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Re.v,u,ta. Coiomb.£ana de. Ma.te.mmc.M VoL XVII (1983), yJlfgJ.J. 31 - 58 o PAIRWISE LINDELOF SPACES by Ali A. FORA and Hasan L HDEIB A B ST RA CT. In this paper we define p:rirwise Lindelof spaces and study their properties and their relations with other topological spaces. We also study certain conditions by which a bitopological space will reduce to a single topology. Several examples are discussed and many well known theorems are generalized concerning Lindelof spaces. RESUMEN. En este articulo se definen espacios p-Lindelof y se estudian sus propiedades y relaciones con otros tipos de espacios topologicos. Tambien se estudian ciertas condiciones bajo las cuales un espacio bitopologico (con dos topologias) se reduce a uno con una sola topologla. Se discuten varios ejemplos y se generalizan varios teoremas sobre espacios d~ Lindelof. 1. I nt roduce pological ion. Kelly [6] introduced space, i.e. a triple T , T are two topologies 1 2 the notion of a bito- (X,T1,T2) where X is a set on X, he also defined pairwise and Haus- 37 dorff, pairwise regular, generalizations pairwise normal spaces, and obtained of several standard results theorem. such as Urysohn's Lemma and the Tietze extension have since considered the problem of defining compactness such spaces: see Kim [7], Fletcher, san [1]. Cooke and Reilly Several authors for Hoyle and Patty [4], and Bir- [2] have discussed the relations be- tween these definitions. In this paper we give a definition bitopological of pairwise Lindelof spaces and derive some related results. We will use p- , s- to denote painw~e ively, e.g. p- compact, and semi-compact s- compact and ~emi-, respect- stand for pairwise compact respectively. The L.-closure, L.-interior l of a set A will be denoted by l Cl.A and IntiA respectively. The product topology of L1 and L2 l will be denoted by L1xT2. Let R, I, N denote the set of all real numbers, val [0,1], and the natural Ll' L numbers respectively. U6ual, ~o~ountable, denote the ~enete, r kigth-~y topologies the inter- Let Ld, L , L , u c lent-~y and on R (or I). , 2. Pairwise Linde15f Spaces. Let us recall known defini- tions which are used in the sequel. 2.1 [4J. A cover ~of is called L 1L 2 -0 pen if 'U.. the bitopological eLl U L 2' If, space (X,L1,L2) in addition, contains u.. at least one non-empty member of L1 and at least one non-empty member of L2, it is_called p-open. 2.2 [4]. A bitopological space lS called p-~ompa~ if every p-open cover of the space has a finite subcover. 38 2.3 [3]. A bitopological ry T T -open 1 2 2.4 space is calle s-~ompact if eve- cover of the space has a finite subcover. space (X.T1.T2) is called T1 ~ompact with nehpect ~o T2 if for each T1-open cover of X there is a finite T2-open subcover. [1]. A bitopological 2.5 [1]. A bitopological space (X. T1.T2) is called B-eompact if it is T1 compact with respect to T2 and T2 compact with respecto to T . 1 If we replace definitions the word "finite" by the word "countable" 2.2, 2.3 and 2.4, then we obtain the definition p-Lindelofi, s-Lindelofi, ~pect ~o and (X,T1,T2) ~ T2, respectively. 2.6 (X,T)is Lindelofi with of ne- space (X,T1,T2) is called B-Lindelon with respecto to T2 and T2 Lindeloff with A bitopological if it is T1 Lindelof respecto to T1. It T1 in clear that (X,T1,T2) is s-Lindelof if and only if Lindelof where T is the least-upper-bound topology lS of T1 and T2. It is also clear that if (X,T1,T2) is B-Lindeloff then each (X,T.) must be a Lindelof space for i 1,2. = l 2.7 Wh~n we say that a bitopological a particular topological property, space (X,T1,T ) has 2 without referring specially to T1 or T2, we shall then mean that both T1 and T have the 2 property; for instance, (X,T1,T2) is said to be Hausdoff if both (X,T1) and (X,T2) are Hausdorff. THEOREM 2.8. The bdopofogiea1. !.lpaee r x, T l' T 2) i!.l e-Lin39 delCi6 -<-6 and only -<-6 d ~ Li.-nde1.o6 and p-Li.-ndelCi6· P~oo6. Necessity follows inmediately from the observation that any p-open,L -open or L2-open cover of (X,Ll,L2) is L 1L 2-0l not pen. Conversely, if a L1L2-open cover of (X,Ll,L2) is p-open, then it is Ll-open or L2-open. EXAMPLE 2.9. The bitopological space (IR,Ld,Lc) is p-Lin- delof but is not s-Lindelof. EXAMPLE 2.10. defined Consider the two topologies L l' L 2 on R by the basis = {(-ro,a):a > O} U {{x}:x > O} , and ~2 = {(a, ro):a < O} U {{x}:x < O} . ~l Then (IR,L ,L ) is p-Lindelof but is not Lindelof. It is also l 2 clear that (R,L ,L2) is not B-Lindelof, for the Ll-open cover l . {(-ro,l)} U {{x}:x ~ l} of R has no countable L2-open subcover. 2.11 [8J. A bitopological space (X,Ll,L2) is called p-Qountably 'Qompact if every countably p-open cover of X pas a finite subcover. 2.12 A bitopological space able Qompact if every countably (X,Ll,L2) L1L2-open is called S-Qount- cover of X has a fin- ite subcover. 2.13 A bitopological space (X,Ll,L2) is called ably Qompact with ~~pect to L2 if for each countably cover of X there 2.14 is a finite A bitopological L2-open space 2 countably compact The following 40 Ll-open subcover. B-QOunt- (X,Ll,L2) is called compact with respect albtj Qompact if it is L1 countably and L l-Qount- L with respecto fact is obvious: to Ll. to L2 THEOREM 2.15. s B)-c.ountably p(Jte-6p. n lo compact and p(Jte-6p. s,B)-countably (ii) EveJty p(Jte-6p. s ,B)-compact EvVty p(Jte-6p. (i) -6pace J.A s B)-Lindelon. compact p(Jte-6p. s,B)-Linde- J.A p (Jte-6P. s, B) - compact. -6pace EXAMPLE p-Lindelof 2.16. The bitopological space (R,Ld,LC) space which is neither p-countably is a compact nor p-com- pact. EXAMPLE 2.17. Let L s denotes the Sorgenfrey topology on R. Then the bitopological space (R,L ,L ) is s-Lindelof u s but is not B-Lindelof, because the L -open cover {[-n,n):n EN} s of R has no L -open countable subcover. It is also clear that u the space (R,L ,L ) is neither u s EXAMPLE (N,Ld,Ld) 2.18. s-countably compact nor s-compact. It is clear that the bitopological is B-Lindelof but is neither B-countably space compact nor B-compact. THEOREM 2.19. -6pace In (x , L l' Let <e J.A a hVteciUaJty Lindelo n n. -then d i-6 s- Lindelo PftO06. T 2) {U : a E i\} U {VS:S e; I"] be a L L -open a 1 2 cover of X, where Ua e: L 1 for each a e: I\. and V S e: L 2 for each BE U = = U{Ua a e: r} is L countable I\.} is L1-Lindelof, there exists a set 1\.1c I\. such that U = U{ua:a e: 1\.1}' Similarly, since V = U{VS:S f. Since E 2-Lindelof, there exists a countable U{VS:S e: f1}.,It is clear that set f1 c f such that V = {U :a € I\. } U {VS:S e: f } is a countable a -6pace 1 1 COROLLARY 2.20. J.A s - Undelo 6. EvVty subcover sesiond countable of ~ for X. bdopologic.al 41 EXAMPLE Let X = RXI and < be the lexicographical 2.21. order on X. Let = ~1 {[x,y):x < y ; x,y E X} and ~2 = {(x,y]:x < y ; x,y EX}. Let T ,T be the topologies on X which generated by the basis ~1 1 2 and ~2' respectively. Then (X,T1,T2) is a Lindelof space which is not p-Linde16f, because the p-open cover { [ ( a , x ) , (1 , x )) , ( ( a , x ) , O,x)] :x of X has no countable neither s-Lindelof Lindelof = subcover. It is clear that (X,T1,T2) nor B-Lindelof. is 2.22. but it is s-Lindelof. EXAMPLE ~2 Id Let X and T1 be the same as in example Then the bitopological space (X,T1,T2) is not hereditary EXAMPLE 2.21. E 2.23. R, '81 = {X,{x}:x E X-{O}} and Let T1,T2 be the topologies on X which are Let X {X,{x}:x E X-{l}}. = by the bases ~1 and '82' respectively. Then (X,T1,T2) is B-Linde16f, for any T1-open cover of X or any T2-open.cover generated of X must contain X as a member. However, (X,T1,T2) is not p-Lin- de16f, for the p-open cover {{x}:x E X} of X has no countable subcover , We may summarize some of the above examples and theorems by the diagram on tre nextpage (T stands for theorem while E stands for example). 2.24 [7J. If T is a topology subset of X then the topology 42 on X and A is a non-empty adjoint topology (denoted by T(A)) is the on X defined by T(A) = {¢,X} U {AUB: BET}. NO E2.23 NO E2.17 o:;:----------------.~ s-Lindelof B-Lindelof YE o T2 8 E 2.21 I NO E2.10 Lindelof ----------------~ p-Lindelof 2.25 A family every member of 2.26 7 of nonvoid subsets of X is T1T2-closed is T1-closed or T2-closed. Y [8]. A family 1of nonvoid T1- or T2-closed sets if 1contains members F1 and F2 such that F1 in X is p-cto~ed is a T1-closed proper subset of X and F2 is a T2-closed subset of X. 2.27 [6]. weakly A set U in a topological ope~ if for any p ing p such E space (X,T) proper lS called U there exists an open set V contain- that V~U is a countable set. ly arosed if X-F is weakly open. then p is called a weak-~ntenio~ weakly open set V containing weak-interior if A set F is called weak- If A is a subset of X and p EX, po~ of A if there exists p such that V cA. The set of a all points of a set A is denoted by WInt A. It is clear that WInt A is the largest weakly open set contained in A. It is also clear that WInt A A is weakly open, and Int B cWInt LEMMA 2.28. be a weakly crosed Let (X, p~op~ T ,T ) 1 2 = A if and only if B for any set B eX. be a p-L~~de1.o6 -6pac.e.a~d c .6ub~et i~ r x, T 1)' The~ c ~ T 2- L~~- de1.°n· 43 P~o06. Let C be a nonempty of X and let ~ = {va:a L1-weakly closed proper subset A} be a L2-open cover for C. For each x E X-C there exists a L -open set H(x) such that x e: H(x) and 1 H(x) n C is a countable set. Since C ~ ~ and C f. X, the n {V :a a E EA} U {H(x):x E X-C}- is a p-open cover for the p-Lindelof space X. A1 c A and a countE A } U {H(x ),H(x2),··} 1 1 Thus, there exists a countable set able set {x ,x , ...} c X-C such that {va:a 1 2 is a countable cover for X. Since H(x. )n C is countable l 00 for all 00 ie: lN, the set Cn(UH(x.)) is countable, say CO(UH(x.)) = i=1 l i=1 l {Y1'Y2, ... L Since Yi e: C, there exists ai e: A such that Yi e: ' It is clear now that {va:a e: A1} U {Va.:i e: ai l able subcover for C. Hence C is L2-Lindelof. V Since every closed lowing corollary W} is a count- set is weakly closed, we have the fol- to Lemma 2.28. COROLLARY A 2.29. 6 delCi 6 -6pac.e £6 L j- LLndelCi L i-uo-6ed (i f. j; i,j Using a similar technique COROLLARY 2.30. ~pac.e i6 Lj-c.ompact It is important j; i,j = -6ub-6eX. 06 a p-L{.n- 1,2). as above, we obtain the following;;; A Li-uo-6ed (i f. p~op~ = pftOP~ -6ub-6eX. 06 a p-c.ompact~~~ 1,2). to note that the word "proper" in Lemma 2.28 can not be removed. not Ld-Lindelof For example, R is L -closed but R is c in example 2.16. We now obtain delof spaces. 44 four alternative characterizations of p-Lin- OoUowing Me. e.qLUVa£e.ru:.: p-Lindel0n. (a) (X,11,12) ~ (b) p-clo~e.d Ev~y oamity with ty ha6 no nempty iru:.~ p~op~- 1, the topology T2(V) ~ Linand oo~ each non-empy ~et v in T2, the topology delao, 11(V) ~ (e) int~eetion eetio n, ~et V in Fo~ each non-empty (c) the. countable. 1 Lindeloo· Each 11-weak1.y clo~e'd p~op~ and each 1 2 ~u.b~et -we.ak1.y clo~ ed p~op~ P~006. The The equivalence 00 X .6u.b~et: fact that (a) is equivalent cs T2-Lindelein, 0 -6 X ~ T 1- to (b) is obvious. of (a), (c) and (d) can be obtained ogous way to the proof of [2,Theorem plies (e) is due to Lemma 2.28. Lindelo -6. in an anal- 2]. The fact that (a) im- The fact that (e) implies (e) is obvious. An easy characterization found in the following THEOREM 2.32. delan io and only io of s-Lindelof spaces can be theorem. A bitopologiea£ e.v~y able inteM e.etio n pM p~y ~pace (X, 1 l' T 1 1 2-clo~ed T 2) oamity w..{;ththe s- Lin- ~ count- hiu. nonempty iru:.~ eetio n. , , THEOREM 2.33. be B- Lindela O. EXAMPLE Let (X,T1,12) be B-compaet and (Y,T1,12) Then (Xxy,11x1~,12X1;) ~ B-Lindela6. 2.34. Let 1f denote the cofinite topology on R. Then (R,lf'ld) is p-compact. Howeve·r, the space 2 (R ,lfX1f' ldX1d) is not even p-Lindelof, for the p-open cover {R x(lR- {O})} U {( x, 0) :x 2.35 [9]. e::: IR} of jR2 A space (X,11,12) has no countable subcover. is said to be p-Ha~do~oi 45 if, for any distinct points x and y, there hood U of x and a l2-neighbourhood We observe a l1-neighbour- V of y such that·U that if (X,ll,l2) and l2 are T1-topologies. p-Hausdorff spaces. is is p-Hausdorff, The following theorem n V = ~. then both l1 characterizes The. ooUow-lng ptwpeJLtie.J.> aJte. e.qtU.val.e.n-t: (a) The. b,[topofog-lQal. ~paQe. (X,l1,l2) ~ p-Ha~do~oO' (b) Fo~ e.a~ x g X, THEOREM 2.36. {x} = n {ci U :U 1 a. a. a. Et:. lS a l2 neighbourhood of x} and {x} = (c) n {Cl a. ez I: The. diagonal. 2 U:U a a is a l1 neighbourhood D = {(x,x):x x} ~ g of x}. a elo~e.d ~ub~e.t ,[n e.ach On the. ptwduct topofog-le.J.> (XXX,l1Xl2) and (XXX,l2Xl1)' P~oo6. (a) implies (b). Let x E X and y E X such that y i- x , By (a) there exists a l1-open that y e: V l' X set V1 and a l2-open set V2 such E V 2 and V 1 n V 2 = 4>. This implies that y e: C11 ~2' This proves the first part of (b). The proof of the second part of (b) is similar to the one we just proved. (b) implies (c). Let (x,y) e: XXX-D. Then x,y e: X and x y. By the second part of (b), there exists al1-open set U1 containing x such that y e: X-C12U1· Let U2 = X-C12U1· Then U2 is a l2-open set and it is easy to check that (x,y) e: U1xU2 c XXX-D. Hence D is a closed set in the topological (XxX, l1Xl2)' space In a similar way we can prove that D is l2xl1- closed subset of XxX. (c) implies EXXX-D. 46 (a). Let x,y e: Since D is a l{l2-closed X such that x i- y. Then (x,y) set, there exists a l1-open i- set U and a T2-open is clear now that x 1 Recall section Then set U2 such that (x,y) E U1xU2 e XXX-D. e:: U , Y e::U U = <jJ • 2 and U1 1 2 that a space (X,T) in which every countable P~006. T. ] Let A be a T.-Lindelof eros ed (i ~ j; subset and x e::X-A. l (\ {Cl.U:U o,e:: I::. l 0, is a T. neighbourhood 0, a T.-open cover of the T.-Lindelof a countable set A. l l set 1::. 1 el::.such that {X - CliUo,:a {X - Cl.U l 0, I::.} :0. e:: Thus there exists e:: 1::. } is a cover 1 n ] ] Using the same technique COROLLARY 2.38. T. - of x} U X-Cl.Uo," Let U = U. Then U is a ae::1::.1 l o,EI::.1a set,contains x and U eX-A. Hence A is T.-closed. for A, i.e. A e f1.Y By ] (i ~ j; i,j = 1,2). Since A e X-{x}, therefore T.-open i , j = 1, 2 ) . 2.36 we have {x} = lS inter- is called a p-~paQe. .s uiis et: M T.-Lindelo6 l Theorem n of open sets is open, ev~y It Qompac.t -6 ub~ l e;t M Le;t as above we obtain (X,T1,T2) T. - ctM ed (i ~ the following. be p-Ha.u6doJz.66. Then j; i, j ] = eve- 1,2). [6] . In a space (X,T ,T2), T1 --L6 ~cu:.d to be Jz.egui.M 1 wUh Jz.e-6 pea to T2 if, for each point x in X and each T1-closed set P such that xl- P, there are a T1-open set U and a T2-open 2.39 set V such that x e::U, P e V and U n V = <jJ (X,T1,T2) is p-Jz.egui.a.Jz. if T1 is regular T2 and vice versa. 2.40 [9J. In a bitopological space with respect to (X,T1,T2), we say that 47 It is interesting to note that if L1 is regular with respect to L2 and L2 is coupled to L1, then L1 CL2. Thus, if (X,L1,L2) is p-regular and Li is coupled to Lj (i I j, i,j = 1,2) then L1 = L2 and the resulting single topology is regular. , r It is also interesting to note that if L1 is coupled to L2 and r r, L1 is regular with respect to L2 then (X,L1) is regular. [6] . 2.41 A space (X,L1,L2) is said to be p-no!U11l:Lt if, given a L -closed set C and a L2-closed set F such that en F = ¢, 1 there are a L1-open set G and a L2-open set V such that F c G, Cc V and V n G = <jJ. THEOREM 2.42. ~pac.e p-Jtegu1.aJt, p-no!U11l:Lt. EveJty p- Li.Yl.deio6 bdopologic.a1. (X, L1,L2) ~ PJtoon. Let A be a nonempty L1-closed set and B be a nonempty L -closed set with An B = <jJ. Since (X,L1,L2) is p-regular 2 for each a E A, there exist a L2-open set Ga and a L1-closed set F with a L G c F c X-B. Also, for each b E B, there exist a a a a L1-open se t- eband a L2-closed set Mb with b e: Cb c Mb eX-A. Let = {Cb:b E B} U {X-B} and ~ = {Ga:a EA} U {X-A}. Since e e and ~ are p-open covers exist countable the p-e Li.ndeLof space X, there for subcollections {C1,C2, ...} of ~ and {G1,G2, ...} 00 of g such that A c; UG. and B c Ve .. Let V1 = C1 and, for i=1 l l=1 l n-1 each positive integer n > 1, let Vn = e - UF.. For each n n i=1 l 00 and UM.. Let V = Uv positive integer n, let H = G n n i=1 l n=1 n co l' H liE L 2 ' A cHand B c V. Furthermore, H = UH . Then VEL n=1 n x E H n V, then x E H n Vn for some m and n, and so m m n-1 x e: (G M.) n (C - U F .). Considering separately the m i=1 l n i=1 l cases m > nand m ~ n yields a contradiction and so H n V = <jJ. 00 U Thus (X,L1,L2) is p-normal. 48 • Let X be a fixed non empty set, and 'BX = {( X,T ,T '):T and T I are topologies ~ on 'BX by: Define the partial ordering Then we have the following Let THEOREM 2.43. on X} . theorem. f = {( X, T ,T') E <EX:(X,T ,T ') b, a p.LLnde1.on P-.6pac.eJ and jf = {( X, T ,T') e: 'Ex: r x, T ,T') b, a p-HalL6don66 p-.6pac.e}. 16 (X,T1,T2) E in 1/, then (X,T1',T2) b, a rniiU.ma1. e1.eme.nt 06 !f and a mlx.Una1. e1.emerz,t 06 i.. Pnoon. Suppose (X,T~,T~) Elf such that (X,T~,T~) ~ (X,T1,T2). Therefore T~ c: T1 and T~ c: T 2' Let G T1-closed proper Corollary e: T 1-H}. Then X·-G is a subset of X. Since (X,T1,T2) is p-Lindelof, by .-. 2.29, X-G is T2-Lindelof. But T; C:T2" Therefore X-G is T~-Lindelof. Since (X,T~,T~) lary 2.37, X-G is T~-closed. is p-Hausdorff P-space, Hence G e::T~. Consequently by corolT1 = T~. In a similar way we can show that T2 = T~. Now, let (X,T~,T~) e::£ such that (X,T1,T2) ~ (X,T~,T;). , , , Then TiC: T 1 and T 2 c: T 2' Let U e:: T i-a}. Then X-U is a T i-closed proper subset of X. Since (X,T~,T~) is p-Lindelof, by Corollary 2.29, X-U is T~-Lindelof. But T2 c: T;. Therefore X-U is T2-Lindeis p-Hausdorff P-space, by corollary 2.37, lof. Since (X,T1,T2) X-U is T i-closed. Hence U , E: T l' Consequently I = lar way we can show that T2 Using the same technique T 1 = T t ' In a simi- T2. as above we obtain the following theorem. Let THEOREM 2.44. pad}, and (X ,T 1 'T 2) E .re = {(X,T,T , I ) E'13X:(X,T,T ) b, p-c.omJv6 p-HalL6don66}. 16 Jf = {(X,T,T') E<EX:(X,T,T') e n Jf, then (X,T 1 'T 2) b, a mLvUmal. e.1emerz,t a 6 jf and a maximal. e1.emerz,t an ~. 2.45 49 (p-op~n, p-cto~~d, iff f:(X,L1) + (Y,0 ) and f:(X,L ) + (Y,02) are continuous(open, closed, homeo1 2 morphism, respectively). 0 VLtornp Let f:(X,L1,L2) 2.46. THEOREM uo lL6 p-hom~omo~phUm,respectively) + (Y,01,02) be. a P-c.on..t.in- . Ifi (X,L1,L2) ~ 16 f ~ p-Lindelofi (s-Lindelo6, B-Lindelo6, ~~p~c.tivefy), (Y,01,02) ~ p-Lindelo6 (s-Lindefo6, B-Lindelo6, ~~p~c.tivefy) . (b) 16 (X,L1,L2) ~ p-c.ompact (s-c.ompac.t, B-c.ompac.t, ~~p~c.tiv~fy), then (Y,01,02) ~ p-c.ompac.t (s-c.ompac.t, B-c.ompac.t, ~~pec.tivefy) . (a) one-to-one., (Y, 01'02) ~ p-HaU6do~fi6 P-~pac.e. and (X,L1 ,L2) ~ p-Lindefo6, then. f u a home.omo~pYU~m. (d) Ifi f ~ one.-to-one., (Y,01,02) ~ p-HalL6do~6fi and (X,L1,L2) ~ p-c.ompac.t, the.n f ~ a homeomo~p~m. PMo6. (a) let e= {va:a e:: Ll} U {ua:a e:: Ll} be a p -open cover (c) of V such that "« e:: 01 and Ua e:: 02 (a e::Ll). Then {f-1 (V ):0'. e::.6. } U {-1 f (U ):0'. e:: Ll} is a p-open cover of X because a a f is p-continuous and onto. exists a countable -1 {f (Va) :ae::Ll1 } U Since (X,L1,L2) Ll such that is p-Lindelof, there set Ll1 C f (Ua) :ae::Ll1 } is a cover for X. Thus {-1 Ll } U {ua:a e:: Ll1} is a countable subcover of <e for X. 1 The remaining parts of the statement (a) are similarly proved. {va:a e: (b) The proof is similar to that in (a). (c) It suffices to show that f is p-closed. proper subset of X. Let A be aLl-closed Then, by Corollary 2.29, A is L2-Lindelof. f:(X,L2) + (Y,02) is continuous. Hence f(A) is O2-Lindelof because By Corollary 2.37, f(A) is 01-closed. Similarly, it can be shown that the image of every L2-closed subset of X is a O2-closed subset of Y. Hence f is p-closed. (d) The proof is similar 50 to that in (c). 3. Conditions reduced under to a which single space is topdldgy. 3.1. Let THEOREM a bitopological be. a HaU6do!l.66 p-Undelo6 (X,L ,T ) 1 2 The.n T1 = L2. P!l.oofi. Let G E L1-{~}. Then X-G is a T -closed proper sub1 set of X. By Corollary 2.29, X-G is T2-Lindelof. By Corollary p-~pace.. 2.37 we have: "every Lindelof (X,T) is closed". Thus X-G is 2-closed, i.e. G we have T2 CL1· Consequently L1 C L2· Similarly THEOREM ~pace.. subset of a Hausdorff Let 3.2. P-space T . Hence 2 T1 = L . 2 L E be. a compact p-HaU6do!l.66 (X,T1,T2) The.n T1 = T2. P!l.oo6. Let G E T1. compact space (X,T1). 2.38, X-G is L2-closed, we have T2 C Then X-G is a T1-closed subset of the Therefore X-G is T1-compact. By Corollary E T2. Hence T1 C T . Similarly 2 Thus T1 = T2. T1· LE~1A 3. 3. Let (X, L l' F be. a T1-we.aQty eio~e.d T 1- Unde..f..o6 . P!l.oo6. Let q G containing i.e. G E ~et L 2) be. a p- Undelo ~uch that WInt2(X-F). fi ~ pace. and .f..e.:t 1 ~. The.n F ~ WInt2(X-F) Then there exists q such that G n F is a countable a T2-open set set. Let ~ = 6} be a T1-open cover for F. For each x E X-F there exists a T1-open set H(x) containi?g x such that H(x) n F is a {Ca:a E countable set. Since {Ca:a cover for the p-Lindelof 61 C 6 and {x x ' ...} 1 2 {ca:a E6 } 1 C E 6} U {G} U {H(x):x E x-rI is p-open space X, there exist two countable sets X-F such that U {G} U {H(X ), H(x2), ... } is a cover for X. 1 00 Since H( x.) l n F is countable, the set F n (U H( x ,» i=1 is countable, l 51 Let ui e 6 such that y.l e::Cu.l say {Y1'Y2' ... }. is a countable {Cu:u e: 61} U {CU{ie: N} L1-Lindelof. We can use the same technique lowing theorem "empty" (We replace in the proof THEOREM Let 3.4. Ld-closed is not Ld-Lindelof. 1 ~" is a necessary .opae.e, and let by the word Let 1 ~. Then F ~ Int2(X-F) set which 3.6. the fol- (X, L l' L 2) be a p-e.ompact .opae.eand let In the p-Lindelof THEOREM p. "countable" 3.5. rrWInt2(X-F) i.e. F is 3.3). F be a L1-c..fo.oed.oet .oue.h that pact. EXAMPLE e, of as above to conclude the word of Lemma subcover (i e:IN). Then (IR,Ld,Lc)' 1R is a This shows that condition in Lemma 3.3. be. a p-Hau.odo~66 p-L~de.fo6 (X,L1,L2) u be a L i-weaU!! space L1-e.om- open .0 et: e.on.t.cUru.ng a 6-ixed po-<.nt Then ( a) thvr.e. ewtoo L 1-open sets that p e::nc. e Fe i=1 l (b) ecthe»: p e::C12(X-U), Ci(i e: N) and a Ll-c..fo.oed.oet F .oue.h U; and; on, thvr.e. ewt and a L 1-c..fo.oed set: F .oue.h that J,2-open.ow QGi p e:: Gi Ci e::lN) e FeU. P~oo6. exists (a) Since p e: U and U is L1-weakly open set, there a Li-open set A such that p e: A and A-U is a countable set. For each x e: X-U there exist G(x) such that x e: G(x), An B(x). D(x)-U a L1-open set B(x) and a L2-open set p e: B(x) and B(x) n G(x) Let D(x) = ~. Then Pe: D(x), D(x) e::Ll, 52 {G(x1),G(x2), set. = <f>, and X-U is a L1-weakly closed proper subset of the p-Lindelof space X, by Lemma 2.28, X-U is L2-Lindelof. Therefore the L2-open cover {G(x):x e: X-U} has a countable subcover is a countable D( x ) n G(x) = Since ... } . Since D(xi)-U (i e: N) is a count- 00 UD( x .)-U lS countable, i=1 l e: IN). Then Ci (ie:~) able set, the set Ci = D(xi)-{yi} (i be~ause (X, T 1) is a T1-space. ((1C .) n G( x .) = ¢ for all j i=1 loo J e: F. i.e. (lC. i=1 F e: U. Hence p nC. e: i=l Therefore e: C12(X-U), p e: T1-Lindelof. 00 (n i=1 J 00 n i=1 Hence e: IN. Since {G(x.):j l (b) If P Let F = say {Yl ,Y , ... }. Let 2 is a T -open set 1 X-G( x. ). Then e: lfr} l C .) n l 00 (U G( x . » i=l is a cover for X-U, then e: F e: U. then we are done. Suppose p ~ C1 (X-U). 2 ~ ¢. By Lemma 3.3, X-U lS Int2U, i.e. Int2(U) For each x e: X-U there exist a T -open 1 set G(x) such that p e: G(x), x e: C(x) set C(X) and The T1-open cover {C(x):x e: X-U} has a count{C(x1),C(X2), ... } . Let G. = G(x.) and F = able subcover 00 l 00 (lx-C(x.). i=1 ¢ l and a T2-open C(x) G(x) = ~. n = J p Then e: l (lG. i=1 l Using the same technique we get the following l e: Fe: U. as in the proof of Theorem 3.6, theorem. Let (X,T1,T2) be. a p-HalL6doJtoo p-c.ompact U be. a T1-ope.n ~et c.ontaining a oixe.d point p. THEOREM 3.7. ~pac.e. and, let The.n (a) the.Jte. e.~t p (b) e: T C e: F e: u, eithe.Jt p T 1- a e: C and a T 2-uMe.d se: F such. that and on: the.Jte. exist: C12(X-U) UM e.d ~ et: se: -ope.n 1 F /.)uc.h that p e: G e: a F C:: 2-ope.n T sei: G and a u. p-HalL6doJt6o p-c.ompac.t ~pac.e. ~ gutaJt (and he.nc.e.~ by The.oJte.m 2.42, ~ p-noJtmal). COROLLARY 3.8. p-Jte.- A PJtooO' Use Theorem 3.7 (a). COROLLARY 3.9. A p-HalL6doJtoo p-Linde.ioo p-Jte.guiaJt (and he.nc.e., by The.oJte.m 2.42~ ~ p-~pac.e. p-noJtmal). ~ 53 P~oo6. Use Theorem COROLLARY ~pac.e.. I6 3.7 (a). Let (x;r1,L2) be. all U E L1-{x}, 3.10. 1 ¢ 6o~ Int2U a p-Ha£L6do~66 p-c.ompae.t then L1 C L2. P~o6. Let U E L1-{X}. Then X-U is a L1-closed set with Int2U 1 ~ . By Theorem 3.4 X-U is L1-compact. Hence, by Corollary 2.38, X-U is L2-closed, be. a p-Ha£L6do~6 6 p-c.ompae.t 6o~ all U EL1-{x}, and Int1V 1 ¢ 6o~ all COROLLARY ~pac.e.. v I6 i.e. U E L2. Let (X, 3.11. L l' L 2) 1¢ The.n. L1 = L2. Int2U 2-{xL « L P~oll' Use corollary It is interesting ed a theorem 3.10. to note that Cooke and Reilly [2, Theorem 4J for B-compact, spaces but did not get any analogous For this reason, Theorem Corollary p-~pac.e.. 6o~ all result for p-compact is an extension I6 1 ~ 6o~ all Int2U the.n. L1 2 1~. Let (X,L1,L2) 3.12. V E L -{X}, P~o6. Int2U = [2, U The.n. th~e CUte - F Midl . L2. Let (X, L l' Unde.lo 6 ~ pac.e.. ( X, L l)a 54 of the result Let U EL1-{xL Then X-U is a L1-closed set with By Lemma 3.3 X-U is Ll-Lindel~f. Hence by Corollary THEOREM 3. 13. ~et spaces. be. a p-Ha£L6do~66 p-Unde.e: L -{x}, and Int V 1- t 1 1 2.37, X-U is a L2-closed set, i.e. U EL2. ly we can prove T2 c T1· Thus T1 = T2. p e:: u. and bicompact 4] . COROLLARY 106 3.11 s-compact [2J obtain- 00 that P E: (\ Let u L i=l l C C L2. Similar- be. a p-Ha£L6do~6 6 ~ pac.e. an.d be. aLl -we.ak£y 0pe.n. ~ et: and L 2) -OPe.n. ~W 2 G. Thus L1 FeU. G i Ci E IN) and a 1-c.lMed T Pnno6. For each x E X-U there exist a T2-open set G(x) and a T1-open set H(x) such that x E H(x). p E G(x) and G(x) n H(x) = ~. Lindelof T1-open Since X-U is a 11-weakly space (X,T1), therefore X-U is 11-Lindelof. Thus the cover {H(x):x ~ X-U}oohas a countable subcover {H(x1),H(x2), ... L Let F = (lX-H(x.). Take G. = G( x . ). FeU. closed set in the l l Using a similar l=l ool Then p E (1 G. i=l l technique Then F is T -closed 1 c FeU. and as above we can prove the fol- lowing theorem. Let (X,T1,T2) be a p-HalL6doftnn .6pac.e and (X,1 ) a c.ompact .6pac.e. Let u be a T -open. sei: and p E u. Then 1 1 tit eJte a.Jte a T 2 - apen .6et: G an.d all - c.1.0.6 ed .6e;t F .6uc.h that THEOREM 3.14. pEG c FeU. COROLLARY p-.6pac.e, then Pftoo6. 3. 15. 1 = T2. Use Theorem In T 2) -i.J.> a Lin.de1..o6 p-HalL6doft6 n 1 3.13. Since every B-Lindelof have the following COROLLARY and UtheJt (x,« l' space is Lindelof, we corollary. 6 As a corollary 16 (X, 1 ,1 ) -i.J.> p-Ha.lL6doft66 P-.6pac.e 1 2 Oft s- Unde1..o 6, then. T 1 = 12, 3. 16. B- Lindelo to Theorem sult (see [2, Theorem COROLLARY (s-Lindel~f) re- 4J). 3. 17. B-c.ompact Oft s-c.ompact, 3.14 we have the following 16 then (x , T 1' T 2) -i.J.> p-HalL6doft6 nand 1 1 WheJt = T2. 55 4• Con c 1 us ion. generalizations As we noted of well-known tension of some theorems , our results classical in this paper are theorems as well as ex- in the literature. Naturally, any result stated in terms of T1 and T2 has a 'dual' in terms of T2 and T1. The definitions of separation and covering properties dorff and p-Lindelof, and covering of two topologies 1 and T2, such as p-Hausof course reduce to the usual separation properties T of one topology T , such as Hausdorff 1 when we take T1 = T2; and the theorems quoted above then yield as corollaries the classical results of which they are generalizations. As an example of theorems results are theorems Theorem 2.15, 2.33, 2.36, 2.42, 2.43, 2.44 and 2.46. 2.8 is an analogue 2.3 (a,c,d) is an analogue that Corollary 2.18J. to [2, Theorem to [2, Theorem 3.11 is an extension 3.7 (a) implies the results Theorem which yield well known classical 2]. We notice also of [2, Theorem 4J. Theorem in [4, Theorem 12 and 13J and [7, It is also clear that Corollary ogue to [7, Theorem 1J while Theorem 2.9J and Corollary 2.30 is an anal- 2.38 is an analogue [7, Lemma 2.11J. It is clear too that Theorems 2.8, 2.42 Corollary 2.20 imply the result to and in [6, Lemma 3.2]. 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Soc.. 32 (1957) 342-354. ~':~': Mathe.matiC6 Ve.paJttme.11t IGi.ng Saud Un.-iVV1.J.lfty, Abha, SAUVI ARABIA. Ve.paJttme.YLt06 MathematiC6 YaJrmO uk Un.-iVV1.J.lfty IJtbid, JORVANIA. (Recibido en abril de 1981). 57