ∈Y with x ≠ y. Then x ≠ y in X. But X
is Pairwise Q *𝑇𝑇1 1⁄2 space,
Hence there exist a τi- Q* open set U and τj- Q* open set V such that U ∩V= φand x ∈ U, y ∈ V or x ∈ V, y ∈ U. But
then x ∈U ∩ Y, y ∈V∩ Y or x ∈V∩ Y, y ∈U ∩ Y with (U ∩ Y) ∩ (V∩ Y) = (U ∩ V) ∩ Y = φ∩ Y = φ.
Hence Y is a pairwise Q*𝑇𝑇11 ⁄2 space,
Theorem 5: The property of being a pairwise Q* T12/3 is a topological invariant.
Proof: Let B be a σi - Q* compact subset of (Y, σ1, σ2). Let h: (X, τ1, τ2) → (Y, σ1, σ2) be a pairwise Q*
homeomorphism. Then h - 1(B) is a τ i - Q* compact subset of (X, τ1, τ2).
Put A = h – 1 (B). But (X, τ1, τ2) is pairwise Q* T12/3.
Accordingly, A is τ j - Q* closed. But, then h (A) is σj - Q* closed because h is a τj - Q* closed map.
That is, h (h – 1 (B)) = B.
Hence B is σ2 - Q* closed.
Consequently, (Y, σ1, σ2) is pairwise Q* T12/3.
This proves the result.
Proposition 4.1: Every pairwise Q* 𝑇𝑇1 1⁄2 space is pairwise Q* T0.
Proof: Suppose that X is pairwise Q* 𝑇𝑇1 1⁄2 space. Let x ≠ y in X. Then there exists a τi - Q* open set U and τj - Q*
open set V such that U ∩ V = φ and x∈U, y∈V or x∈V, y∈U.
⇒ X is pairwise Q* 𝑇𝑇0 .
Definition 4.1: A bitopological space X is said to be a pairwise Q* 𝑻𝑻𝟐𝟐 𝟏𝟏⁄𝟐𝟐 space if x and y are distinct points in X then
there exist a τi - Q* open neighborhood U of x and τj - Q* open neighborhood V of y such that τj - Q* cl (U)∩τi - Q* cl
(V) = φ, where i, j = 1, 2 and i ≠ j.
Example 4.1: Let X = {a, b} and τ1 = {φ, X, {b}}, τ2 = {φ, X, {a, c}}. Then τ1 - Q* open neighborhood U of x is {b
}and τ2- Q* open neighborhood V of y is {a, c} such that τ2 - Q* cl (U) ∩τ1 - Q* cl (V) = {b} ∩ {a, c} = φ . Hence X is
pairwise Q*𝑇𝑇2 1⁄2 space.
Theorem 4.1: The property of being a pairwise Q*𝑇𝑇2 1⁄2 space is hereditary.
Proof: Let X be a pairwise Q* 𝑇𝑇2 1⁄2 space. Let Y be a subspace of X, Let x, y ∈Y with x ≠ y. Then x ≠ y in X. But X
is Pairwise Q *𝑇𝑇2 1⁄2 space,
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⇒ there exist a τi- Q* open neighborhood U of X and τj- Q* open neighborhood V of Y such that U∩V= φ . But Y ⊂X.
� ∩V
� = φ.
U of X and τj - Q* open neighborhood𝑉𝑉� of Y such that U
⇒there exist a τi - Q* open neighborhood���
Hence Y is a pairwise Q*𝑇𝑇2 1⁄2 space.
Theorem 4.2: Every pairwise Q*𝑇𝑇2 1⁄2 space is pairwise Q* 𝑇𝑇2 .
Proof: The proof is obvious.
Theorem 4.3: Every pairwise Q* 𝑇𝑇2 1⁄2 space is pairwise T2 ½ space.
Proof: The proof is obvious.
Definition 4.2: A Q* T1 - space X is said to be a pairwise Q*𝑻𝑻𝟒𝟒 𝟏𝟏⁄𝟐𝟐 space if for each τi- Q* closed set A and τj - Q*
closed set B with A∩B = φ there exists a τi- Q* open set V ⊃ B and a τj- Q* open set U ⊃ A such that τi - Q*cl (U ) ∩τj
- Q* cl (V) = φ, where i, j = 1, 2 and i ≠ j.
Example 4.2: Let X={a, b, c} and τ1={φ, X, {c}}, τ2 = {φ, X, {a, b}}. Then τ1 -Q* closed sets are φ, {a, b} and τ2 - Q*
closed sets are φ, {c}. Take A = {a, b} and B = {c}. Then A is τ1 - Q*closed and B is τ2 - Q* closed. Also A ∩ B = φ.
Then there exists a τ2- Q*open set U = {a, b} & a τ1- Q*open set V = {c} such that
U ⊂ A and V ⊂ B ,τ1 - Q* cl (U) ∩τ2- Q* cl (V) = τ1-Q*cl ({a, b}) ∩τ2 - Q* cl ({c}) = {a, b} ∩ {c}= φ.
Hence X is pairwise Q* 𝑇𝑇4 1⁄2 space.
Theorem 4.4: Every pairwise Q* 𝑇𝑇4 1⁄2 space is pairwise Q* 𝑇𝑇4 space.
Proof: The proof is obvious.
Theorem 4.5: Every pairwise Q* 𝑇𝑇4 space is pairwise Q* 𝑇𝑇4 1⁄2 space.
Proof: The proof is obvious.
Definition 4.3: A Q* T1 - space X is said to be a pairwise Q* 𝑻𝑻𝟓𝟓 𝟏𝟏⁄𝟐𝟐 space if for every subsets A and B of X such that
τi- Q* cl (A) ∩ B = φ and A ∩τj - Q* cl (B) = φ there exists a τj - Q* open set U & a τi- Q* open set V such that A ⊂ U
and B ⊂ V,τi- Q* cl (U ) ∩τj- Q* cl (V) = φ, where i , j = 1 , 2 and i ≠ j.
Example 4.3: Let X = {a, b, c} and τ1 = {φ, X, {c}, {b, c}} τ2 = {φ, X, {a},{a, c}}. Then τ1- Q* closed sets are φ, {a},
{a, b} and τ2 - Q* closed sets are φ, {b, c}, {b}. Take A = {a} and B = {b}.
Then τ1 - Q* cl (A) ∩ B = τ1 - Q*cl ({a}) ∩ {b} = {a} ∩ {b} = φ
A ∩τ2- Q* cl (B) = {a} ∩τ2 - Q* cl ({b}) = {a} ∩ {b} = φ
Then there exists a τ2- Q* open set U = {a} & a τ1-Q* open set V = {c} such that A ⊂ U and B ⊂ V,τ1- Q* cl (U) ∩τ2Q* cl (V) = {a} ∩ {b, c}= φ.
Hence X is pairwise Q* 𝑇𝑇5 1⁄2 space.
Theorem 4.6: Every pairwise Q* 𝑇𝑇5 1⁄2 space is pairwise Q*𝑇𝑇5 space.
Proof: Let X be a pairwise Q*𝑇𝑇5 1⁄2 space. Then X is Q* T1 - space. Let A and B are disjoint subsets of X. Then A =
τi- Q* cl (A) and B = τj- Q* cl (B).
⇒τi- Q* cl (A) ∩ B = A ∩ B = φ and A ∩τj - Q* cl (B) = A ∩ B=φ.
Hence A and B are Q* separated sets in X.
Since X is pairwise Q* 𝑇𝑇5 1⁄2 space, we have
τi - Q* open set U and τj- Q* open set V such that
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A ⊂ U and B ⊂V, τi - cl (U) ∩τj - cl (V) = φ .
⇒U∩V=φ.
Theorem 4.7: Every pairwiseQ* 𝑇𝑇5 1⁄2 space is pairwise𝑇𝑇5 1⁄2 space.
Theorem 4.8: Every pairwiseQ* 𝑇𝑇5 1⁄2 space is pairwise Q* 𝑇𝑇4 1⁄2 space.
Proof: Let X be a pairwise Q* 𝑇𝑇5 1⁄2 space. Then X is Q* T1 - space. Let A and B are disjoint closed subsets of X.
Then A = τi - Q* cl (A) and B = τj - Q* cl (B).
⇒τi - Q* cl (A) ∩ B = A ∩ B = φ and A ∩τj - Q* cl (B) = A ∩ B =φ.
Hence A and B are Q*separated sets in X. But X is pairwise Q*𝑇𝑇5 1⁄2 space, then there exists a τi - Q*open set V ⊃ B
and τi - Q*open set U ⊃ A such that τi - Q* cl ( U ) ∩τj - Q* cl ( V ) = φ.
⇒U∩V=φ.
Therefore, pairwise Q* 𝑇𝑇5 1⁄2 space is pairwise Q*𝑇𝑇4½ space.
Theorem 4.9: Every pairwise Q* 𝑇𝑇4 1⁄2 space is pairwise𝑇𝑇5 1⁄2 space.
Proof: The proof is obvious.
Remark: The following implications are obvious.
pairwise Q*𝑇𝑇2 1⁄2 space ⇒ pairwise Q* T2 space ⇒ pairwise Q* 𝑇𝑇1 1⁄2 ⇒pairwise Q* T0 .
pairwise Q* 𝑇𝑇4 1⁄2 space ⇒pairwise Q* 𝑇𝑇4 space
pairwise Q* 𝑇𝑇5 1⁄2 space ⇒pairwise Q* 𝑇𝑇5 space.
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