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Kyungpook Matbematical Journal Volume 30, Numbcr 2, December, 1990

THEORY OF EXTENSIONS OF BICLOSURE SPACES AND BASIC QUASI-PROXIMITIES

K.C. Chattopadhyay and R.N. Hazra

1. Introduction

B잉 ng motivated by the work of Wilson [27J on quasi-metric spaces

,

J.C. Kelly [13J in 1963

,

introduced the concept of bitopological spaces and investigated some interesting generalizations of Urysohn

s lemma

,

Urysohn

s metrization theorem

,

Tietze extension theorem and the Baire category theorem. Since then the concept has been studied by [8], [14], [15], [19J.

Since no compatible (Efremovic) proxirnity [7J is available for an arbi trary topological space unless it i8 completely regular

,

Pervin [2이 intro duced the concept of qu잃i-proximities to show that each topological space can be induced by a qua8i-proximity. I-Ie defined quasi-proximities just by ornitting the symmetry condition from the definition of (Efremovic) prox

1II삐es. Later Steiner [23J added an additional condition to the definition of quasi proximities of Pervin [20J to show that quasi-proximities induce a topological 8pace. Quasi proximities were aJso investigated by Gastl [8], E.P. Lane [14

,

15], Mattson [17J and Sing따 and Sunderlal [22J among others. Quasi-proximities and bitopological spaces are investigated in [8],

[14

,

15], [19J and [22J

Classical proximities and their generalizations were studied in [10], [16], [24J. Each such generalization is a basic proximity. Basic proximities were introduced by Cech [2J. A basic proximity always induces a closure operator which need not be idempotent but must satisfy the Ro-axiom So all closure spaces cannot be subsumed under the scheme of proxim ities. Being motivated by this fact and Pervin

s introduction of quasi proximities and the c∞o아rπrectiorαonno여f Perπv피1I따 definition by St잉 ner [23], Hazra

R.eceived May 30, 1988

137

(2)

138 K.C. Chattopadhyay and

R.N .

Hazra

and Chattopadhyay [11] introduced the concept of basic quasi-proxim띠es

and investigated their properties. Thron [25] pointed out the importance of grills in the theory of proximity spaces. The importance of grills has further been supplemented by the present authors in the theory of basic quasi-proximities also

1n this paper we introduce a theory of extension of biclosure spaces Some separation axioms for biclosure spaces are studied. Compactness and linkage compactness for biclosure spaces are defined.

We have defined R1-quasi-proximities

,

invcstigated some of their prop- erties and proved that each R

,

-biclosure space admits a compatible RI

qu잃 i-proximity.

It is well known that proximities, contiguities and nearnesses have been used to construct extensions of underlying spaces on which they are defined (see [1], [3]

,

[6]

,

[9] and [26]). The underlying sel of each such extensic'n consists of certain maximal families (maximal clans

,

ends

,

clusters) asso- ciated with the structures Even though

,

due to lack of symmetry

,

the choice of such maximal families associated with quasi-proximities seems to be diffìcult, we have nevertheless been able to introduce a concept of maximal clans in the context of quasi-proximi ties and used them succes fully to construct a class of extensions of biclosure spaces. VVe conclude the paper with our main result: Given a R" T

,

-biclosure space, there exists a 1-1 correspondence between the class of all RI-quasi-proximities and the class of all principal

,

linl

age compact extension of the space. To make our investigation meaningful we draw the reader’s attention to the observations made in Remark 3.5, the results of earlier investigators in the area of bitopological spaces and the fact that if a bitopological space can be

embedded

in a meaningful way into a compact bitopological space then many of the nice properties of the latter space can to some extent be brought to bear on the study of the former space. 1n this connection it may be mentioned that we have reformulated a problem of F. lliesz and indicated its solution for all separated RI-quasi-proximities

2. Preliminaries

Thron [25] has shown that the concept of grills plays an important role in the theory of proximities. Since then the properties of grills have extensively been used by many authors in the studies of extension theory and nearness spaces. Present authors also used the properties of grills to develop a theory of basic quasi-proximities in [11]

(3)

. Theory of Extensions of Biclosure Spaces 139

We begin by recalling the definition of a grill. A 9πII 9 on X is a collection of subsets of X satisfying

A :::) B E

9

=} A E

9 ,

AUBE9=수 A E

9

or B E

9

and

0

rf-

9

For our purpose it is sufIìcient to note that grills are exactly unions of ultrafilters. The relation between grills and filters has been investigated by Thron [25]. We shall use the symbol

f(

X)

,

and í!(X) to denote the set of grills and the set of ultrafilters on a set X

,

respectively. Though an element x in X is conceptually different from a singleton {x} they are not distinguished

Following Herrlich [12] we define sec A

,

for each subfamily A of the power set P(X) of the set X by

Sec A = {A C X : A

n

B

# 0

for all B E A}

The function A sec A establishes a 1-1 correspondence between the set of filters and the set of grills on the same set

If F is a fi1ter and 9 is a grill then sec F is a grill and sec 9 is a filter. Also sec2 F = F and sec29 = 9. For any ultra-filte1' U, sec U = U.

Moreover if F is a filter and 9 is a grill on a set X and F C 9 then there exists an ultrafilter U on X such that F

c

U

c 9.

A subset

9

1- of í!(X) is defined for every 9 in r ( X) as follows

9

1-

=

{U E í!(X) : U C g}

From the definition of grills it follows that the empty collection of subsets of a set is a grill. In what follows we shall always assume that g1'ills are nonempty unless stated otherwise.

A (éech) closure operator c on a set X is a function from the power set P( X) of X into itself satisfying the following conditions:

c(Ø) = 0

,

c(A) :::) A and c(A U B) = c(A) U c(B) fo1' all A

,

B C X.

A pair (X

,

c)

,

w he1'e c is a closure operator on X

,

is called a closure space.

Closure spaces are studied in detail in δech [2]. A closure space (X

,

c) is said to be a topological space if in addition c satisfies the condition :

c(c(A)) C c(A) for all A C X

(4)

140 K.C Chattopadhyay and R.N. Hazra

For each element x and a closure space (X

,

c) the familty Q(c,τ) = {A C X : X E c(A)} is a grill on X and is called the adherence grill of x. A grill Q on the closure space (X

,

c) is called a c-grill if

c(A) E Q =추 A E Q.

Note that the closure space (X

,

c) is a topological space iff Q( c

,

x) is a

c一grill for each τ in X (see [4])

A point x in X is said to be an adhe

,

ymce poinl of a grill Q on (X

,

c)

if Q

c

Q( c

,

x). The grill Q is said to conve''!Je to a point x in X if Q

n

(c, x) ¥

O

A closure space (X

,

c) is said to be a 낀o-space if for eacll X

,

ν in X Q(c

,

x) = Q(c

,

y) =} x = y

The closure space (X

,

c) is said to be a T

,

-space iffor each x in X

,

c(x) x.

Note that a T

,

-closure space is a To-space but the converse may not be true. The closure space (X

,

c) is said to be an R

,

-space if for each x in X and A

c

X

,

c(x)

n

c(A)

o

=} x E c(A).

Every T

,

-space is an R

,

-space

,

the converse need not hold. A function

J

from a closure space (X

,

c) into another (Y

,

k) is said to be conlinuous if J(c(A))

c

k

(J

(A)) for 때 1 A

c

X. It is easy to verify that J is continuous iff J(Q(c

,

x))

c

Q(k,f(x)) for each x in X.

Following Cech [2] we define a closure space (X

,

c) to be compact if

,

for every filter F on X

n{c(F): F E F}

0.

lt is well known that (X

,

c) is compact iff {Q.L( c

,

x) x E X} covers f1(X) iff each grill on X converges to a point (see [4], [6]). It can be easily verified that a continuous image of a compact closure space is compact

A closure space (X

,

c) is said to be linkage compacl provided each linked grill has an adherence point. By a linded grill we mean a grill Q on X such that c(A)

n

c(B)

f 0

for each pair A

,

B of Q. For details of these concepts we refer to Chattopadhyay, Njåstad and Thron [6].

3. Biclosure spaces

An order triple (X

,

c"

C 2)

is called a biclosure space if c" are two closure operators on X. (X

,

Cl

,

C2) is called a bitopological space if C" C2

are Kuratowski closure operators on X

(5)

Theory of Extensions of Biclosure Spaces 141

The cartesian product Q1 x Q2 of two grills Q1

,

Q2 on

X

is called a bigrill on X. For every point X of the biclosure space (X

,

Cl

,

C2) the cartesian product Q(Cl'X) X Q.(C2

,

X) is called the adherence bigrill of ε A bigrill Q1 X Q2 is called a (c\, c2)-bigπ11 provided Q1 is a ci-grill for each i

=

1

,

2.

A function

f

from (X

,

c" C2) into (Y

,

k" k2) is said to be continuous provided

f

is continuous from (X

,

Ci) into (Y

,

ki ) for each i = 1

,

2. Furthel

f

is said to be a homeomorphism if / is a bijection and /, /-1 are both continuous. Also (X

,

C1

,

C2) is called homeomorphic to k1

,

k2) if there exists a home아norphism from one to the other.

A biclosure space (X

,

Cl

,

C2) is called a 매 space if

,

for each x

,

y in X Q(c\,X) X Q(C2'X) = Q(c\,y) X Q(C2

,

y) =추 X=y

The biclosure space

(X ,

Cl

,

c,) is called a T1-space if C1(X) nC2(X) = X for all X in X

It can be verifìed that a T1-space is a To-space but the converse may not be true. The biclosure space (X

,

Cl

,

C2) is called an R1-space if

,

for each X in X and A

c

X

,

Cj(X) n c2(A)

ø

=추 X E c2(A)

and

c1(A) nC2(X)

f ø

=} x ε c1(A)

Theorem 3.1. Let (X, c" C2) be an RJ-space. Theη the space is To iff the space is T1.

Proof From the definitions it is immediate that a TJ-space is 피b

Now suppose that the space is To and let y E cJ(X)nC2(X) for some x,y in X. Since the space is R j we have y E Cj(x)

n

C2(X) iff X E Cl(y)

n

C2(y) Hence

(A

,

B) E Q(cJ, x) X Q(C2

,

x) iff :1: E c1(A)

n

c2(B) iff C1(y)

n

c2(B) m and C2(y)n C1(A)

f ø

iffy E c1(A) nC2(B) iff (A

,

B) E Q(cJ , ν) X Q(C2

,

y) Thus Q(C1'X) X Q(C2'X) = Q(cJ

,

y) X Q(C2 , ν) ‘ Since the space is To it follows that X

=

Y and hence Cl(X)

n

C2(X)

=

X

Hence the space (X

,

Cl

,

C2) is T J. Below we give an example of a bi- closure space (X

,

C1

,

C2) which is T1 and R 1 but not a bitopological space

(6)

142 K.C. Chattopadhyay and R.N. Hazra

Example 3.2. Set Xn

=

{(1 , lJn)(2 lJη) ,

.. . }

for n

=

1

,

2

, ... ,

J {1

,

2

, . ..

} Y = U얻 Xi , Z= JU{ω} 、애ere ω is an element not belonging to Y U J

,

X = Y U Z

Define cy(A) = A U {n E J : A

n

Xn is infinite

},

if A

c

Y

,

and f01 each A

c

Z

,

cz(A) = A

,

if A is finite

= AU }, if A is infinite Finally for each A C X

c

,

(A) = Cy (A

n

Y) U cz(A

n

Z)

and

c2(A) = A.

It can be easily verified that (X

,

Ct, C2) is a nonbitopological biclosure space which is also T

,

and R

,

Let (X

,

c" C2) be a biclosure space and

ç ,

x Ç2 be a bigrill on X. A point x of (X

,

Ct,C2) is called an adherence point of

ç ,

x Ç2 if

ç ,

x Ç2 C

ç (

Ct, x) x

ç (

C2

,

x)

The biclosure space (X

,

Ct, C2) is said to be compact if for each U E !1(X)

,

U x U has an adherence point. A bigrill

ç ,

x Ç2 on a biclosure space

(X

,

CI , 이) is said to be linked bigrill if

c

,

(A)

n

c2(B)

"1 ø

for all A ε

ç"

and B EÇ2

The biclosure space (X

,

c" C2) is said to be linkage compact if each linked bigrill bas an adherence point

Theorem 3.3. A continuous image

0/

a compact space is compact.

Proof Let (X

,

C"C2) beacompadspaceand let (Y씨

,

k2) be a continuous image of (X

,

c" C2) under the function /

Let μ be an ultrafilter on Y. Then F

= U-

1(A) : A E U} is a filter on X. Let V be an ultraftlter on X such that F

c

V. Since (X

,

C" C2) is compact space

,

it foUows that there exists x in X such that V

c

Ç(

C;,

x)

for i 1

,

2. Further since / is continuous

,

for each B E U and each i = 1

,

2

,

we have

r 1(B) E V =수 B:::)

/U-

1(B)) E Ç(k;,/(x))

~

B E Ç(k

‘’

/(x))

(7)

Theory of Extensions of Biclosure Spaces 143

Hence it follows that for each i

=

1

,

2.

u c

9(k;,f(x)).

Therefore i t follows

U x U

c

9(k,,J(x)) X 9(k2,J(X)).

Thus (Y

,

k" k2 ) is compact.

Theorem 3.4. A linkage compact biclosure space is compact

The proof follows from the fact that μ x U is a linked bigril1 for each ultrafilter U

Remark 3.5. It is easily verified that if (X

,

cd and (X

,

C2) are cor때 act

closure spaces then it may not be true that

(X , c"

C2) is compact and the same conclusion remains valid w hen the word ‘compact’ is replaced by the word

linkage compact'. But if (X

,

Cr

,

C2) is compact then (X

,

cr) and (X

,

C2) are cor매act. Whether the corresponding statement holds for linkage compactness is not known to the authors

,

however this does not any way affect our investigation. Also it is interesting to note that a closure space (X

,

c) is compact iff the biclosure space (X

,

c

,

c) is compact Also if (X

,

c

,

c) is linkage compact then (X

,

c) is linkage compact but the converse is not known to us

,

however this also does not stand in the way of our investigation

Below we give an example in support of our assertions made in the remarl

Example 3.6. Let X be an infinite set and Y

=

X U {ψ1

,

ω2} such that

ψ,

'"

W2 and X

n

{ω, , ω2}

0.

Define two closure operators on Y a8

f0110W8:

c

,

(A) = c2(A) = A for all finite subset A of Y

,

c,(A)

= Au

{ψ,}, c2(A)

=

A

u

{ω2}

for al1 infinite subset A of Y

Let 9 be a linked grill on (Y

,

c

,).

Then either 9 contains no finite subset of Y or there is exactly one Yo in Y such that {yo} E 9 and hence either 9

c

9(c"w,) or 9

c

9(c"yo). Thus (Y

,

c,) is linkage compact

Similarly it follows that (Y

,

C2) is also linkage compact. Hence in par ticular both (Y

,

c,) and (Y

,

C2) are compact. Let U be a no때rincipal

ultrafilter on Y then w

,

is the unique point in Y such that μ C 9(ct

,

wr)

(8)

144 K.C. Chattopadhyay and R.N. Hazra

and W2 is the unique point in Y such that U C Q( C2, ω2). Hence (Y, C.) and (Y

,

C2) are both compact but the biclosure space (Y

,

Cj

,

C2) is not compact and in particular (Y

,

Cj

,

C2) is not linkage compact.

4. Extensions of biclosure spaces

1n this section we introduce extensions of bioclosure spaces. Results will be used to investigate the relation between basic quasi-proximities and certain principal extensions

4.1. E = (ψ, (Y

,

kj

,

k2)) is said to be an exlension of a biclosure space (X

,

Cj

,

C2) if (Y

,

kj

,

k2 ) is a biclosure space and ψ form X into Y is an injection such that for each A C X

ψ (ci(A))

=

ki(ψ(A))

n

ψ(X)

and

ki(ψ (X)) = Y for all i = 1

,

2

For an extension E ψ , (Y,kjk2)) of (X

,

Cj

,

C2) we define the tmce o[

each point y in Y by T(E , ν)

=

7j(y) x T,(y)

,

where for each i

=

1

,

2

,

τ (y) = {A C X : y E ki(ψ (A))}. We shall simply write T(y) for T(E

,

y) when no confusion is likely to arise. The collection XE

=

{T(ν) ν E Y}

is called the tmce system of the exte떠on E of (X

,

Cl

,

C2)

Note that (ψ , (Y, kjk2)) is an extension of (X

,

Cj

,

C2) iff , (Y, 싸)) is an extension of (X

,

c‘) for each i = 1

,

2

,

(For the extension of c10sure spaces we refer to Chattopadhyay and Thron [4] in which an extension theory of closure spaces has been investigated)

Two extensio따 Ej

=

(rp

,

(Y

,

kj

,

k2 ))

,

E2

=

(ψ , (Z, k3

,

k4)) of the biclo- sure space (X

,

c] , C2) are said to be equivalenl if there exists a homeomor.

phism

J

of (Y

,

kj

,

) onto (Z

,

k3

,

μ) such that

J

0 rp = ψ on X. Tbe extension E j is said to be greater tban E2 if tbere exists a continuous function

J

from Y onto Z such that

J

0 rp = ψ on X

An extension E = (ψ , (Y

,

k], k2 )) of a biclosure space (X

,

Cj

,

C2) is said to be To(Tj

,

Rj) provided the biclosure space (Y

,

k" k2) is 낀o(T1R

Il

Theorem 4.2. Let E (Y

,

k], k2)) be an extension oJ a biclosure space (X, C] , 아). Theπ

(i) T(y) is a big써 on X J01' each y in Y

,

(ii) T(ψ(x))

=

Q(C1'X) X Q(C2'X) J01' each x in X Proof is straight forward.

(9)

Theory of Extensions of Biclosure Spaces 145

The following result ;s easily established.

Theorem 4.3. Two equivalent extensions oJ a biclosu1e space have iden- tical trace systems

4.4. Construction of extensÎons of To-biclosure spaces. Let (X

,

c]

,

C2) be a To-biclosure space and let X' denote a collection of bigrills on X such that 9(c], x) X 9(C2, x) E X' for each x in X. Define

cp: X X‘ by cp(x) = 9(c"x) X 9(C2

,

X) for each x E X A'ó = {Ç] X 92 E X' : A E 9;} for eacb A C X for each i = 1

,

2

Since the elements of X. are bigrills on X it foUows that (A U B)C, =

ACU BCó for each i = 1

,

2

,

and for each pair of subsets A

,

B of X. Hence for each i = 1

,

2

, {A"

A C X} is a base for the closed sets of some topology on X' For each i = 1

,

2

,

let Ti : P(X' - ψ(X)) P(X*) be a function satisfying the conditions

η(0)

=

0

,

Ti(O) :::) 0 and Ti(O U β)

=

T‘(0) U Ti(ß)

Then for each i = 1

,

2 the function hró : P(X.) P(X*) defined by h

,

Jo) = [cp-I(O)]Có U η (O- <p (X))

is a closure operator on Hence (X*

,

h"

,

hr2 ) is a biclosure space.

Note that to prove h

,

ó is a closure operator on X。ne requires that (A U

B)"A'ó U BC' for each i = 1

,

2 and A

,

B

c

X

,

in add.ition to the properties of η as prescribed above.

Below we state and prove our main result of this section

Theorem4.5. Let(X

,

c]

,

C2) be a

To

-biclosU1-e space. Then(

<p,

(X

’,

h"

,

h

,,))

is a To-extension oJ (X

,

c]

,

C2) JOT aLI T], T2

,

such that T(Ç] x 92)

=

91 X 92 Jor all 91 x 92 E X' . M01"eover iJ (ψ , (X' , k] , k2)) is an exteηsion oJ (X

,

c]

,

C2) such that T(9] x 92) = 9 1 X 9 2 Jor aII 9 1 x 9 2 E X

,

lhen -.þ =

<p

and k= h

,

ó Jor some sltitable choice oJ η (i=1 , 2).

Proof Since the space (X

,

C1

,

C2) is To

,

the mapping

<p

from X into X' is an injection. It is clear that for each A

c

X and for each i

=

1

,

2

,

h,ó(씨 A)) = A'ó and hence h

,,

(cp(X)) = X. for eacb i = 1

,

2

Again since

<p

(x) E Nó iff9(cI'x) X 9(C2'X) E ACó iff A E 9(c;,x)

(10)

146 K.C. Chattopadhyay and R.N. Hazra

iff x E c

(A) iff cp(x) E cp(c;(A)), it foUows that

,

for each i = 1

,

2 and A

c

X

,

hr• (ψ(A))

n

cp(X) = cp(c(A))

Thus (cp

,

(X

,

hr

, ,

hr

,))

is an extension of C1

,

C2)

Next suppose that 9(hr,,91 x

92)

x 9(hr,,91 x

92)

= 9(hr"

9 3

x

9.)

x 9(hr2'

93

x

9.).

Then (A

,

B) E

9 ,

X

92

iff

9 ,

x

92

E

N' n

BC' = hr

,

(cp(A))nhη (cp(B)) iff (cp(A

)),

cp(B)) E

9(

hr!1

9 ,

x

92)

x 9(hr2'

9 ,

x

92)

=

9(h,,,93 x

94)

x 9(hr"

93

x

9. )

iff

93

x

94

E hr

,

(cp(A))

n

hr,(cp(B)) = A"

n

B'2 iff (A

,

B) E

93

X

94

Hence (cp

,

(X

,

hr!1 h

,,))

is a extension of (X

,

C"C2)' Now for each

9 ,

X

92

E X'

,

we have

(A

,

B) E T

(Ç,

x

92 )

= T,

(Ç,

x

92)

X T

(Ç,

x

92)

iff

9 ,

X

92

E hr

,

(cp(A))

n

hr

,

(cp(B)) = A"

n

B" iff (A

,

B) E

9 ,

x

92

Thus for eaιh

9 ,

X

9 2

E X'

,

T

(Ç,

X

92)

=

9 ,

X

92

If (ψ , (X'

,

k" k2)) is an extension of (X

,

c" C2) satisfying the condition stated in the theorem then from Theorem 4.2 (ii) and from the given condition we have

,

for each x in X

,

ψ(x)

=

T(ψ (x))

=

9(C1'X) X 9(딩 , x)=cp(x).

Note that for A

,

B

c

X we have

9 ,

X

92

E k ,(ψ(A))

n

k2(cp(B)) iff (A

,

B) T

(Ç,

x

92)

9 ,

X

92

iff

9 ,

x

92

E A"

n

BC'

Since the above relation holds for arbitrary subsets A

,

B of X

,

we have

k

,

(cp(A))

=

k

,

(cp(A))

n

k2(cp(X))

=

AC'

k2(cp(B)) = k

,

(cp(A))

n

k2(cp(B)) = B"

Hence for each a

c

X and for each i

=

1

,

2

,

k;(a) = k;(cp('f'-' (a))) U k;(a - cp(X))

= [cp-1(a

)J"

U k;(a ψ(X))

(11)

Theory of Extensions of Biclosure Spaces 147 Put ki(o:)

=

ri(O:) for all 0: C X' - c,o(X) and for each i

=

1

,

2. Then each k

is of the form hr ; and hence

(!þ,

(X'

,

k1

,

kz)) is an extension of the form (cp

,

(X'

,

hr p hr2 ))

Remark

4-

6. If (ψ , (Y

,

k1

,

k2 )) is a To extension of a To.biclosure space (X

,

Cl

,

C2) V따h trace system X' then the trace function T from Y into X' may not be a bijection. Such extensions are not included among the ones discussed above. Now if T is a bijection and if we define

hi(o:) = T(ki(T-1(0:))) for all 0: C X' and i = 1

,

2

,

then T is a homeomorphism of k], k2 ) onto (X", h" h2 ). Also (T ψ, (X‘ , h" h2 )) is an extension of (X

,

Cl

,

cz) such that the trace T

(Ç,

x

92) = 91

X

92

for each

91

x

92

E X' and hence by the last part o[ the above Theorem 4.5 it follows that (T 。 ψ(X' , h1hz)) is an extension of the type (cp

,

(X'

,

hr " )) and hence so is (Y

,

k1

,

k2 )) if we ignore the difference between homeomorphic spaces

4.7. Some special choices of ri and topological extensions and principal extensions

An extension E = (ψ , (Y

,

k" k2 )) is called a topological exteηsion of (X

,

Cl

,

C2) provided (Y

,

k1

,

k2 ) is a bitop이ogical space.

For each i = 1

,

2 if we define

r; , r;'

two functions from P(X' - c,o(X)) into P(X') by

and

<(0:) = 0:

J X' if 0: 0

’‘ (a)

= 1

lf a

= 6

then 서 and r‘ are possible choices of ri

,

defined at the beginning of this section. They also satisfy the following conditions

ri(o:) C η(0:) C

7-;

’(0:) for each i = 1

,

2

,

for each 0: C X‘ - <p(X) and for every ri as defìned earlier

,

and hence we have

hr;(Q)

c

hr,(o)

c

hr”(Q) for all a

c x-

and for z

=

1

,

2, Thus we have the following result:

E = (p

,

(X·

,

hr;

,

hr;)) is the lamest and EII = (v

,

(X*

,

hr;’ hr; ) ) is the smallest 01 all extensions E = (cp

,

(X

,

hr

"

hr,)) 01 (X

,

Cl

,

cz)

(12)

148 K.C. Chattopadhyay and R.N Hazra

For a biclosure space (X

,

C]

,

C2) it has already been pointed out that for each i = 1

,

2

,

{A"

,

AC X}

is a base for the closed sets of a topology on X' and hence for each i = 1

,

2

,

이 (0) = n{Aι A" 그 ll'} for a11 ll' C X'

,

defines a Kuratowski closure operator on X'. Note that '1'" (A) C A' for each i

=

1

,

2 and for each A C X

,

hence d

(<p

(A)) C A". The equality need not hold in general.

However

,

if (X

,

C2) is a bitopological space then for each x in X

,

9(C1'X) X 9(C2'X) is a (c1

,

c2)-bigrill on X and if in addition tbe rest of the elements of X’ are also (c1

,

c2)-bigrills then for eacb i = 1

,

2.

d;(

<p

(A))

=

A" for all A C X

This equality follows from tbe following facts :

(i) Ao; :::> cp(A) for all A C X

,

(ii) Bo; :::>

<p

(A) iff c;(A)

c

c;(B)

,

(i

ii) since elements of X' are (c]

,

c2)-bigrills

,

it follows that

[e;

(D)]'= Do; for all D

c

X Thus in tbis situation we also ha、 e the following

hdo) = ψ-](ll' )]'i U d;(ll'-ψ(X))

= d;(cp(

<p

-1(ll'))) U d;(o -

<p

(X)) d‘(ll') for all 0

c

X' and i = 1

,

2

,

and hence

(<p,

(X'

,

d]

,

d2)) is a T

o一

topological extension of the bitopolog- ical space (X

,

c], C2) such tbat the traces T(91 x 92) 9] x 92 for all 9] X 92 E X'

If

(<p ,

(X'

,

h1

,

h2 )) is anotber topological extension of the bitopological

space (X

,

C1

,

C2) satisfying T(91 x 92)

=

9] X 92 for all 9] x 92 E X'

,

where X' is the same collection of (c], C2) bigrills then we must have A"

=

h;(ψ (A))

=

h;(h

(cp(A)))

=

h;(A'i) for all A C X and i

=

1,2 Consequently ACi are closed in (X'

,

h

‘)

and bence the identity function is a continu。야 function from (X

,

h]

,

h2) onto (X‘

,

d" d2 ). Thus we have the following result :

(13)

Theory of Extensions of Biclosure Spaces 149

Theorem 4.8. If

(X ,

c" C2) is a To-bitopological space and η X !s a collection of (c\,c2)-big1'ills such that g(c\,x) x g(C2'X) E X' fo1' all x in X then

(<p,

(X

‘,

d" d2)) is the smallest of all topological exlensions

(<p,

(X'

,

h" h2)) satisfying lhe condition that the trace of each gl x g2 is g

,

X g2

Looking the equality hd. = di for i = 1

,

2 in the above topoJogical situ ation and being motivated by the usuaJ definition of principal extensions used in the literature by various authors in the topological situation

,

fo

,

an arb따ary To-biclosure space (X

,

c" C2) we define (X'

,

hd" hd

,))

to be a prin cipal βtπ ct) extension of (X

,

C"C2) 、애ere X' is a colIection of bigrills on X containing all g(c"x) X g(C2'X)

In this connection i t is interesting to note that

(<p ,

(X

,

k" k2 )) is a principal extension of the To-biclosure space (X

,

C" C2) iff [0110"1l1g two conditions are met :

(1) B

c

X'- ψ(X)A C X

,

B

c

ki('P(A)) =} k;(B)

c

ki(ψ(A)) ,

(II) for each B C X. -

<p

(X) there exists a famiJy A

c

P(X) such that k(B) = n{k‘

(<p

(A)) : A E A}.

Remark 4.9. In the bitopological situation a principal extension is a min- imal one but in the case of biclosure spaces a principal extension satisfies two extremal properties

,

it is a minimal one with respect to the above property (1) and is a maximal one with respect to the above property (II)

From Theorem 4.5 it immediately folJows that a prillcipal extension of a To-biclosure space is To But a principal extellsion of T

,

-biclosure space

need not be τ1 in general.

Example 4.10. A principal extension of a T

,

-biclosure space need not

be T

,

and in particular a principal extension of R

,

-biclosure space need

not be R

,

Let (X

, c"

C2) be the biclosure space of Example 3.2. Now define X'

to be a set of bigrills such that for each x E X

,

both U(x) x μ (x) and g(Cl

,

X) X g(C2'X) E X‘. Here U(x)

=

{A C X : x E A}, the principal 미­

trafilter containing x. Consider the principal extension (X' , hdl

,

hd

,))

of the space (X

,

c" C2)' Recall thal (X

,

c\, C2) is a nontopological T" R

,

biclosure space. Obviously by Theorem 4.5

,

(X'

,

hdJ> hd

,)

is a To space It is easy to verify that U(ω) x U(ω )

=J

g(c\,x) X g(C2'X)

=

<p(ω) and

U(ω) x U(ω) E hd1 ('P(ω)) n hd,(<p(ω)). Thus (X

,

hd

, ,

hd

,)

is not T

,

and

(14)

150 K.C. Chattopadhyay and R.N. Hazra

hence in ,~ew of Theorem 3.1, (X‘, hd " hd,) is not RJ. H。、、앙er the fol lowing holds :

Theorem 4.11. A principal extension

('1',

(X'

,

hd" hd,)) of a To-biclosurc space (X

,

C1

,

c,) is T1 i

fJ

each element

Çh

x 9, of

x' -

'P(X) is !1ηw따xzml

!nπ X' and satisfies the conditioπ that 9J

n

9

,

contains no singleton Proof Suppose that (X',hd"hd,) is T1. Let x ε X. Then 'P(x) E 'P(cJ(X) n c,('x)) C hd, ('P(x)) n hd,(ψ(X)) = rp(x). Since 'P is 1-1, Cl(X) n c

,

(x) = x. Thus (X

,

C"C2) is

Let 91 X 9

,

ε X' - cp(x). If 93 X 94 E X' and 9

,

x 9

,

C 93 X αl then 93 x 94 E hd,(QJ X 9,)

n

hd,(Ql x 9,). Since (X , hd, , hd,) is 1’"

93 X 9. = 9 1 X 9

,

and hence each element of (X' cp(x)) is maximal in X“ Further if there exists x E X such that x E 91

n

9

,

then 91 x 9

,

E

hd,('P(x))

n

hd,(<p(x)) = <p(x)-a contradiction. Thus for each 91 X 9, E X' - cp(X) there is no singleton in 91

n

9

,

Conversely suppose that the conditions of the theorem hold. Then fOl eachx EX

hd,(<p(x))

n

hd,('P(x)) = [X)Cl

n

[x)C'

Since (X

,

C1'C,) is Tt, [X)Cl n [x)C' nψ (X) = ψ(τ). Also [X)Cl n [x)C' n (X' cp(X)) =

ø

for 91

n

9

,

contains no singleton for 91 X 9

,

E X. - ψ(X)

Hence we have hd1 (<p(X))nhd,(ψ (x)) = cp(x) for all x E X. Also if 9

,

x9

,

E

X cp(X) and 93 X 94 E X' such that 93 x 94 ε hd1 (91 x9,)nhd,(Q1 x9,) then 9J x 9

,

C 93 X 9. and hence by the maximality of the elements of X. - cp(X) in X'

,

9 1 X 9

,

= 93 X 9.

Thus the space (X'

,

hd" hd,) is T1

5. Basic quasi-proximity structures

5.l. A binary relation π 。n P(X) is said to be a basic quasi-p1"Oximity on X if the following conditions hold

QP 1 : (1) (A,BUC) E π lκ (A,B) E 7r or (A,C) E π,

(1I)(AUB,C)E π iff (A, C) E π 。r (B ‘ C) E π

QP

, :

A

n

B

-1 0

=수 (A,B)E 7r;

QP3 : (A

,

B) E π =추 A

-I 0 -1

B

The pair (X

,

π) is called a basic quas!-p1"Oxim때 space. If in addition π

satisfies

QP4 : (A

B) E π =? (8

,

씨) ε π7r, A t비he애n π i섭5 called a b따as앙1κc pψ1"0‘oximit

。on X and (αX’ 7πr샤) is called a basic proximity space. For each basic quasi- proximity π on X We defìne a binary relation γ on P(X) by the rul

(15)

Theory of Extensions of Biclosure Spaces 151

π"=

{(A , B)

E

P(X)

x

P(X) : (B , A)

E π} ‘

lt can be verified that 7r" is also a basic quasi-proximity 00 X and (π‘)* 二 π

,

τ‘ is called the conjuafe of π Each basic quasi-proximity π on X induces a closure operator c~ 00

X

defined by

c.(A) = {x E X : (A

,

x) E π} . For each A C X we define

π (A) = {B

c

X : (A

,

B) E 샤,

It is easily verified that π (A) is a grill 00 X. Further for each filter F on X and for each basic quasi-proximity π ooX

,

π (F) defined by

π (F) = n{π (F) : F E F },

is a grill on X

The basic quasi-proximity π is said to be separaled if for each x

,

y in

x

(x, ν) E π

n

π‘ =?x=y

It is easily verified that if τ is a quasi-proximity on X theo

πis separaled i

.fJ

(X

,

cπ .)isT1

Basic proximities were introduced by èech [2]. It is known that the closure operator induced by a basic proximity satisfies the symmetry axiom (x E μ(ν) =추 Y E c.(x)). Thus there are closure spaces for which there is no basic proximity compatible with the given closure space. Being motivated by this fact

,

in the paper [11] we have introduced basic quasi- proximities and shown that for any closure space (X

,

c) there is a basic quasi-proxirnity π on X such that = c. Heoceforth we shall drop the prefix ‘basic’ and just talk of quasi-proxirnities. For details on quasi proximities we refer to Hazra and Cbattopadbyay [11].

Definition 5.2. A quasi-proximity π on X is called an Rl-quasi-proximitν

iffor A

,

B

c

X

,

c.(A)

n

.(B)

i ø

=? (A,B) E 7r

(16)

152 K.C. Chattopadhyay and R.N. Hazra

It should also be noted that π is an RI -quasi-proximity iff π‘ is an RI

qu싫 i-proximi ty.

Theorem 5.3. A quasi-proximity τ oπ X is an RI.quasi-proximitν i

fJ

for each x in X

π .(x) x π(x)C 1r

Proof Suppose for each x in X

,

π.(x) x π (x) C 1r. Then for A

,

B

c

X

,

cπ (A)

n

Cπ.(B)

ø

implies that there exists an x in X such that x

E

π (A)

n

1r-(B). Then (A, B)

E

π .(x) x π (x) C π Hence π 15 an RJ-quasi-procimity on X

Conversely

,

suppose π is an RI-quasi-proximity on X. Then for each x in X.

(A

,

B) E 1r.(x) X r.(x) =추 x E Cπ (A)

n

cπ.(B) =추 (AB) E π

Theorem 5.4. For each RI -quasi-pmxim때 π on Xκ, (Xχ’~,~껴사r R1-biκclμosure space

Proof Since π is an RI -quasi-proximity on X it follows from the definition that

Cτ (A)

n

cπ· ¥

ø =추

(A

,

x) E π =양 x E cT,(A)

,

and

Cτ .(A)

n

cπ ¥ 0=}(A

,

x) E π -=} xEcπ. (A)

Definition 5.5. Let (X

,

Cl

,

C2) be a biclosure space. A quasi-proximity

1r on X is called (Cl, C2) compatible if

~

=

Cl and .

=

C2

Theorem 5.6. A biclosure space (X

,

CJ, C2) R1 i

fJ

there is a (Cl

,

C2)- compatible RI -quasi-proximity on X

Proof Define a binary relation 71"( Cl

,

C2) on the power set of X by the rule π(CJ, C2) = {(A

,

B) E P(X) x P(X) : cl(A)

n

c2(B)

0}

It can be ea.sily verified that π (Cl , C2) is a quasi-pro져mity on X. Now

.ve

have

x E ~(C},c2)(A) iff (A,x) E π (ClC2) i:ff cl(A)

n

C2(X)

ø

iff x E cl(A),

and

yEc써

(17)

Theory of Extensions of Biclosure Spaces

Hence i t follows that π(C1' C2) is (c] , c2)-compatible.

Further since for each A

,

B

c

X

,

Cπ(ct 찌 (A) n C~

'

(cl ,C2)(B) ¥

m

=추 c[(A)

n

~(B) ¥

ø

=} (A

,

B) E π (C1C2)

Hence it follows that π (C1C2) is an RI-quasi-proxirnity on χ

The converse follows frorn Theorem 5.4.

153

Theorem 5.7. Let π be a (ct, c2)-compatible quasi-pl'Oximily a biclo SU7'e space (X

,

ct, C2)' Then

τ "(x) X π (x)

=

9(c[

,

x) X 9(C2'X)

,

for all x E X

It is a matter of simple varifìcation using appropriate defìnitions and hence we omit it.

Remark 5.8. If (X

,

C[

,

C2) is an R[-biclosure space then π( Ct, C2)

,

as defì ned above

,

is the smallest (c], c2)-compatible RI-quasi-proxim따 on X If in addition C1C2) is (hence T1 Theorem 3.1) then π (C1, C2) is als。

separated

Definition 5.9. Let π be a quasi proximity on a set X . Then a bigrill 9

,

X 92 on X is said to be a π-clan if

9[ X 92

The bigrill 9[ x 92 is said to be ".-closed if for each A

,

B

c

X

,

91 X 9 2

c

".*(B) x π (A) =추 (A

,

B) E 91 x 92

Finally the bigrill 9[ x 9 2 is said to be a π c1uste7' if it is a τ clan which is also ".-closed

Note that if A

,

B

,

C are subsets of X such that

π*(A) x ".(A)

c

".-(C) x τ (B)

then i t is easily verifìed that (B

,

C) E π'(A) x π (A)

Thus π -(A) x π (A) is ".-c1osed for each A C X. In view of this fact and the result of Theorem 5.3

,

we have the following

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