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Acta Math. Hungar., 1 1 9 ( 1 - 2 ) ( 2 0 0 8 ) , 1 9 7 - 2 0 0 .DOI: 10 .1007/s l0474-007-7023-4

First published online October 4, 2007

R E M A R K S O N Q U A S I -T O P O L O G I E SA. CSASZAR*

Eotvos Lorand Univers i ty, Depar tm ent of Analys is , H-1117 Bud apes t , Pa zma ny P. se tany 1 /C

(Received February 6, 2007; accepted April 3, 2007)

A b s t r a c t . A qu as i - t op o logy i s a gen e ra l i z e d t opo lo g y ( cf . [3] closed for fin i t e i n t e r s e c t i o n s . T h e p a p e r d i s c u s s e s t h e q u e s t i o n w h e t h e r s t a t e m e n t s v a l i d f o rt opo log i e s c a n b e ge ne ra l i z e d fo r q ua s i - t o po log i e s .

1 . In t roduc t ion and p re l imina r i e s

Let X be a non-empty set with power set exp X . According to [3], acollection /x C exp X is said to be a generalized topology (briefly GT) iff 0 G /xand each unio n of a non- em pty subse t of /x belongs to /x. The eleme nts of /xare said then to be /i-open and their complements /x- closed

A function 7 : exp X > exp X is said to be monotonic iff A c B C Ximnlies . The collection of all monotonic functions is denoted bv

ents of T are said to be operations. If 7 e T tin>e 7 - open iff A c 7 A T he collection of all 7-opi

sets is a GT an d is den oted b y /x(7) (see [1]). An op eratio n 7 e T is said

T (see [1]) and the elements of V are said to be operations. If 7 e T the na. s e t i s s a i d t o b e -nnp.n \f? . T h e c o l l e c t i o n o f a l l - n n f t n

'Rese arch suppor t ed by Hung ar ian Fou ndat ion for Scienti f ic Research , grant No. T 49786.Key words and phrases: generalized topology, mo noton ic ope ratio n, / i-friendly oper atio n, quasi-

topology.2000 Mathematics Subject Classification: 54A05.

0236-5294 /$ 20 .00 2007 Akademiai K iado, B udape s t

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19 8 A. CSASZAR

to be restricting (enlarging, idempotent) iff A c X implies 7 A c A (jA D A,^A = A).

If /x is a GT and A

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REMARKS ON QUASI-TOPOLO GIES 19 9

THEOREM 2 .3 . If /x is a QT and A G a{p), B G a(/x) (B G cr(/x); B G7r(/x); B G /3 /X)) t/ien A n B G a(/x) (A n B G cr(/x), A n B G 7r(/x); A n BG /?(//)) a* we//.

PROOF. Assume first that A G a(/x) an d -B G a(/x). Let us w rite, forbrevity, i for iM and c for cM. Then A n B C idA n iciB c d A n i d> C c(iAn id>) C c(iA n d>) C cc(iA n iB) = ci(A n B) (see 2.1 and 2.2). Henceir >)' R i //-open and contained in ci(A n B), consequently A n -B C idA

n B).Assume now A G a(/x) an d -B G cr(/x). Then Af)B C id A n d> C c(iciA

see 2.1 and 2.2

n iciB c id(A n B).3ume now

n iB) c c(dA n i >) C cc(iA n iB) = d (A n B) (see 2.1 and 2.2)s s i i m n t l i A n f l i n tAssume then that A G a(/x) an d > is 7ic-open where 7 G T is //-friendly.

We show tha t A n -B is 7ic-open. In fact, AnBc idA n ^icB c ^(iciA n ic>)= 7i ( dAn i c i3 ) C 7ic( iAn i ci3) C jic(iAC\cB) C 7ic c( iAnI3) C jic(AnB)(cf. 2.1 and 2.2).

Now 7 = id produces the case B G 7r(/x) and 7 = c (cf. 2.1) gives the caseB G /?(//).

In particular, if /x is a QT then so is a(/x). Of course, all results aboveare well-known in the case when /x is a topology.

Similar state me nts a re contained in [6], but instead of the condition tha t/x is a QT , it is supp osed th at /x = /x(7) and 7 G T4. However, the followingexam ple shows tha t /x can be a Q T (or even a topology) and /x = /x(7) where7 is restricting without being /x-friendly.

EXAMPLE 2.4. Let X = N U {p} where N is the set of all natural numbers, and p ^ N. We define an operation 7 on exp X.

Let 70 = 0, 7N = N, 7X = X. If A c X, p G A and A / X, let 7A =. Ifhe sn

Clearly 7 is restricting. We show that 7 is monotonic.

A n N . \{ p ^ A (i.e. A c N) and A^ , A / N , let 7 A = A {m } wherem is the smallest element of the non-empty set AcN.

mot

Let A c B C X. The cases A = 0 and B = X are obvious. Consider firstthe case p e i c B / I . Then 7A = 4 n N c 5 n N = 7 ?.

Let now | / i 4 c N ,A c 5 / I , j ) e 8 . Then 7A c A = AnNc5nN= jB

In the case A c 5 = N we have 7A c A c B = 7U .Finally if 0 / A C B C N , > / N , let us den ote by a and 6 the smallest

elements of A an d >, respe ctively. In the case a = 6, we have 7A = A {a}

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20 0 A. CSASZAR: REMARKS ON QUASI-TOPO LOGIES

However, 7 is not //-friendly. In fact, consider A = {l p} and N e /x.Then jA = Af]N = {l} so tha t 7 A n N = {l} g 7 A nN) = 7 { 1 } = 0.

eferences

[1] A. Csaszar, Generalized open sets, Acta Math. Hunger. 75 1997), 65-87 .[2] A. Csaszar, On the 7-interior and 7-closure of a set, Acta Math. Hungar. 80 1998),

89-93.[3] A. Csaszar, Generalized topology, generalized continuity, Acta Math. Hungar. 96

(2002), 351-357.[4] A. Csaszar, Generalized open sets in generalized topologies, Acta Math. Hungar. 105

(2005), 53-66.[5] A. Csaszar, Further remarks on the formula for 7-interior, Acta Math. Hungar. to

appear ) .[6] P. Sivagami, Remarks on 7-interior, Acta Math. Hungar. to appear).

Acta Mathematica Hungarica 119 2008