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Introducing a topology by an operation of a set-theoretic boundary

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A C T A U N I V E R S I T A T I S L O D Z I E N S I S FO LIA M ATHEM ATICA 7, 1995

A lina Chądzyńska and Mirosław Pustelnik

I N T R O D U C I N G A T O P O L O G Y B Y A N O P E R A T I O N O F A S E T - T H E O R E T I C B O U N D A R Y

A d efin itio n of a topology by a b o u n d a ry o p e ra tio n is co n sid ered .

Let X be an a rb itra ry set. It is known th a t in th e set X one can define a topology uniquely when the closure o p eratio n is given ([1],

p - 36) ’ . r • j •

In this pap er, we shall give a way of introducing a topology in th e set X by m eans of an operation Fr : 2A -+ 2X satisfying th e conditions: ( F I) (F2) (F3) (F4) (F5)

T h e o r e m 1. Let X be an arbitrary set, and F r : 2X —> 2X an operation satisfying conditions (F l)-(F 5 ). P u ttin g

(1) A = A \J F r A, F r(0) = 0, F r (A ) = F r(X \ A), F r (A U B ) C Fr(A ) U Fr ( B) , Fr(Fr(A )) C Fr(A), A C B => A U Fr(A ) C B U F r (B ).

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we obtain th e closure operation satisfying the conditions:

( C l) 0 = 0,

(C 2) A C A,

(C 3) A U B = A U B ,

(C 4) ( I ) = A.

In th e topological space so obtained, the set-th eo retic b o u n d a ry of th e set A is F r A

Proof. E q u ality ( C l) follows im m ediately from (1) an d ( F I ), w hereas (C 2) - from (1). M aking use of (F3), we get

A U B = A U B U Fr(A U B ) C i U B U Fr( A) U F r(B ) = A U B . O n th e o th e r h a n d

A U B = A U Fr(A ) U B U F r(B ) C A U B l J F r ( A U B ) = A~UB because A c A U B , B c i U B , thus one can use (F 5). So, (C3)

holds. _ ___

T h e inclusion A C (A ) follows directly from (C 2), an d (Z ) = (A U Fr(A ) =_A U Fr(A ) U Fr(A U Fr(A )) C A U Fr(A ) U F r(F r(A )) C A U Fr(A ) = A on th e basis of (1), (F3) and (F4).

Now, denote by fr(A ) th e b o u n d ary of th e set A in th e topological space X , i.e.

fr(A ) = A D Y \ A . So, for any A C X , we have

fr(A ) = (A U Fr(A )) n ((X \ A) U F r(X \ A)) = (A U F r(A )) n ((X \ A) U Fr(A ))

— ( Af l Fr(A )) U (Fr(A ) n (X \ A)) U Fr(A ) = Fr(A )

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C onditions ( F I ) - (F 5) are independent.

Exam ple 1. Let X be an a rb itra ry nonem pty set and Fr(A ) — X for any A C X . T h e op eratio n thus defined satisfies conditions (F 2 ) - (F 5) an d, of course, does not satisfy (F I).

Exam ple 2. In any nonem pty set X let us p u t Fr(A ) = A for each A C Ar . It is easy to verify th a t conditions (F I ), (F 3), (F 4) an d (F 5) are satisfied, while (F 2) is not.

Exam ple S. Let X = {0,1,2}. Define th e o p eratio n F r in th e set X as follows :

Fr(0) = Fv( X) = 0,

Fr({0}) = F r ({ l,2 , }) = {0}, F r({ l} ) = Fr({0,2}) = {1}, Fr({2}) = F r({ 0 ,1}) = {2}.

T h e o p eratio n so defined satisfies conditions ( F I ) , (F 2), (F 4) an d (F 5 ), an d does no t satisfy (F3), for we have

F r ( { 0 ,l ) } ) £ F r ( { 0 } ) U F r ( { l } ) . Exam ple 4- Again, let X = {0,1,2}. P u t

Fr(0) = Fr ( X) = 0,

Fr({0}) = F r ( { l ,2 ,}) = {0,1}, F r({ l} ) = F r({0,2}) = { 0 ,1 ,2 }, Fr({2}) = F r({ 0 ,1}) = {1,2}.

T h e o p eratio n Fr th u s defined satisfies conditions ( F l ) - (F 3) an d (F 5 ), an d does no t satisfy (F4). Indeed,

Fr({0}) = {0,1} w hereas Fr(Fr({0})) = {1,2}. Exam ple 5. In A' = { 0 ,1 ,2 }, p u t Fr(0) = Fr ( X) = 0, Fr({0}) = F r ( { l ,2, }) = {1,2}, F r({ l} ) = Fr({0,2}) = {0,2}, F r({2}) = Fr({0,1}) = {0,1}.

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T his o p eratio n satisfies conditions ( F I) - (F4) a n d does no t satisfy (F 5) since {0} C {0,2}, b u t {0} U F r({0}) = X and

{0,2} U F r({ 0 ,2}) = {0,2}.

C onditions ( F I ) - (F5) characterizing the b o u d aris of sets m ay be replaced by th e equivalent system of three conditions. Namely, th e following theorem holds :

T h e o r e m 2. Let X be an arbitrary set. Each operation F r : 2 X —> 2A satisfies conditions ( FI ) - (F5) i f and only i f it satisfies the syste m o f conditions (A ) Fr(0) = 0, (f2) Fr(A ft jB) C A U Fr(A ), ( f 3 ) Fr(A U B ) U F r(X \ A) U Fr(Fr(A )) C Fr( A) U Fr( B) , w here A , B C X . Proof. “=>” is im m ediate.

It is easily seen th a t conditions ( F I) an d (F3) follow a t once from (f l) an d (f3). P u ttin g B — 0 in (f3) and m aking use of (f l), we get (F 4). S u b stitu tin g B = A in (f3), we have

F r(A) U F r(X \ A) U Fr(Fr(A )) C Fr(A ), th u s

(2) F r(X \ A)

C

Fr(A).

Sim ilarly, replacing in (f3) th e sets A and B by th e set X \ A, we o b ta in

F v(X \ A) U Fr(A) U F r(F r(X \ A))

C

F r(X \ A)

an d , consequently, Fr(A ) C F r(X \ A), which, to g eth er w ith (2) gives (F2).

If A C B , th e n A fl B = A and, by (f2),

Fr(A n B ) = Fr(A ) C B U fr(B).

Hence A U F r(A ) C i?U Fr(J5) and, therefore, im plication (F 5) is true. Exam ples 1, 5 and 3 allow us to find th a t each of conditions (fl), (f2), f3 is independent of th e rem aining ones.

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Re f e r e n c e s

[1] R. E ngelking, Topologia ogólna, P W N , W arszaw a, 1976.

A lina Chądzyńska i Mirosław Pustelnik

W P R O W A D Z E N I E T O P O L O G I I P R Z E Z O P E R A C J Ę B R Z E G U

W p racy rozw aża się topologię wprow adzoną przez operację brzegu.

I n s titu te o f M a th e m a tic s Lódź U n iv ersity ul. B an a c h a 22, 90 - 238 Lódź, P o la n d

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