A Unified Theory on Some Basic Topological Concepts

A Unified Theory on Some Basic T op ological Concepts T. Hatice Y alv a¸ c No v em b er 20, 2018 Abstract Sev eral mathematicians, includ ing m yself, ha v e studied some unifications in general top o logical spaces as w ell as in fuzzy top olo gical spaces. F or in- stance in our earlier works, using op erati ons on top ologic al sp a ces, we ha ve tried to unify some co ncepts similar to con tin uity , op enness, closedness of functions, compactness, filter con v ergence, closedness of graphs, countable compactness and L in del¨ of p r o p ert y . In this article, to obtain further unifica- tions, w e will study ϕ 1 , 2 -compactness and r e lations b et w een ϕ 1 , 2 -compactness, filters and ϕ 1 , 2 -closure op erator. 1 In t r o duction Sev eral unifications ha ve b een studie d in [1, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1 8 , 19, 20, 21 , 22]. Some unifications w ere studied in [10] and [1 2] by using op erations for fuzzy top ological spaces. It w as claimed there and it can be e asly seen that most of the definitions and results can b e applied to top olo g ical spaces. As fa r as p ossible we do not rep eat definitions related of kno wn concepts. Because one aim of us to reduce the confusions caused b y so mu c h definitions. How ev er, man y suc h definitions will b e clear f rom the sp ecial op erations considered. In a top olo gical space ( X , τ ) in t, cl, scl etc. will stand f or in terior, closure, semi- closure op erations so on and A o , A will stand for the in terior of A, the closure of A for a subset A of X resp ectiv ely . Definition 1.1 L e t ( X , τ ) b e a top olo gi c al sp ac e. A mapp ing ϕ : P ( X ) → P ( X ) is c al le d an op er ation on ( X , τ ) if A o ⊂ ϕ ( A ) for al l A ∈ P ( X ) and ϕ ( ∅ ) = ∅ . The class of all op erations on a top o logical space ( X , τ ) will b e denoted by O ( X , τ ). 1 A partial order ” ≤ ” on O ( X , τ ) is defined as ϕ 1 ≤ ϕ 2 ⇔ ϕ 1 ( A ) ⊂ ϕ 2 ( A ) for eac h A ∈ P ( X ). An op eration ϕ ∈ O ( X , τ ) is called monotonous if ϕ ( A ) ⊂ ϕ ( B ) whenev er A ⊂ B ( A, B ∈ P ( X )). Definition 1.2 L et ϕ ∈ O ( X , τ ) and A, B ⊂ X . A is c a l le d ϕ -op en if A ⊂ ϕ ( A ) . B is c al le d ϕ -close d if X \ B is ϕ -op en. Op er ation ˜ ϕ ∈ O ( X , τ ) is c al le d the dual op er ation of ϕ if ˜ ϕ ( A ) = X \ ϕ ( X \ A ) for e ach A ∈ P ( X ) . If ϕ is monotonous, then the family o f all ϕ -op en sets is a supratop ology ( U ⊂ P ( X ) is a supratop ology on X means that ∅ ∈ U , X ∈ U a nd U is closed under arbitrary union [2]) . Let ( X, τ ) b e a top ological space, ϕ ∈ O ( X , τ ), U ⊂ P ( X ) , x ∈ X . W e use the follo wing notations. U ( x ) = { U : x ∈ U ∈ U } , ϕO ( X ) = { U : U ⊂ X , U is ϕ − o p en } , ϕC ( X ) = { K : K ⊂ X, K is ϕ − cl osed } , ϕO ( X , x ) = { U : U ∈ ϕO ( X ) , x ∈ U } . N ( U , x ) = { N : N ⊂ X and ther e exists a U ∈ U ( x ) such that U ⊂ N } . Definition 1.3 L et ϕ ∈ O ( X , τ ) , U ⊂ P ( X ) . ϕ is c al le d r e gular with r esp e ct to (shortly w.r.t.) U if x ∈ X and U, V ∈ U ( x ) , ther e exists a n W ∈ U ( x ) such that ϕ ( W ) ⊂ ϕ ( U ) ∩ ϕ ( V ) . F or any op eration ϕ ∈ O ( X , τ ) , τ ⊂ ϕ O ( X ), and X , ∅ are b oth ϕ -open and ϕ -closed. Definition 1.4 L et ϕ 1 , ϕ 2 ∈ O ( X , τ ) , A ⊂ X . a) x ∈ ϕ 1 , 2 intA ⇔ ther e exists an U ∈ ϕ 1 O ( X , x ) such that ϕ 2 ( U ) ⊂ A . b) x ∈ ϕ 1 , 2 clA ⇔ f or each U ∈ ϕ 1 O ( X , x ) , ϕ 2 ( U ) ∩ A 6 = ∅ . c) A is ϕ 1 , 2 - open ⇔ A ⊂ ϕ 1 , 2 intA . d) A is ϕ 1 , 2 - cl osed ⇔ ϕ 1 , 2 clA ⊂ A . If A ⊂ B then ϕ 1 , 2 intA ⊂ ϕ 1 , 2 intB . Clearly f or an y set A, X \ ϕ 1 , 2 intA = ϕ 1 , 2 cl ( X \ A ) and A is ϕ 1 , 2 -op en iff X \ A is ϕ 1 , 2 - cl osed . ϕ 1 , 2 O ( X ) ( ϕ 1 , 2 C ( X )) will stand for the family of all ϕ 1 , 2 -op en subs ets ( ϕ 1 , 2 - closed subsets) of X. 2 Theorem 1.5 ([12]). L et ϕ 1 , ϕ 2 ∈ O ( X , τ ) . a) ϕ 1 , 2 O ( X ) is a supr atop olo gy on X. b) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) then ϕ 1 , 2 O ( X ) is a top olo gy on X and a subset K of X is close d w.r.t. t his top olo gy iff ϕ 1 , 2 clK ⊂ K . c) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) an d ( ϕ 2 ≥ ı or ϕ 2 ≥ ϕ 1 ) then ϕ 1 , 2 O ( X ) is a top olo gy o n X and a set K is close d w.r.t. this top olo gy iff ϕ 1 , 2 clK = K (her e ı is the idendity op er ation). Example 1.6 L et the fo l lowing op er ations b e define d on a top olo gic al sp ac e ( X , τ ) . ϕ 1 = int , ϕ 2 = cl oint , ϕ 3 = cl , ϕ 4 = scl , ϕ 5 = ı , ϕ 6 = intocl . ϕ 1 ≤ ϕ 2 ≤ ϕ 3 , ϕ 1 ≤ ϕ 5 ≤ ϕ 4 ≤ ϕ 3 , ϕ 1 ≤ ϕ 6 ≤ ϕ 4 . ϕ 1 O ( X ) = τ , ϕ 2 O ( X ) = S O ( X ) = the family of se mi-op en sets. ϕ 3 O ( X ) = ϕ 5 O ( X ) = ϕ 4 O ( X ) = P ( X ) = p ow er set of X. ϕ 6 O ( X ) = P O ( X ) = the family of pr e-op en sets. ϕ 1 , 3 O ( X ) = τ θ = the top olo gy of a l l θ -op en sets. ϕ 2 , 4 O ( X ) = S θ O ( X ) = the family of semi- θ - o p en se ts. ϕ 1 , 6 O ( X ) = τ s = the semi r e gularization top olo gy of X. I t is, the top olo gy with the b ase R O(X) which c onsists of r e gular op en se ts = the family of δ -op e n sets. ϕ 2 , 3 O ( X ) = θ S O ( X ) = the family of al l θ -semi-op en sets. ϕ 1 , ϕ 3 ( ϕ 2 , ϕ 6 ) ar e the dual op er a tion s of e ach other. ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 , ϕ 6 ar e r e gular w.r.t. ϕ 1 O ( X ) . SC(X) (PC(X), R C(X), S θ C(X), θ SC ( X) r esp e ctively) wil l stand for the family semi-close d (pr e-close d, r e gular close d, sem i- θ -close d , θ -sem i-close d) sets. SR(X)=SO(X) ∩ SC(X)= the family of semi-r e gular sets. F or op erations ϕ 1 , ϕ 2 ∈ O ( X , τ ) , clearly if ϕ 1 is monotonous and ϕ 2 = ı fo r ϕ 1 , ϕ 2 ∈ O ( X , τ ) then ϕ 1 , 2 O ( X ) = ϕ 1 O ( X ) and ϕ 1 , 2 C ( X ) = ϕ 1 C ( X ). If ϕ 1 O ( X ) is a top ology and ϕ 2 is monoto nous then ϕ 2 is regular w.r.t. ϕ 1 O ( X ), so ϕ 1 , 2 O ( X ) is alw a ys a top ology . 2 ϕ 1 , 2 -closure Op e r ato r and Filters Along of the pap er it will b e accepted that op erations ϕ i , i = 1 , 2 , ... are defined on a top olog ical space ( X , τ ). Lemma 2.1 If ( ϕ 2 ≥ ϕ 1 or ϕ 2 ≥ ı ) then ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) . But the c o n verse is not true. Example 2.2 L et ϕ 1 = cl oint , ϕ 2 = semi-int b e define d on R with the usual top olo gy. F or A = (0 , 1] , ϕ 1 ( A ) = [0 , 1] , ϕ 2 ( A ) = (0 , 1 ] a nd ϕ 1 ( A ) 6⊂ ϕ 2 ( A ) . i.e. ϕ 1 6≤ ϕ 2 . F or the set of r ational numb ers Q , Q 6⊂ ϕ 2 ( Q ) = ∅ . i.e. ϕ 2 6≥ ı . B ut ϕ 1 O ( R ) = S O ( R ) = ϕ 2 O ( R ) . 3 Theorem 2.3 L et B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } . a) I f ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) an d ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) then ϕ 1 , 2 -cl op er ator defines the same (pr e)top olo gy given in The or em 1.5 (c). L et τ ϕ 1 , 2 b e stand for this top olo gy. A ⊂ ϕ 1 , 2 clA ⊂ τ ϕ 1 , 2 clA for any subset A of X. b) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) , ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) an d B ⊂ ϕ 1 , 2 O ( X ) , then ϕ 1 , 2 -cl op er ator is a Kur atowski closur e op er ator and ϕ 1 , 2 clA = τ ϕ 1 , 2 clA for any sub- set A o f X. Definition 2.4 ([20]). L et F b e a filter (or filterb ase) in ( X , τ ) and a ∈ X . F is said to b e: a) ϕ 1 , 2 -ac cumulates to a if a ∈ ∩{ ϕ 1 , 2 clF : F ∈ F } . b) ϕ 1 , 2 -c onver g es t o a if for e ach U ∈ ϕ 1 O ( X , a ) , ther e exists a n F ∈ F such that F ⊂ ϕ 2 ( U ) . Theorem 2.5 ([20]). 1) A filterb ase F b ϕ 1 , 2 -ac cumulates ( ϕ 1 , 2 -c onver g es) to a iff filter ge n er ate d by F b ϕ 1 , 2 -ac cumulates ( ϕ 1 , 2 -c onver g es) to a. 2) I f ϕ 2 is monotonous, we c an get the family N ( ϕ 1 O ( X ) , a ) ins te ad of ϕ 1 O ( X , a ) in the ab ove de finitions. 3) A filter F ϕ 1 , 2 -c onver g es to a iff { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X , a ) } ⊂ F . 4) If F ϕ 1 , 2 -c onver g es to a then F ϕ 1 , 2 -ac cumulates to a. 5) L et F ⊂ F ′ for the filters F and F ′ a) If F ′ ϕ 1 , 2 -ac cumulates to a, then F ϕ 1 , 2 -ac cumulates to a. b) If F ϕ 1 , 2 -c onver g es to a then F ′ ϕ 1 , 2 -c onver g es to a. 6) I f ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) then a filter F ϕ 1 , 2 -ac cumulates to a ∈ X iff ther e exists a fi lter F ′ such that F ⊂ F ′ and F ′ ϕ 1 , 2 -c onver g es to a. 7) If F is a maxim al fi lter w hich ϕ 1 , 2 -ac cumulates to a ∈ X , then F ϕ 1 , 2 - c onver g e s to a. 8) If ϕ ′ 1 O ( X ) ⊂ ϕ 1 O ( X ) and ϕ ′ 2 ≥ ϕ 2 for the op er a tions ϕ 1 , ϕ 2 , ϕ ′ 1 , ϕ ′ 2 ∈ O ( X , τ ) , then a filter (or a filterb ase) F ϕ ′ 1 , 2 -ac cumulates ( ϕ ′ 1 , 2 -c onver g es) to a w hen- ever F ϕ 1 , 2 -ac cumulates ( ϕ 1 , 2 -c onver g es) to a. Pro of. 6) Let ϕ 2 b e regular w.r.t. ϕ 1 O ( X ) and F ϕ 1 , 2 -accum ulates to a. Then the family F b = { ϕ 2 ( U ) ∩ F : U ∈ ϕ 1 O ( X , a ) , F ∈ F } is a filterbase and the filter 4 F ′ generated b y F b is finer tha n F and it ϕ 1 , 2 -con v erges to a. The other part of the pro of is cle ar. 7) Let F b e a maximal filter and ϕ 1 , 2 -accum ulates to a ∈ X . F or eac h U ∈ ϕ 1 O ( X , a ), F b u = { ϕ 2 ( U ) ∩ F : F ∈ F } is a filterbase. The filter F u generated b y F b u is finer tha n F . So, for eac h U ∈ ϕ 1 O ( X , a ), F = F u . Now it is clear t ha t F ϕ 1 , 2 -con v erges to a. A space ( X , τ ) is called ϕ 1 , 2 - T 2 if f o r each x, y ∈ X ( x 6 = y ) there are ϕ 1 -op en sets U x and U y suc h that x ∈ U x , y ∈ U y and ϕ 2 ( U x ) ∩ ϕ 2 ( U y ) = ∅ ([12],[20]). Theorem 2.6 ([20]) L et ( X , τ ) b e ϕ 1 , 2 - T 2 sp ac e. If a filter F ϕ 1 , 2 -c onver g es to some p oint a ∈ X and ϕ 1 , 2 -ac cumulates to some p oint b ∈ X then a = b . Pro of. Let’s accept t ha t a filter F b e ϕ 1 , 2 -con v ergent to a and ϕ 1 , 2 -accum ulate to b and a 6 = b . Then there exists U ∈ ϕ 1 O ( X , a ) and V ∈ ϕ 1 O ( X , b ) suc h that ϕ 2 ( U ) ∩ ϕ 2 ( V ) = ∅ . But ϕ 2 ( U ) ∈ F and F ∩ ϕ 2 ( V ) 6 = ∅ fo r eac h F ∈ F . It m ust b e ϕ 2 ( U ) ∩ ϕ 2 ( V ) 6 = ∅ . This con tradiction complete s the pro o f. Example 2.7 L et a ∈ X and F b e a filter in ( X , τ ) . a) L et ϕ 1 = cl oint , ϕ 2 = cl . F ϕ 1 , 2 -c onver g es ( ϕ 1 , 2 -ac cumulates) to a iff F r c-c onver ges (r c-ac cumulates) to a sinc e { V : V ∈ τ , x ∈ V } = { U : x ∈ U ∈ S O ( X ) } [8] iff F s-c onver ges (s- ac cumulates) to a [5 ]. b) L et ϕ 1 = int , ϕ 2 = cl . F ϕ 1 , 2 -c onver g es ( ϕ 1 , 2 -ac cumulates) to a iff F r-c onve r ges (r-ac cumulates) to a [7] iff F almost c onver ges to a (a is an almost adher ent p oint of F )[4]. ( X , τ ) is ϕ 1 , 2 - T 2 iff ( X, τ ) is Urysohn. c) L et ϕ 1 = int , ϕ 2 = ı . F ϕ 1 , 2 -c onver g es to a iff F c on v er ges to a in ( X , τ ) . ( X , τ ) is ϕ 1 , 2 - T 2 iff ( X, τ ) is Hausdorff. It is wel l known that in Hausdorff sp ac es, a c onver gent filter c an not have mor e than one ac cumulation p oint ([6], p age 220). Theorem 2.8 ([20])L et ϕ 1 , ϕ 2 ∈ O ( X , τ ) . 1) If ϕ 1 O ( X ) is close d under finite interse ction and ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) , then for e ach a ∈ X , the family Φ a = ϕ 1 O ( X , a ) is a filterb ase, and Φ a ϕ 1 , 2 -c onver g es to a. 2) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) and ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) , then for e ach a ∈ X the family Φ a = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X , a ) } is a filterb ase and ϕ 1 , 2 -c onver g es to a. 5 Theorem 2.9 ([20]) L et A ⊂ X and a ∈ X . 1) If ther e exis ts a filter which c ontain s A and ϕ 1 , 2 -ac cumulates to a, then a ∈ ϕ 1 , 2 clA . 2) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) and a ∈ ϕ 1 , 2 clA , then ther e exists a filter c on- taining A an d ϕ 1 , 2 -c onver g ing to a. 3) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) , then a ∈ ϕ 1 , 2 clA iff ther e exists a filter F c ontaining A and ϕ 1 , 2 -c onver g ing to a. 4) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) , then A is ϕ 1 , 2 -close d iff whenever ther e exists a filter c ontaining A and ϕ 1 , 2 -c onver g ing to a p oint a in X, then a ∈ A . Pro of. 2) Let a ∈ ϕ 1 , 2 clA . Then ϕ 2 ( U ) ∩ A 6 = ∅ for each U ∈ ϕ 1 O ( X , a ). Φ = { ϕ 2 ( U ) ∩ A : U ∈ ϕ 1 O ( X , a ) } is a filterbase. The filter generated b y Φ con tains A and ϕ 1 , 2 - conv er g es to a. The pro ofs of the others are easy . Theorem 2.10 L et us define cl ∗ : P ( X ) → P ( X ) as cl ∗ A = { x : ther e exists a filter F c ontainin g A and ϕ 1 , 2 -c onver g ing to x } for A ⊂ X . 1) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) then cl ∗ A = ϕ 1 , 2 clA and cl ∗ op er ator defines the fol lowing (pr e)top olo gy. τ ∗ = { U ⊂ X : cl ∗ ( X \ U ) ⊂ X \ U } = { U ⊂ X : ϕ 1 , 2 cl ( X \ U ) ⊂ X \ U } = τ ϕ 1 , 2 . 2) If ϕ 2 is r e gular w. r. t. ϕ 1 O ( X ) and ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) then cl ∗ op er ator defines the fol lo wing top olo gy. τ ∗ = { U ⊂ X : cl ∗ ( X \ U ) = ( X \ U ) } = { U ⊂ X : ϕ 1 , 2 cl ( X \ U ) = X \ U } = τ ϕ 1 , 2 . 3) If ϕ 2 is r e gular w.r.t. ϕ 1 O ( X ) , ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) , and for e ach U ∈ ϕ 1 O ( X ) ϕ 2 ( U ) ∈ ϕ 1 , 2 O ( X ) , then cl ∗ op er ator is a Kur atowski closur e op er ator and again τ ∗ = τ ϕ 1 , 2 . Example 2.11 L et ϕ 1 = int , ϕ 2 = cl . A filter F ϕ 1 , 2 -c onver g es to a iff F is θ -c onver ges to a. θ -c o nver genc e define s a pr etop olo gy [8]. 3 ϕ 1 , 2 -compactnes s Definition 3.1 ([20]) L et ϕ 1 , ϕ 2 ∈ O ( X , τ ) , X ∈ A ⊂ P ( X ) , A ⊂ X . 6 a) If A ⊂ ∪U for a subfamily U of A , then U is c al le d an A - c over of A. If an A -c over U of A is c ountable (finite) then we c al l U as c ountable A -c over (finite A -c over). b) If e ach A -c over U of A has a finite subfamily U ′ such that A ⊂ ∪{ ϕ 2 ( U ) : U ∈ U ′ } , then w e c al l A is ( A - ϕ 2 ) -c omp act r elative to X (shortly ( A - ϕ 2 )-c omp act set). c) We c al l an ( A - ı )-c omp act set r elative to X as A - c omp a c t set s hortly. d) I f we take A = ϕ 1 O ( X ) in (b), then w e c al l A as ϕ 1 , 2 -c omp a ct r ela tive to X (shortly ϕ 1 , 2 -c omp a ct se t). If we take A = ϕ 1 , 2 O ( X ) in (c) we ge t the de fi nition of a ϕ 1 , 2 O ( X ) -c omp act set. If X is ϕ 1 , 2 -c omp a ct set r elative to itself, then X wil l b e c al le d ϕ 1 , 2 -c omp a ct sp ac e. If X is ϕ 1 , 2 O ( X ) -c omp act set r elative to itself, then X wil l b e c al le d ϕ 1 , 2 O ( X ) - c omp a c t sp ac e. ϕ 1 , 2 -Lindel¨ of sets r elat ive to X, ϕ 1 , 2 -Lindel¨ of spaces and ϕ 1 , 2 -coun table compact sets relativ e to X, ϕ 1 , 2 -coun table compact spaces w ere defined in a similar w ay as in [21], [22]. Theorem 3.2 ([20]) If ϕ ′ 1 O ( X ) ⊂ ϕ 1 O ( X ) , ϕ 2 ≤ ϕ ′ 2 (henc e if ϕ ′ 1 ≤ ϕ 1 , ϕ 2 ≤ ϕ ′ 2 ), then e ach ϕ 1 , 2 -c omp a ct se t is ϕ ′ 1 , 2 -c omp a ct se t. Example 3.3 L et A ⊂ X . a) L et ϕ 1 = int , ϕ 2 = cl . A is ϕ 1 , 2 -c omp a ct se t iff A is an H-set. b) L et ϕ 1 = int , ϕ 2 = intocl . A is ϕ 1 , 2 -c omp a ct se t iff A is an N-set. c) L et ϕ 1 = cl oint , ϕ 2 = scl . A is ϕ 1 , 2 -c omp a ct se t iff A is an s-set. d) L et ϕ 1 = cl oint , ϕ 2 = cl . A is ϕ 1 , 2 -c omp a ct se t iff A is an S-set. e) L et ϕ 1 = int , ϕ 2 = ı . A is ϕ 1 , 2 -c omp a ct se t iff A is c omp a c t. By using Theorem 3.2, w e get that, each N-set is an H- set, eac h s-set is an S- set and eac h S-set is an H-set. 7 Theorem 3.4 ([20]) The fol lowing ar e e quivalent for a n y subset A of X. a) A is ϕ 1 , 2 -c omp a ct se t. b) Every fi lterb ase in X which me ets A, ϕ 1 , 2 -ac cumulates in X to some p oi nt in A. c) Every maximal filterb ase in X which me ets A, ϕ 1 , 2 -c onver g es in X to some p oint in A. d) Every filterb ase in A ϕ 1 , 2 -ac cumulates in X to some p oint in A. e) Every ma x imal filterb ase in A, ϕ 1 , 2 -c onver g es in X to some p oint in A. f ) F or any family W of non empty sets with A ∩ ( ∩{ ϕ 1 , 2 clF : F ∈ W } ) = ∅ , ther e exists a fi nite subfamily W ′ of W such that A ∩ ( ∩ W ′ ) = ∅ . g) F or any family of nonempty sets such that for e ach finite subfa m ily W ′ of W we have A ∩ ( ∩{ F : F ∈ W ′ } ) 6 = ∅ , then A ∩ ( ∩{ ϕ 1 , 2 clF : F ∈ W } ) 6 = ∅ h) If F is a fi l terb ase such that A ∩ { ϕ 1 , 2 clF : F ∈ F } = ∅ then ther e exis ts an F ∈ F such that F ∩ A = ∅ . If ˜ ϕ 2 is the dual of ϕ 2 , then the fol lowing statements (i) and (j) ar e e quivalent to e ach one of the ab ove statements. i) F or any family Φ of ϕ 1 -close d sets with A ∩ ( ∩ Φ) = ∅ , ther e exists a finite subfamily Φ ′ of Φ such that A ∩ ( ∩{ ˜ ϕ 2 ( F ) : F ∈ Φ ′ } ) = ∅ . j) If Φ is a fam i l y of ϕ 1 -close d sets such that for e ach finite subfamily Φ ′ of Φ we have A ∩ ( ∩{ ˜ ϕ 2 ( F ) : F ∈ Φ ′ } ) 6 = ∅ , then A ∩ ( ∩ Φ) 6 = ∅ . By c ho osing X instead of A in the abov e Theorem 3.4, w e get the equiv alen t statemen ts for a space ( X , τ ) to b e ϕ 1 , 2 -compact space. Theorem 3.5 L et B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } . a) If ϕ 2 ( U ) ∈ ϕ 1 O ( X ) , ϕ 2 ( ϕ 2 ( U )) ⊂ ϕ 2 ( U ) for e ac h U ∈ ϕ 1 O ( X ) then B ⊂ ϕ 1 , 2 O ( X ) ∩ ϕ 1 O ( X ) . b) If ϕ 1 O ( X ) ⊂ ϕ 2 O ( X ) an d if B ⊂ ϕ 1 , 2 O ( X ) , then B is a b ase fo r the supr atop olo gy ϕ 1 , 2 O ( X ) . 8 c) If ( ϕ 2 ≥ ϕ 1 or ϕ 2 ≥ ı ) and if B ⊂ ϕ 1 , 2 O ( X ) , then B is a b ase for the supr atop olo gy ϕ 1 , 2 O ( X ) [17]. d) If ( ϕ 2 ≥ ϕ 1 or ϕ 2 ≥ ı ) and, ϕ 2 ( U ) ∈ ϕ 1 O ( X ) , ϕ 2 ( ϕ 2 ( U )) ⊂ ϕ 2 ( U ) for e ach U ∈ ϕ 1 O ( X ) , then B is a b ase for the supr atop olo gy ϕ 1 , 2 O ( X ) [18]. Pro of. a) L et U ∈ ϕ 1 O ( X ) a nd x ∈ ϕ 2 ( U ). x ∈ ϕ 2 ( U ) ∈ ϕ 1 O ( X ) a nd ϕ 2 ( ϕ 2 ( U )) ⊂ ϕ 2 ( U ). So x ∈ ϕ 1 , 2 intϕ 2 ( U ). W e hav e ϕ 2 ( U ) ⊂ ϕ 1 , 2 intϕ 2 ( U ). Hence ϕ 2 ( U ) ∈ ϕ 1 , 2 O ( X ) for eac h U ∈ ϕ 1 O ( X ) and B ⊂ ϕ 1 , 2 O ( X ) ∩ ϕ 1 O ( X ). b) Let A ∈ ϕ 1 , 2 O ( X ) and x ∈ A . T here exists a U ∈ ϕ 1 O ( X , x ) suc h that ϕ 2 ( U ) ⊂ A . W e hav e x ∈ U ⊂ ϕ 2 ( U ) ⊂ A , ϕ 2 ( U ) ∈ ϕ 1 , 2 O ( X ) and ϕ 2 ( U ) ∈ B . Pro ofs of (c) and (d) are clear from Lemma 2.1 and (a), (b). Theorem 3.6 L et B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } . If ( ϕ 2 ≥ ϕ 1 or ϕ 2 ≥ ı ), and for e ach U ∈ ϕ 1 O ( X ) we have ϕ 2 ( U ) ∈ ϕ 1 O ( X ) , ϕ 2 ( ϕ 2 ( U )) ⊂ ϕ 2 ( U ) then the fol lowing ar e e quivalent for any subset A of X. a) A is ϕ 1 , 2 -c omp a ct se t. b) A is B -c omp act set. c) A is ϕ 1 , 2 O ( X ) -c omp act set. d) If W is an y subfa m ily o f { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } such that for e ach finite subfamily W ′ of W we have A ∩ ( ∩ W ′ ) 6 = ∅ then A ∩ ( ∩ W ) 6 = ∅ . e) If W is any subfamily of { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } with A ∩ ( ∩ W ) = ∅ then ther e exists a fi nite subfamily W ′ of W such that A ∩ ( ∩ W ′ ) = ∅ . f ) If W is any su bfamily of { X \ U : U ∈ ϕ 1 , 2 O ( X ) } such that f o r e ach finite subfamily W ′ of W we have A ∩ ( ∩ W ′ ) 6 = ∅ then A ∩ ( ∩ W ) 6 = ∅ . g) I f W is any subfa m ily of { X \ U : U ∈ ϕ 1 , 2 O ( X ) } such that A ∩ ( ∩ W ) = ∅ , then ther e exists a fini te subfamily W ′ of W such that A ∩ ( ∩ W ′ ) = ∅ . Pro of. Let us see that a ⇔ c is true. In the case A = ∅ pro ofs are clear. Let A b e a nonempt y ϕ 1 , 2 -compact set and U a ϕ 1 , 2 O ( X )-cov er of A. F or eac h x ∈ A , there exists a U x ∈ U s.t. x ∈ U x . There exis ts a ϕ 1 -op en set V x con taining x s.t. V x ⊂ ϕ 2 ( V x ) ⊂ U x .Since A is ϕ 1 , 2 -compact set, there exis ts a finite subset { x 1 , ..., x n } o f A s.t. A ⊂ ∪ n i =1 ϕ 2 ( V x i ) ⊂ ∪ n i =1 U x i . Hence A is ϕ 1 , 2 O ( X )-compact set. 9 Let A b e ϕ 1 , 2 O ( X )-compact set and U a ϕ 1 O ( X )-cov er of A. W e hav e A ⊂ ∪{ ϕ 2 ( U ) : U ∈ U } . Since for eac h U ∈ U , ϕ 2 ( U ) ∈ B ⊂ ϕ 1 , 2 O ( X ), there exis ts a finite subfamily { U 1 , ..., U n } o f U s.t. A ⊂ ∪ n i =1 ϕ 2 ( U i ). Hence A is ϕ 1 , 2 -compact set. Under the giv en conditions, since B is a base of the sup ratop olo g y ϕ 1 , 2 O ( X ), b ⇔ c is clear. Now the other pro ofs are easy . Under the hy p othesis of The orem 3 .6 b y joining Theorems 3.4 and 3.6 we g et the equiv alen t statemen ts for a set to b e ϕ 1 , 2 -compact set. Theorem 3.7 Under the hyp othesis of The or em 3.6, the fol low i n g ar e e quivalent. a) X is ϕ 1 , 2 -c omp a ct sp ac e. b) X is B -c omp act sp ac e. c) X is ϕ 1 , 2 O ( X ) -c omp act sp a c e. d) F or e ach U ∈ ϕ 1 O ( X ) , X \ ϕ 2 ( U ) is ϕ 1 , 2 -c omp a ct se t. e) F or e ach U ∈ ϕ 1 O ( X ) , X \ ϕ 2 ( U ) is ϕ 1 , 2 O ( X ) -c omp act set. f ) F or e ach U ∈ ϕ 1 O ( X ) , X \ ϕ 2 ( U ) is B -c omp act set. g) Each ϕ 1 , 2 -close d set is ϕ 1 , 2 -c omp a ct se t. h) Each ϕ 1 , 2 -close d set is ϕ 1 , 2 O ( X ) -c omp act set. i) Each ϕ 1 , 2 -close d set is B - c omp a c t set. No w, by using the The orems 3.4, 3.6 and 3.7, w e get the equiv alen t forms for a space ( X , τ ) to b e ϕ 1 , 2 -compact space under the h ypot hesis of Theorem 3.6. Example 3.8 L et A ⊂ X . a) L et ϕ 1 = int , ϕ 2 = intocl as in Example 3 . 3 (b). ϕ 1 O ( X ) = τ . ϕ 2 ≥ ϕ 1 . F or e ach U ∈ ϕ 1 O ( X ) = τ , we have ϕ 2 ( U ) = U o ∈ ϕ 1 O ( X ) and ϕ 2 ( ϕ 2 ( U )) = ( U o ) o = U o = ϕ 2 ( U ) . B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = { U o : U ∈ τ } = RO ( X ) . ϕ 1 , 2 O ( X ) = τ s . We know that R O(X) is a b a se for τ s . { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = RC ( X ) . ϕ 1 , 2 C ( X ) = the family o f τ s -close d sets = the family of δ -close d sets. ˜ ϕ 2 = cl oint is the dual of ϕ 2 . A is ϕ 1 , 2 -c omp a ct set i ff it is N-set iff A is c omp act in ( X , τ s ) iff A is R O(X)- c omp a c t set. ( X , τ ) is ϕ 1 , 2 -c omp a ct iff ( X , τ s ) is c omp act. T hese ar e v e ry wel l known r esults. 10 b) L et ϕ 1 = cl oint , ϕ 2 = scl as in Example 3.3 (c). ϕ 2 ≥ ı . ϕ 1 O ( X ) = S O ( X ) , ϕ 1 , 2 O ( X ) = S θO ( X ) , ϕ 1 , 2 C ( X ) = S θC ( X ) . F o r e ach U ∈ ϕ 1 O ( X ) we have ϕ 2 ( U ) = scl U ∈ S R ( X ) ⊂ S O ( X ) , and ϕ 2 ( ϕ 2 ( U )) = scl ( s c l U ) = scl U = ϕ 2 ( U ) . B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = S R ( X ) is a b ase for the supr atop olo gy S θ O ( X ) . { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = S R ( X ) . ˜ ϕ 2 = semi-int is the dual of ϕ 2 . A is ϕ 1 , 2 -c omp a ct set iff it is ( S O(X)-scl)-c om p act set iff it is S θO ( X ) -c omp act set iff it is SR(X)-c omp act set. No w, by using the Theorems 3.4, 3 .6 w e can write the equiv alen t statemen ts for a set to b e ( τ - in to cl)-compact set, or to b e (SO(X)-scl)-compact set, and b y using the Theorems 3.4, 3.6, 3.7 w e can write t he equiv alen t stateme n ts for a space t o be ( τ -in to cl)-compact sp ace or to b e (SO(X)-scl)-compact space. Some equalities related to closure types can b e o btained b y using op eratio ns and some of them were giv en in [18]. F o r example for any op en set T in a topo logical space ( X , τ ), we hav e T = θ c l T = τ s - cl T . Theorem 3.9 L et A ⊂ X . If ϕ 2 ( U ) = ϕ 3 ( U ) for e ach U ∈ ϕ 1 O ( X ) , then ϕ 1 , 2 O ( X ) = ϕ 1 , 3 O ( X ) , and A is ϕ 1 , 2 -c omp a ct ( ϕ 1 , 2 O ( X ) -c omp act ) set iff it is ϕ 1 , 3 -c omp a ct ( ϕ 1 , 3 O ( X ) -c omp act) set. Example 3.10 L et ϕ 1 = int , ϕ 2 = scl , ϕ 3 = intocl . F or U ∈ ϕ 1 O ( X ) = τ , ϕ 2 ( U ) = sclU = U ∪ U o = U o = ϕ 3 ( U ) , (sclU = U ∪ U o , [3]). A is τ s -c omp a ct se t iff A is ( τ -into cl)-c o mp act set iff A is RO(X)-c omp a ct se t iff A is ( τ -scl)-c omp act set. Theorem 3.11 If ϕ 1 is mon otonous and , fo r e ach p air U, V ∈ ϕ 1 O ( X ) , ϕ 2 ( U ∪ V ) = ϕ 2 ( U ) ∪ ϕ 2 ( V ) , then the fol low ing ar e e quivalent. a) A is ϕ 1 , 2 -c omp a ct se t. b) Each filterb ase F b which is a subfamily of { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } and me ets A, ϕ 1 , 2 -ac cumulates to some p oint a ∈ A . Pro of. (a ⇒ b). It is clear from Theorem 3.4. (b ⇒ a). Let A b e not ϕ 1 , 2 -compact set under the assumption of (b). There is a subfamily U = { U i : i ∈ I } of ϕ 1 O ( X ) s.t. A ⊂ ∪U but f or eac h finite subset J of I A 6⊂ ∪ j ∈ J ϕ 2 ( U j ) = ϕ 2 ( ∪ j ∈ J U j ). F or eac h finite subset J of I w e hav e A ∩ ( X \ ϕ 2 ( ∪ j ∈ J U j )) 6 = ∅ . Since ϕ 1 is monotono us, ϕ 1 O ( X ) is a supratop olo gy and hence F b = { X \ ϕ 2 ( ∪ j ∈ J U j ) : J ⊂ I , J f inite } is a filterbase s.t. F b ⊂ { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } and F b meets A. There exists a p oin t a in A s.t. F b ϕ 1 , 2 -accum ulates to a. T here exists a U i a ∈ U s.t. a ∈ U i a . X \ ϕ 2 ( U i a ) ∈ F b , ϕ 2 ( U i a ) ∩ ( X \ ϕ 2 ( U i a )) 6 = ∅ . This contradiction completes the pro o f. 11 Example 3.12 a) L et A ⊂ X , ϕ 1 = cl oint , ϕ 2 = cl as in Exa mple 3.3 (d) . ϕ 1 O ( X ) = S O ( X ) , ϕ 2 ≥ ϕ 1 . ϕ 1 is m onotonous. F or e a ch U ∈ ϕ 1 O ( X ) = S O ( X ) , ϕ 2 ( U ) = U ∈ S O ( X ) and ϕ 2 ( ϕ 2 ( U )) = ϕ 2 ( U ) . If U, V ∈ ϕ 1 O ( X ) , then ϕ 2 ( U ∪ V ) = U ∪ V = U ∪ V = ϕ 2 ( U ) ∪ ϕ 2 ( V ) . B = { ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = RC ( X ) . ϕ 1 , 2 O ( X ) = θ S O ( X ) . A is ϕ 1 , 2 -c omp a ct set iff it is B -c o m p act set iff it i s ϕ 1 , 2 O ( X ) -c omp act set. X i s ϕ 1 , 2 -c omp a ct an d Hausdorff iff X is S-close d sp ac e. ˜ ϕ 2 = int is the dual of ϕ 2 . { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = RO ( X ) = { ˜ ϕ 2 ( X \ U ) : U ∈ ϕ 1 O ( X ) } = { intK : K ∈ S C ( X ) } . b) L et ϕ 1 = int , ϕ 2 = cl as in Example 3.3 (a). ϕ 1 is mo n otonous, ϕ 2 ( U ∪ V ) = ϕ 2 ( U ) ∪ ϕ 2 ( V ) for U, V ∈ ϕ 1 O ( X ) = τ . { X \ ϕ 2 ( U ) : U ∈ ϕ 1 O ( X ) } = RO ( X ) . F or a fi l terb ase F b ⊂ RO ( X ) , we h ave ∩{ ϕ 1 , 2 clF : F ∈ F b } = ∩{ θ cl F : F ∈ F b } = ∩{ F : F ∈ F b } = ∩{ τ s - cl F : F ∈ F b } . X is ϕ 1 , 2 - c omp c t and Hausdorff iff X is H-close d sp ac e. No w w e can get man y kno wn results that some o f them can b e seen from [4,5,7 ,8 ] and man y unkno wn results b y special c hoices of o p erations. And w e will ha v e many results b y using the pap ers [18,19,2 0]. REFEREN CES 1. M.E.Ab d El-Monse f, F .M.Zey a da and A.S.Mashhour, Op erations on top olo- gies and their a pplications on some t ypes of cov erings. Ann. So c. Sci. Brux- elles, 97, No. 4 (1983), 155-172. 2. M.E.Ab d El-Monsef and E.F.Lashien, Lo cal discrete extensions of supratop ologies. T amk ang J. Math. 21, No .1 (1990), 1- 6. 3. D . Andrijevi ´ c, On the top ology g enerated b y pre-op en sets, Mat. V esnik, 39 (1987), 367-3 7 6. 4. R . F. D ickman, JR and Jack R. P orter, Θ-p erfect a nd a bsolutely close d func- tions, Illinois J. Math. 21 (1977),42 -60. 5. R . F. Dick man, JR and R. L. Krysto c k, S-sets and s-p erfect mappings, Pro c. Amer. Math. So c. 80, 4 (1980), 687-69 2. 6. J. Dugundji, T op olog y , (1966). 7. L. L. Herrington and P . E. Long, Chracterizations of H-closed spaces, Pro c. Amer. Math. So c. 48, 2 (1975), 469-47 5. 8. R . A. Herrmann, R C-Con vergence , Pro c. Amer. Math. So c. 75, 2 (1979), 311-317 . 12 9. D . S. Jank ovi ´ c, On functions with α -closed graphs, G lasnik Matematı ˇ ckı, 18, No. 38(1983), 141- 148. 10. A. Kandil, E. E. Kerre and A. A. Nouh, Op eratio ns and mappings o n fuzzy top ological spaces, Ann. So c. Sci. Bruxelles 10 5, No. 4. (19 91), 165-188. 11. S. Kasahara, Op eration- compact space s, Math. Jap onica 24, No . 1 (19 79), 97-105. 12. E. E. Kerre, A. A. Nouh a nd A. Kandil, Op era t io ns on the class of fuzzy sets on a univ erse endo w ed with a fuzzy to p ology , J. Math. A nal. Appl. 180 (1993), 325-3 4 1. 13. J. K. Kohli, A class of mappings containing all con tin uous and all semicon- nected mappings, Pro c. Amer. Math. So c. 72, No. 1 (1978), 175-181 . 14. J. K. Kohli, A fr a mew ork including the theories of contin uous and certain non-con tin uous functions, Note di Mat. 10, No. 1 (199 0), 37-45 . 15. J. K. Kohli, Change of top o lo gy , ch aracterizations and pro duct theorems for semilo cally P-spaces, Houston J. Math. 17 , No . 3 (1991), 33 5-349. 16. M. N. Mukherjee and G. Sengupta, On π - closedness : A unified theory , Anal. Sti. Univ. ”Al. I. Cuza” Iasi, 39, s.l.a, Mat. No. 3 (1993), 1-10. 17. G. Sengupta and M. N. Mukherjee, O n π -closedness a unified theory I I, Boll. U. M. I. 7, 8-B (1994), 999-1 0 13. 18. T. H. Y alv a¸ c, A unified theory for con tin uities, submitted. 19. T. H. Y alv a¸ c, A unified theory on op enness and closedne ss of functions, sub- mitted. 20. T. H. Y alv a¸ c, A unified approac h to compactness and filters, appear . 21. T. H. Y alv ac . , On some unifications, The F irst T urkish In ternational Confer- ence on T op ology a nd Its Applications, Aug ust (2000), submitted. 22. T. H. Y alv a¸ c, Unification of some concepts similar to Lindel¨ of prop ert y , sub- mitted. 13