More generalizations of pseudocompactness

MORE GENERA LI ZA TIONS OF PSEUDOCOMP A CTNESS P AOLO LIPP ARINI Abstract. W e int ro duce a cov er ing notion dep ending o n tw o car- dinals, which we call O -[ µ, λ ]-compactness, and which enco mpasses bo th pseudo compactness and man y other generalizatio ns o f pseu- do compactness. F or Tyc honoff spaces, pseudo co mpactness turns out to b e equiv a le nt to O -[ ω , ω ]-compac tness. W e provide several characterizations o f O -[ µ, λ ]-compactness, and we discuss its connection with D -pseudo compac tness, for D an ultrafilter. W e analyze the behaviour of t he ab ove notio ns with resp ect to pro ducts. Finally , we show that o ur res ults ho ld in a more gener al frame- work, in which c ompactness prop erties ar e defined relative to an arbitrar y family o f subsets of s ome top o logical space X . 1. Intr oduction As we ll-known, there are many equiv alen t refor m ulations of pseudo- compactness. See, e. g . [St]. V arious generalizations and extensions of pseudo compactness ha v e b een in tro duced b y man y authors; see, among others, [Ar, CoNe, F r, Ga, GiSa, Gl, Ke, Li4, Re, Sa, SaSt, ScSt, St, StV a]. W e in tro duce here some more pseudo compactness-lik e prop- erties, fo cusing mainly on notio ns related to cov ering prop erties and ultrafilter con v ergence. The most general form of our notion dep ends on t w o cardinals µ and λ ; w e call it O -[ µ, λ ]-compactness. It generalizes and unifies sev- eral pseudo compactness-lik e notions app eared b efor e. See Remark 2.3. 2010 Mathematics S ubje ct Classific ation. Primary 54 D20; Secondary 5 4 A20, 54B10 . Key wor ds and phr ases. Pseudo compa ctness, O - [ µ, λ ]-compactness, D - pseudo compactness, D -limits, in pro ducts, r egular ultrafilter , family of s ubsets of a top ological space. The author ha s r eceived supp or t from MPI a nd GNSAGA. W e wish to express our g r atitude to X. C a icedo a nd S. Ga rcia-F err eira for stimulating disc us sions and corres p o ndence. 1 2 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS In a sense, O -[ µ , λ ]-compactness is to pseudo compactness what [ µ, λ ]- compactness is to coun table compactness . See Remark 2.2. In par - ticular, for Ty chonoff spaces, O -[ ω , ω ]-compactness turns out to b e equiv alen t to pseudo compactness. W e find man y conditions equiv alent to O -[ µ , λ ]-compactness. In par- ticular, a characterization by means of ultrafilters, Theorem 3.2, pla ys an imp ortant role in this pap er. It provides a connection b etw een O - [ µ, λ ]-compactness and D -pseudo compactness, f o r D a ( µ, λ )-regular ultrafilter. The not ion of a ( µ, λ )-regular ultrafilter arose in a mo del- theoretical setting, a nd has pro v ed useful also in some areas of set- theory , and ev en in top olog y . See [Li3, Li2] for references. More sophisticated results are inv o lv ed when w e deal with pro ducts, since D - pseudo compactness is pro ductive , but O - [ µ , λ ]-compactness is not pro ductiv e, as w ell kno wn in the sp ecial case µ = λ = ω , that is, pseudo compactness. W e sho w that if D is a ( µ, λ ) - regular ultr a filter, then ev ery D -pseudo compact top ological space X is O -[ µ, λ ]-compact, hence all (Ty chonoff ) p ow ers of X are O - [ µ , λ ]-compact, to o (Corollary 3.7). The situation is in part parallel to the relationship b et w een the more classical notions of D -compactness a nd [ µ, λ ]-compactness. In this latter case, an equiv alence holds: all p ow ers of a top ological space X are [ µ, λ ]-compact if and o nly if there is some ( µ , λ )-regular ultr a- filter D suc h that X is D -compact. W e show tha t an ana logous result holds fo r D -pseudo compactness, pro vided w e deal with a notio n sligh tly stronger tha n O -[ µ, λ ]-compactness. See D efinition 4 .1 and Theorem 4.6. In particular, we pro vide a c haracterization o f those spaces whic h are D -pseudo compact, fo r some ( µ , λ )-regular ultrafilter D . In the final section of this note w e men tion that our results gener- alize to the abstract framew ork presen ted in [Li4]. That is, our pro ofs w ork essen tially unc hanged b oth for pseudo compactness-lik e notions and fo r the corresp onding compactness notions. In [Li4 ] each compact- ness prop ert y is defined relativ e t o a family F of subsets o f some top o- logical space X . The pseudo compactness case is obtained when F = O , the family of all nonempty o p en sets of X . Wh en F is the family of all singletons of X , we obtain results related to [ µ, λ ]-compactness. Our notation is fairly standard. Unless explicitly men tioned, we as- sume no separation axiom. Ho w ev er, the r eader is w arned that there are many conditions equiv a lent to pseudo compactness, but t he equiv- alence holds o nly assuming some separation axiom (t hey are all equiv- alen t o nly for Tyc hono ff spaces). F o r T yc honoff spaces, the particular case µ = λ = ω of our definitions of O - [ µ, λ ]-compactness (Definition 2.1) turns out to b e equiv alen t t o pseudo compactness, but this is not GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 3 necessarily the case for spaces with lo w er separation prop erties. See Remark 2.3. 2. A tw o cardinal generaliza tion of pseudocomp actness The following definition originally a pp eared in [Li4] in a more gen- eral framew ork. The letter O is in tended to denote the family of all the nonempt y o p en sets of some top ological space X . In this sense, the definition o f O - [ µ, λ ]-compactness is the particular case F = O of the definition of F -[ µ, λ ]-compactness in [Li4, D efinition 4.2]. See also Section 5. Definition 2.1. W e sa y that a top ological space X is O - [ µ, λ ]- c o m p act if and only if the follo wing holds. F o r ev ery sequence ( C α ) α ∈ λ of closed sets of X , if , for eve ry Z ⊆ λ with | Z | < µ , there exists a nonempt y op en set O Z of X s uc h that T α ∈ Z C α ⊇ O Z , t hen T α ∈ λ C α 6 = ∅ . Clearly , in the ab o ve definition, w e can equiv alen tly let O Z v ary among the (nonempty ) elemen ts of some base of X , rather than a mo ng all no nempt y o p en sets. Also, b y considering complemen ts, w e hav e that O -[ µ, λ ]-compactness is equiv alen t to the follo wing statemen t. F o r eve ry λ -indexed op en cov er ( Q α ) α ∈ λ of X , there exists Z ⊆ λ , with | Z | < µ , suc h that S α ∈ Z Q α is dense in X . R e m ark 2 .2 . The notion o f O -[ µ, λ ]-compactness should b e compared with the more classical notion of [ µ, λ ]-compactness. A t o p ological space X is [ µ, λ ]- c omp act if and only if, for ev ery se- quence ( C α ) α ∈ λ of closed sets of X , if T α ∈ Z C α 6 = ∅ , for ev ery Z ⊆ λ with | Z | < µ , then T α ∈ λ C α 6 = ∅ . Th us, in the definition of [ µ, λ ]-compactness, we require only the w eak er assumption that T α ∈ Z C α is nonempt y , for ev ery Z ⊆ λ with | Z | < µ , rather than requiring t ha t T α ∈ Z C α con tains some nonempt y op en set. In par t icular, ev ery [ µ, λ ]-compact space is O -[ µ, λ ]-compact. Th us, [ ω , ω ]-compactness is the same as coun table compactness, whic h is the analogue of pseudo compactness for O -[ µ, λ ]-compactness. Man y of the results presen ted here are vers ions for O -[ µ, λ ]-compactness of kno wn results ab o ut [ µ, λ ]-compactness . Indeed, a simu ltaneous metho d of pro of is a v ailable fo r b o th cases, and shall b e men tioned in Section 5. Notice that [ µ , λ ]-compactness is a notio n whic h encompasses b oth Lindel¨ ofness (more generally , κ -final compactness) a nd countable com- pactness (more generally , κ -initial compactness). See, e. g ., [Ca2, G´ a, Li1, Li2, V a] and references t here for further inf o rmation ab o ut [ µ, λ ]- compactness. 4 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS R e m ark 2.3 . F or Tyc honoff spaces, O - [ ω , ω ]-compactness is equiv a len t to pseudo compactness. Without assuming X to b e T yc honoff, O -[ ω , ω ]- compactness turns out to b e equiv alen t to a condition whic h is usually called fe eble c omp ac tnes s . See [Li4, Theorem 4.4(1 ) and Remark 4.5] and [St]. More generally , the particular case µ = ω of Definition 2.1, that is, O -[ ω , λ ]-compactness, has b een in tro duced and studied in [F r], where it is called almost λ -c omp actness . The notion of O -[ ω , λ ]-compactness has a lso b een studied , under differen t names, in [SaSt ], as we ak- λ - ℵ 0 - c omp actness , and in [Re, StV a] as we ak initial λ -c omp actness . Moreo v er, [F r ] in tro duced also a notion whic h corresp o nds to O - [ µ, λ ]-compactness for all cardinals λ , calling it almost µ -Lindel¨ ofness . Assuming that X is a T yc honoff space, a prop ert y equiv alen t to O -[ κ, κ ]-compactness, has b een in tro duced in [CoNe] under the name pseudo- ( κ, κ ) -c om p actness . See [Li4, Theorem 4 .4]. Definition 2.1 generalizes all the ab o v e men tioned notio ns. See [Ar, CoNe, F r, Ga, G iSa, Gl, Ke, Li4, Re, Sa , SaSt, ScSt, St, StV a] for the study of further related notions. F o r λ , µ infinite cardinals, S µ ( λ ) denotes the set of all subsets of λ of cardinalit y < µ . W e put λ <µ = sup µ ′ <µ λ µ ′ . Th us, λ <µ is the cardinalit y of S µ ( λ ). In the next prop o sition w e presen t some useful conditions equiv alen t to O -[ µ , λ ]-compactness. A further imp o rtan t c haracterization will b e presen ted in Theorem 3.2. Prop osition 2.4. F or every top olo g i c al sp ac e X and infinite c ar din als λ and µ , the fol lowi ng ar e e q uivalent. (1) X is O - [ µ , λ ] -c omp act. (2) F or every se quenc e ( P α ) α ∈ λ of subsets of X , if, for every Z ⊆ λ with | Z | < µ , ther e exists a nonempty op en set O Z of X such that T α ∈ Z P α ⊇ O Z , then T α ∈ λ P α 6 = ∅ . (3) F or every se quenc e ( Q α ) α ∈ λ of op en sets of X , if, for e v ery Z ⊆ λ w i th | Z | < µ , ther e exists a n o nempty op en set O Z of X such that T α ∈ Z Q α ⊇ O Z , then T α ∈ λ Q α 6 = ∅ . (4) F or eve ry se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of X , it happ ens that T α ∈ λ S { O Z | Z ∈ S µ ( λ ) , α ∈ Z } 6 = ∅ . (5) F or every se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of X , the fol lo wing holds. If, for every fi nite subset W of λ , we put Q W = S { O Z | Z ∈ S µ ( λ ) an d Z ⊇ W } , then T { Q W | W is a finite subse t of λ } 6 = ∅ . GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 5 (6) F or every se quenc e { C Z | Z ∈ S µ ( λ ) } of close d se ts of X , such that e ach C Z is pr op erly c on tain e d in X , if we let, for α ∈ λ , P α b e the interior of T { C Z | Z ∈ S µ ( λ ) , α ∈ Z } , then we have that ( P α ) α ∈ λ is not a c over of X . Pr o of. (1) ⇒ (2) Just tak e C α = P α , for α ∈ λ . (2) ⇒ (3) is trivial. (3) ⇒ (5) The sequence { Q W | W is a finite subset o f λ } is a se- quence of λ op en sets of X , since there are λ finite subsets of λ . F o r ev ery ν < µ , if ( W β ) β ∈ ν is a sequence of finite subsets of λ , then Z = S β ∈ ν W β has cardinality ≤ ν , and th us b elongs to S µ ( λ ). Moreo v er, f or eac h β ∈ ν , w e hav e tha t Z ⊇ W β , hence Q W β ⊇ O Z . This implies that T β ∈ ν Q W β ⊇ O Z . W e hav e prov ed that the sequence { Q W | W a finite subset of λ } is a sequence of λ op en sets of X suc h that the interse ction of < µ mem b ers of the sequence contains some nonempt y op en set of X . By applying (3) to this sequenc e, w e ha v e that T { Q W | W is a finite subset o f λ } 6 = ∅ . (5) ⇒ (4) is trivial. (4) ⇒ (1 ) Supp ose that ( C α ) α ∈ λ and O Z , for Z ⊆ λ with | Z | < µ , are as in the premise of the definition of O -[ µ, λ ]-compactness. F o r α ∈ λ , let C ′ α = S { O Z | Z ∈ S µ ( λ ) , α ∈ Z } . Since C α is closed, and C α ⊇ O Z whenev er α ∈ Z , we hav e that C α ⊇ C ′ α . By (4), T α ∈ λ C ′ α 6 = ∅ , hence also T α ∈ λ C α 6 = ∅ . Thus we hav e prov ed (1). W e shall also giv e a direct pro of of ( 3 ) ⇒ (4), since it is v ery simple. Giv en the sequence { O Z | Z ∈ S µ ( λ ) } then, for ev ery α ∈ λ , put Q α = S { O Z | Z ∈ S µ ( λ ) , α ∈ Z } . F or ev ery Z ∈ S µ ( λ ), and ev ery α ∈ Z , we ha v e that Q α ⊇ O Z . Hence, for ev ery Z ∈ S µ ( λ ), w e get T α ∈ Z Q α ⊇ O Z , so that w e can apply (3 ). (4) ⇔ (6) is immediate by taking complemen ts.  In the particular case when µ = λ is regular, there are many more conditions equiv alen t to O - [ λ, λ ]-compactness . Theorem 2.5. Supp ose that X is a top o l o gic al sp ac e, and λ is a r e gular c ar dinal. Then the fol low ing c onditions a r e e quivalent. (a) X is O - [ λ, λ ] -c omp act. (b) Supp ose that ( C α ) α ∈ λ is a se quenc e of close d sets of X such that C α ⊇ C β , whenever α ≤ β < λ . If, for every α ∈ λ , ther e exists a nonempt op en set O of X such that C α ⊇ O , then T α ∈ λ C α 6 = ∅ . (c) Supp ose that ( C α ) α ∈ λ is a se quenc e of close d sets of X such that C α ⊇ C β , when ever α ≤ β < λ . S upp o se further that, for every α ∈ λ , C α is the closur e of some op en set of X . I f, for every α ∈ λ , ther e exists a non e mpt op en set O of X such that C α ⊇ O , then T α ∈ λ C α 6 = ∅ . 6 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS (d) F or every se quenc e ( O α ) α ∈ λ of n onempty op en sets of X , ther e exists x ∈ X such that |{ α ∈ λ | U ∩ O α 6 = ∅}| = λ , for every neigh b or- ho o d U of x in X . (e) F or ev ery se quenc e ( O α ) α ∈ λ of n onempty op en sets of X , ther e exists some ultr afi lter D uniform over λ such that ( O α ) α ∈ λ has a D - limit p oint (se e De fi nition 3. 1). (f ) F or every λ - indexe d op en c over ( O α ) α ∈ λ of X , such that O α ⊆ O β whenever α ≤ β < λ , ther e exists α ∈ λ such that O α is den s e in X . In al l the ab ove statements we c an e quivalently r e quir e that the ele - ments of the se quenc e ( C α ) α ∈ λ , r esp e ctively, ( O α ) α ∈ λ , ar e al l dis tinc t. Pr o of. By [L i4 , Theorem 4.4], taking F there to b e t he family O of all the nonempt y op en sets o f X . Since λ is regular, the last statemen t is trivial, as fa r as conditions (b), (c) and (f ) are concerned. It f ollo ws fro m [Li4 , Prop o sition 3.3(a)] in case (d). Then apply [Li4, Prop osition 4.1 ] in order to get (e).  R e m ark 2.6 . A t this p oin t, w e should men tion a significan t difference b et w een O -[ µ, λ ]-compactness and [ µ, λ ]-compactness . It is true that a top ological space is [ µ, λ ]-compact if and only if it is [ κ, κ ]-compact, for ev ery κ suc h that µ ≤ κ ≤ λ . Though simple, the ab ov e equiv alence has prov ed v ery useful in man y circumstances. See, e. g ., [Li2]. It is trivial that ev ery O - [ µ, λ ]-compact space is O -[ µ ′ , λ ′ ]-compact, whenev er µ ≤ µ ′ ≤ λ ′ ≤ λ . In particular, ev ery O -[ µ, λ ]-compact space is O -[ κ, κ ]-compact, fo r ev ery κ suc h that µ ≤ κ ≤ λ . On the con tra r y , the condition o f b eing O -[ κ, κ ]-compact, for ev ery κ suc h that µ ≤ κ ≤ λ , is not alw ays a sufficien t condition in order to get O -[ µ, λ ]-compactness. See R emark 4.13. This fact limits the usefulness of Theorem 2.5 in the presen t contex t. 3. A characteriza tion by means of ul trafil ters The first theorem in this section, Theorem 3.2, furnishes a c harac- terization of O -[ µ, λ ]-compactness by means of t he existence of D - limit p oin ts of ultrafilters. T his c haracterization is the k ey f or the study o f the connections b et we en O - [ µ, λ ]-compactness and D -pseudo compact- ness, for D a ( µ, λ )-regular ultrafilter and shall b e used in the next section in connection with prop erties o f pro ducts. Definition 3.1. Suppo se that D is an ultrafilter o v er some set I , and X is a top o lo gical space. If ( Y i ) i ∈ I is a sequence of subsets of X , then x ∈ X is called a D - limit p oint of ( Y i ) i ∈ I if and only if { i ∈ I | Y i ∩ U 6 = ∅} ∈ D , for eve ry neighborho o d U of x in X . The notion of a D - limit GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 7 p oin t is due to [GiSa, Definition 4.1] f or non-principal ultrafilters ov er ω , and app ears in [Ga] for uniform ultrafilters o v er arbitra ry cardinals. W e sa y t ha t an ultrafilter D o v er S µ ( λ ) c overs λ if and only if, for ev ery α ∈ λ , it happ ens that { Z ∈ S µ ( λ ) | α ∈ Z } ∈ D . This notion is connected with ( µ, λ )-regularit y , a s we shall see in Definition 3.5. Theorem 3.2. F or every top olo gic al sp ac e X and infinite c ar din a ls λ and µ , the fol lowing ar e e quivalent. (1) X is O - [ µ , λ ] -c omp act. (2) F or eve ry se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of X , ther e exists an ultr afilter D over S µ ( λ ) which c overs λ and such that { O Z | Z ∈ S µ ( λ ) } has a D -lim i t p oint. Pr o of. (1) ⇒ (2) Supp ose that { O Z | Z ∈ S µ ( λ ) } is a sequence of nonempt y op en sets of X . F or ev ery finite subset W of λ , let Q W = S { O Z | Z ∈ S µ ( λ ) and Z ⊇ W } . By O - [ µ, λ ]-compactness, and Condi- tion (5) in Prop osition 2.4, w e ha ve that T { Q W | W a finite subset of λ } 6 = ∅ . Supp ose t hat x ∈ T { Q W | W a finite subset of λ } . F o r ev ery ne ighborho o d U of x in X , let A U = { Z ∈ S µ ( λ ) | U ∩ O Z 6 = ∅} . F or ev ery α ∈ λ , let [ α ) = { Z ∈ S µ ( λ ) | α ∈ Z } . W e are going to show that the f amily A = { [ α ) | α ∈ λ } ∪ { A U | U a neigh b orho o d of x in X } has the finite in tersection prop ert y . Indeed, let U 1 , . . . , U n b e neigh b orho o ds of x , and α 1 , . . . , α m b e elemen ts of λ . Let U = U 1 ∩ · · · ∩ U n , W = { α 1 , . . . , α m } , and [ W ) = [ α 1 ) ∩ · · · ∩ [ α m ) = { Z ∈ S µ ( λ ) | Z ⊇ W } . Since x ∈ Q W , w e get tha t U ∩ Q W 6 = ∅ , that is, U ∩ O Z 6 = ∅ , for some Z ∈ S µ ( λ ) with Z ⊇ W . Hence Z ∈ A U , and also Z ∈ A U 1 , . . . , Z ∈ A U n , since U 1 ⊇ U , . . . , U n ⊇ U . In conclusion, Z ∈ A U 1 ∩ · · · ∩ A U n ∩ [ α 1 ) ∩ · · · ∩ [ α m ), hence the ab ov e intersec tion is not empt y . W e hav e show ed that A has the finite in tersection prop ert y . Hence A can b e extended to some ultra filt er D ov er S µ ( λ ). By construction, [ α ) ∈ D , for ev ery α ∈ λ , hence D co v ers λ . Again by construction, A U ∈ D , fo r ev ery neighborho o d U o f x in X , and this means exactly that x is a D -limit p o in t of { O Z | Z ∈ S µ ( λ ) } . Th us, (2) is pro v ed. In order to prov e (2) ⇒ (1), it is sufficien t to pro v e that (2) implies Condition (4) in Prop osition 2.4. Let { O Z | Z ∈ S µ ( λ ) } b e a sequence of nonempt y op en sets of X . Letting C α = S { O Z | Z ∈ S µ ( λ ) , α ∈ Z } , for α ∈ λ , we need to o sho w that T α ∈ λ C α 6 = ∅ . Let D b e an ultrafilter as giv en by (2), a nd supp ose t ha t x is a D - limit p oin t of { O Z | Z ∈ S µ ( λ ) } . W e are going to sho w that x ∈ T α ∈ λ C α . Supp ose b y contradic- tion that, for some α ∈ λ , it happ ens that x 6∈ C α . Since C α is closed, x has some neighborho o d U disjoint from C α . Not ice that, if Z ∈ S µ ( λ ) 8 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS and α ∈ Z , then C α ⊇ O Z . Hence { Z ∈ S µ ( λ ) | U ∩ O Z 6 = ∅} ∩ [ α ) = ∅ , hence { Z ∈ S µ ( λ ) | U ∩ O Z 6 = ∅} 6∈ D , since D is an ultra filter, and [ α ) ∈ D b y assumption, since D is supp osed to cov er λ . But { Z ∈ S µ ( λ ) | U ∩ O Z 6 = ∅} 6∈ D con tradicts the assumption that x is a D -limit p oin t of { O Z | Z ∈ S µ ( λ ) } . Hence x ∈ T α ∈ λ C α , th us T α ∈ λ C α 6 = ∅ , and the pro of is complete.  R e m ark 3.3 . Theorem 3.2 is inspired by results b y X. Caicedo f rom his seminal pap er [Ca2]. See also [Ca1]. Caicedo prov ed results similar to Theorem 3.2 for [ µ, λ ]-compactness . The result a na logous to the implication (1) ⇒ (2) in Theorem 3.2 is Lemma 3.3 (i) in [Ca2]. A common generalization and strengthening of b oth Theorem 3 .2 and [Ca2, Lemmata 3.1 and 3.2] holds. See Theorem 5.2 (1) ⇒ (7) b elo w. Notice t ha t, b ecause of the w ell kno wn result ab out [ µ , λ ]-compact- ness mentioned in Remark 2.6, essen tially all applications of results in [Ca2] can b e obtained using only the particular case λ = µ of [Ca2, Lemmata 3.1 a nd 3.2]. Ho we ve r, suc h a reduction is not p ossible in the case of O -[ µ, λ ]-compactness, b y Remark 4 .1 3. Hence it is necessary to deal with the more general case in whic h λ 6 = µ is allo w ed. The idea from [Ca1, Ca2] of treating the full general case is th us w ell-justified Definition 3.4. If D is an ultrafilter ov er I , then a top ological space X is said to b e D - pseudo c om p act ([GiSa, Ga]) if and only if ev ery sequence ( O i ) i ∈ I of nonempt y op en subsets o f X has some D -limit p oint in X . Definition 3.5. An ultr afilter D o v er some set I is said to b e ( µ, λ ) - r e gular if and only if there is a function f : I → S µ ( λ ) suc h that { i ∈ I | α ∈ f ( i ) } ∈ D , for ev ery α ∈ λ . See , e. g., [Li3] for equiv alen t definitions and for a surv ey of results on ( µ , λ )-regular ultrafilters. If D is a n ultrafilter ov er I , and f : I → J is a function, the ultrafilter f ( D ) ov er J is defined by the following clause: Z ∈ f ( D ) if and only if f − 1 ( Z ) ∈ D . With the a b ov e notat ion, it is trivial to see that D o ve r I is ( µ, λ )- regular if and only if there exists some f unction f : I → S µ ( λ ) suc h that f ( D ) co v ers λ . In passing, let us men tion that the ab ov e definitions in v olve the so- called Rudin-Keisler order. If D a nd E are tw o ultrafilters, resp ectiv ely o v er I a nd J , then E is said to b e less than or equal to D in the Rudin- Keisler (pre-) or der, E ≤ RK D fo r short, if and only if there exis ts some function f : I → J suc h that E = f ( D ) . If b oth E ≤ RK D and D ≤ RK E , then E and D a r e said to b e (Rudin-Keisler) e quivalent . The next fact is trivial, but ve ry useful. GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 9 F act 3.6. If D is an ultr afilter over I , X is a D -pse udo c omp act top ol o g- ic al sp ac e, and f : I → J is a function, then X i s f ( D ) -pseudo c omp act. Corollary 3.7. Supp ose that D is a ( µ , λ ) -r e gular ultr afilter. If X is a D -pseudo c omp act top olo gic al sp ac e, then X is O - [ µ, λ ] - c omp act. Mor e gener al ly, if ( X j ) j ∈ J is a se quenc e of D -pseudo c omp a ct top o- lo gic al sp ac es, then the T ychonoff pr o duct Q j ∈ J X j is O - [ µ, λ ] -c omp act. Pr o of. By ( µ, λ )-regularit y , there is f : I → S µ ( λ ) suc h that f ( D ) co v ers λ . By F act 3.6, X is f ( D )-pseudo compact, hence O -[ µ , λ ]- compactness of X follows from Theorem 3 .2 with f ( D ) in place of D . Notice that here f ( D ) w orks “uniformly” for ev ery sequence , while, in the statemen t of Theorem 3.2(2), the ultrafilter, in general, dep ends on the sequen ce. The last statemen t follows from the kno wn fact ([GiSa, Theorem 4.3]) that D -pseudo compactness is preserv ed under taking pro ducts.  A result analogous to Corollar y 3.7 for [ µ, λ ]-compactness is prov ed in [Ca2 , Lemma 3.1]. W e now presen t a nice c haracterization of D -pseudo compactness. Theorem 3.8. Supp ose that D is an ultr afilter over some set I , and X is a top olo gic al sp a c e. Then the fol lowing ar e e quivalent. (1) X is D - p seudo c omp act. (2) F or every se quenc e { O i | i ∈ I } of no n empty op en sets of X , if, for Z ∈ D , we put C Z = S i ∈ Z O i , then we have that T Z ∈ D C Z 6 = ∅ . (3) Whenever ( C Z ) Z ∈ D is a se quenc e of clo se d sets of X with the pr op erty that, for ev ery i ∈ I , T i ∈ Z C Z c ontains some nonempty op en set of X , then T Z ∈ D C Z 6 = ∅ . (4) F or every op en c over ( Q Z ) Z ∈ D of X , ther e is some i ∈ I such that S i ∈ Z Q Z is den s e in X . (5) F or every se quenc e { C i | i ∈ I } o f close d sets of X , such that e ach C i is p r o p erly c ontaine d in X , if, for Z ∈ D , we le t Q Z b e the interior of T i ∈ Z C i , then we have that ( Q Z ) Z ∈ D is not a c over of X . Pr o of. (1) ⇒ (2 ) By D - pseudo compactness, the sequence { O i | i ∈ I } has some D -limit p oint x in X , that is, { i ∈ I | U ∩ O i 6 = ∅} ∈ D , for ev ery neigh b o rho o d U of x in X . W e are going to sho w that x ∈ T Z ∈ D C Z . Indeed, let Z b e any set in D . If U is a neighborho o d of x , then Z ′ = Z ∩ { i ∈ I | U ∩ O i 6 = ∅} is still in D , th us is nonempt y . Let i ∈ Z ′ . Then U ∩ O i 6 = ∅ , and 10 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS C Z ⊇ O i , since i ∈ Z . Hence U ∩ C Z 6 = ∅ . Since the ab ov e arg ument w orks f or ev ery neigh b o rho o d U of x , w e hav e that x ∈ C Z , sinc e C Z is a closed set. W e hav e sho w ed that x ∈ C Z , for ev ery Z ∈ D , hence x ∈ T Z ∈ D C Z . (2) ⇒ (3) F or ev ery i ∈ I , let O i b e some nonempty op en set of X suc h that T i ∈ Z C Z ⊇ O i . F or ev ery Z ∈ D , put C ′ Z = S i ∈ Z O i . By Clause (2), w e ha v e t ha t T Z ∈ D C ′ Z 6 = ∅ . Since, for ev ery i ∈ Z , C Z ⊇ O i , w e ha v e that C Z ⊇ C ′ Z , for ev ery Z ∈ D . Hence, T Z ∈ D C Z ⊇ T Z ∈ D C ′ Z 6 = ∅ . (3) ⇒ (1) Supp ose that ( O i ) i ∈ I is a sequence of nonempt y op en sets of X . F or Z ∈ D , let C Z = S i ∈ Z O i . Henc e, for ev ery i ∈ Z , C Z ⊇ O i , and, for ev ery i ∈ I , T i ∈ Z C Z con tains the nonempt y op en set O i . By (3), t here is some x ∈ X suc h that x ∈ T Z ∈ D C Z . It is enough to sho w that x is a D - limit p oin t of ( O i ) i ∈ I . If not, x has some neigh b orho o d U suc h that { i ∈ I | U ∩ O i 6 = ∅} 6∈ D , that is , { i ∈ I | U ∩ O i = ∅} ∈ D . Letting Z = { i ∈ I | U ∩ O i = ∅} , w e hav e that U ∩ S i ∈ Z O i = ∅ , but this con tradicts x ∈ C Z = S i ∈ Z O i . (3) ⇔ (4) and (2) ⇔ (5 ) are obta ined by considering complemen ts.  4. Theorems about pr oducts In this section w e consider, f or a pro duct space Q j ∈ J X j , a v arian t of O -[ µ, λ ]-compactness, a v ariant whic h tak es in to accoun t all the op en sets in the b o x top ology on the set Q j ∈ J X j . This notion shall b e used in order to prov ide a c haracterization of a ll those spaces X whic h are D -pseudo compact, for some ( µ, λ )-regular ultrafilter D (Theorem 4.6). W e shall need to consider the se t Q j ∈ J X j endo w ed b oth with the T yc honoff top ology and with the b ox to p ology . A base for the latt er top ology is given b y al l the pro ducts Q j ∈ J O j , eac h O j b eing an op en set of X j . When w e write Q j ∈ J X j , w e shall alw ays assume that the pro duct is endo w ed with t he T yc honoff top o lo gy , while ✷ j ∈ J X j shall denote the pro duct endo we d with the b o x top o logy . Definition 4.1. Supp ose that ( X j ) j ∈ J is a sequence o f to p ological spaces. W e sa y that the top olo gical space Q j ∈ J X j is O ✷ -[ µ, λ ]- c omp act if and only if the follo wing holds. F o r ev ery sequenc e ( C α ) α ∈ λ of closed sets of Q j ∈ J X j , if, for eve ry Z ⊆ λ with | Z | < µ , there exists a nonempt y op en set O Z of ✷ j ∈ J X j suc h that T α ∈ Z C α ⊇ O Z , then T α ∈ λ C α 6 = ∅ . GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 11 Notice that O ✷ -[ µ, λ ]-compactness is a notion stronger than O - [ µ, λ ]- compactness, that is, ev ery O ✷ -[ µ, λ ]-compact pro duct Q j ∈ J X j is O - [ µ, λ ]-compact. The t w o notions are distinct, in general, as we shall see in Remark 4.8. Notice also that ev ery [ µ, λ ]-compact pro duct is O ✷ -[ µ, λ ]-compact. R e m ark 4.2 . Notice that O ✷ -[ µ, λ ]-compactness is not an intrinsic prop- ert y o f the top olo gical space Y = Q j ∈ J X j . That is, O ✷ -[ µ, λ ]-compact- ness do es not only dep end on the top o lo gy on Y , but dep ends also on the w a y Y is realized as a pro duct. There might b e tw o homeomorphic spaces, say , Y = Q j ∈ J X j and Z = Q h ∈ H Y h suc h that Y , as a pr o duct Q j ∈ J X j , is O ✷ -[ µ, λ ]-compact, while Z , as a pr o duct Q h ∈ H Y h , is no t . Just to consider a simple case, if Y = Q j ∈ J X j , a nd Z is a homeomor- phic cop y of Y , and we consider Z “as itself ”, that is, as the pro duct of just a single factor, then Z is O ✷ -[ µ, λ ]-compact if and only if it is O - [ µ, λ ]-compact. On the con trary , as we shall see, O ✷ -[ µ, λ ]-compactness and O - [ µ, λ ]-compactness are distinct notions, in general. The ab o v e remark will cause no problem here, since w e will alwa ys b e dealing with a space Y = Q j ∈ J X j together with just one single realization of Y as Q j ∈ J X j . In other w ords, w e shall nev er deal with the homeomorphism equiv alence class of Y , but w e shall alw ays deal with Y = Q j ∈ J X j just in its concrete realization. Of course, O ✷ -[ µ, λ ]-compactness can b e characterized in a w a y sim- ilar to the c haracterizations of O - [ µ, λ ]-compactness giv en in Propo- sition 2.4 . Clause (7) in the next pro p osition is prov ed as the last statemen t of Definition 2 .1. Prop osition 4.3. F or every se quenc e ( X j ) j ∈ J of top olo gic al sp ac es, and λ , µ infinite c ar dinals, the fol lowing ar e e quivalent, whe r e , in i tems (2)-(5), clos ur es ar e c ompute d in Q j ∈ J X j . (1) Q j ∈ J X j is O ✷ - [ µ, λ ] -c omp act. (2) F or every se quenc e ( P α ) α ∈ λ of subsets of Q j ∈ J X j , if, for every Z ⊆ λ with | Z | < µ , ther e exists a no n empty op en set O Z of ✷ j ∈ J X j such that T α ∈ Z P α ⊇ O Z , then T α ∈ λ P α 6 = ∅ . (3) F or every se quenc e ( Q α ) α ∈ λ of o p en sets of ✷ j ∈ J X j , if, for every Z ⊆ λ with | Z | < µ , ther e exists a no n empty op en set O Z of ✷ j ∈ J X j such that T α ∈ Z Q α ⊇ O Z , then T α ∈ λ Q α 6 = ∅ . (4) F or eve ry se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of ✷ j ∈ J X j , it happ ens that T α ∈ λ S { O Z | Z ∈ S µ ( λ ) , α ∈ Z } 6 = ∅ . (5) F or every se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of ✷ j ∈ J X j , the fol lowin g holds. If, for every finite subset W 12 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS of λ , we put Q W = S { O Z | Z ∈ S µ ( λ ) and Z ⊇ W } , then T { Q W | W is a finite subset of λ } 6 = ∅ . (6) F or every se quenc e { C Z | Z ∈ S µ ( λ ) } of close d sets of ✷ j ∈ J X j , such that e a c h C Z is pr op erly c ontaine d in X , if we let, for α ∈ λ , P α b e the interior (c ompute d in Q j ∈ J X j ) of T { C Z | Z ∈ S µ ( λ ) , α ∈ Z } , then w e have that ( P α ) α ∈ λ is not a c over of X . (7) F or every λ -indexe d op en c over ( Q α ) α ∈ λ of Q j ∈ J X j , ther e exists Z ⊆ λ , with | Z | < µ , such that S α ∈ Z Q α is a dense s ubse t in ✷ j ∈ J X j . The pro of of Theorem 3.2 carries ov er essen tially unc hanged in order to get the f ollo wing useful t heorem. Theorem 4.4. F or every se quen c e ( X j ) j ∈ J of top olo gic al sp ac es, an d λ , µ i n finite c ar dinals, the f o l lowing ar e e quivalent. (1) Q j ∈ J X j is O ✷ - [ µ, λ ] -c omp act. (2) F or eve ry se quenc e { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of ✷ j ∈ J X j , ther e exists an ultr afilter D ove r S µ ( λ ) which c overs λ and such that { O Z | Z ∈ S µ ( λ ) } has a D - l i m it p oint in Q j ∈ J X j . Theorem 4.4 can b e used to improv e the last statemen t in Corollary 3.7. Corollary 4.5. Supp ose that D is a ( µ , λ ) -r e gular ultr afilter. If ( X j ) j ∈ J is a se quenc e of D -pseudo c o mp act top olo gic al s p ac es, then Q j ∈ J X j is O ✷ - [ µ, λ ] -c omp act. W e are no w going to sho w that a top ological space X is D - pseu- do compact fo r some ( µ , λ )-regular ultrafilter D if and only if all (T y- c honoff ) p ow ers of X are O ✷ -[ µ, λ ]-compact. W e shall denote by X δ the T yc honoff pro duct of δ -man y copies o f X . Theorem 4.6. F or every top olo g i c al sp ac e X , and λ , µ infinite c ar di- nals, the fol lowing ar e e quivalent. (1) Ther e exis ts some ultr afilter D over S µ ( λ ) wh i c h c overs λ , an d such that X is D -pseudo c omp act. (2) Ther e e x ists so me ( µ, λ ) -r e gular ultr afilter D (over any set) such that X is D - pseudo c omp act. (3) Ther e exists some ( µ , λ ) -r e gular ultr afilter D such that, for ev- ery c a r d i nal δ , the sp ac e X δ is D -pseudo c omp act. (4) The p ower X δ is O ✷ - [ µ, λ ] -c omp act, for every c ar d i n al δ . (5) The p ower X δ is O ✷ - [ µ, λ ] -c omp act, for δ = min { 2 2 κ , ( w ( X )) κ } , wher e κ = λ <µ . GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 13 Pr o of. (1) ⇒ (2 ) is trivial, since if D is ov er S µ ( λ ) and co ve rs λ , then D is ( µ, λ )-regular. (2) ⇒ (3) follow s from the men tioned result from [GiSa, Theorem 4.3], asserting that a pro duct of D -pseudo compact space s is still D - pseudo compact. (3) ⇒ (4) follow s fro m Corollary 4 .5 . (4) ⇒ (5) is trivial. (5) ⇒ (1) W e first consider the case δ = ( w ( X )) κ . Let B b e a base of X of cardinality w ( X ). Th us, there are δ -many S µ ( λ )-indexed sequenc es of elemen ts of B , sinc e | S µ ( λ ) | = κ . Let us en umerate them as { Q β ,Z | Z ∈ S µ ( λ ) } , β v arying in δ . In X δ consider the sequence { Q β ∈ δ Q β ,Z | Z ∈ S µ ( λ ) } . F or ev ery Z ∈ S µ ( λ ), the set Q β ∈ δ Q β ,Z is op en in the b ox top olog y on X δ . By the O ✷ -[ µ, λ ]- compactness of X δ , and by Theorem 4.4 ( 1 ) ⇒ (2) , there exists an ultrafilter D ov er S µ ( λ ) whic h co ve rs λ and suc h that { Q β ∈ δ Q β ,Z | Z ∈ S µ ( λ ) } has some D -limit p o in t x in X δ . W e a re going to sho w tha t X is D - pseudo compact. So, let { O Z | Z ∈ S µ ( λ ) } b e a sequence of nonempt y op en sets of X . Since B is a base for X , then, for eve ry Z ∈ S µ ( λ ), there is a nonempt y B Z in B suc h that O Z ⊇ B Z . Cho ose one suc h B Z for each Z ∈ S µ ( λ ). The sequence { B Z | Z ∈ S µ ( λ ) } is a n S µ ( λ )-indexed sequences of elemen ts of B . Since, b y construction, all suc h sequence s are en umerated b y { Q β ,Z | Z ∈ S µ ( λ ) } , there is some β 0 ∈ δ suc h tha t B Z = Q β 0 ,Z , fo r ev ery Z ∈ S µ ( λ ). By what we hav e prov ed b efore, the sequence { Q β ∈ δ Q β ,Z | Z ∈ S µ ( λ ) } has some D -limit p oin t x in X δ , sa y x = ( x β ) β ∈ δ . A trivial prop erty of D -limits implies that, for ev ery β ∈ δ , w e hav e that x β is a D -limit of { Q β ,Z | Z ∈ S µ ( λ ) } . In particular, b y taking β = β 0 , w e get that x β 0 is a D -limit p oint of { B Z | Z ∈ S µ ( λ ) } . Since O Z ⊇ B Z , fo r ev ery Z ∈ S µ ( λ ), w e g et that x β 0 is also a D - limit p o in t of { O Z | Z ∈ S µ ( λ ) } . W e ha v e prov ed that ev ery sequence { O Z | Z ∈ S µ ( λ ) } of nonempty op en sets of X has some D -limit p oin t in X , that is, X is D -pseudo compact. No w w e consider the case δ = 2 2 κ . W e shall prov e that if δ = 2 2 κ and (1 ) f a ils, then (5) fails. If (1) fails, t hen, for ev ery ultra filter D o v er S µ ( λ ) whic h co v ers λ , there is a sequence { O Z | Z ∈ S µ ( λ ) } of nonempt y op en sets of X whic h has no D -limit p oint. Since there a re δ - man y ultrafilters ov er S µ ( λ ), w e can en umerate the ab ov e sequences as { O β ,Z | Z ∈ S µ ( λ ) } , β v arying in δ . Now, giv en an y ultrafilter D ov er S µ ( λ ) a nd co v ering λ , it is not the case that the sequenc e { Q β ∈ δ O β ,Z | Z ∈ S µ ( λ ) } has some D - limit p oint. Indeed, w ere x = 14 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS ( x β ) β ∈ δ a D -limit p oint of { Q β ∈ δ O β ,Z | Z ∈ S µ ( λ ) } , then, by a trivial prop erty of D - limits, f or ev ery β ∈ δ , x β w ould b e a D - limit p o int of { O β ,Z | Z ∈ S µ ( λ ) } . This is a con tradiction since, by construction, for ev ery ultrafilter D o v er S µ ( λ ) cov ering λ , there exists some β ∈ δ suc h that { O β ,Z | Z ∈ S µ ( λ ) } has no D -limit p o int. W e hav e sho w ed that for no ultrafilter D ov er S µ ( λ ) and cov ering λ the sequence { Q β ∈ δ O β ,Z | Z ∈ S µ ( λ ) } ha s some D -limit p oint. Since, for ev ery Z ∈ S µ ( λ ), Q β ∈ δ O β ,Z is an op en set of the b o x top o lo gy on X δ , w e get that, by Theorem 4.4 ( 1 ) ⇒ (2), X δ is not O ✷ -[ µ, λ ]- compact, that is, (5) fails.  R e m ark 4.7 . Condition (5) in Theorem 4.6 can b e impro v ed to the effect that w e can take κ t here to b e equal to the cofinalit y of the partial order S µ ( λ ). A subset H of S µ ( λ ) is said to b e c ofinal in S µ ( λ ) if and only if , for ev ery Z ∈ S µ ( λ ), there is Z ′ ∈ H suc h that Z ⊆ Z ′ . The c ofinality cf S µ ( λ ) of S µ ( λ ) is the minimal cardinality of some subse t H cofinal in S µ ( λ ). Notice that if λ is regular, then cf S λ ( λ ) = λ and, more generally , cf S λ ( λ + ) = λ + . Highly non trivial results ab o ut cf S µ ( λ ) a re consequenc es o f Shelah’s p cf-theory [Sh]. F o r the rest of this remark, let us fix some subset H cofinal in S µ ( λ ). All the definitions and results inv olving S µ ( λ ) can be mo dified in order to a pply to H , to o. In particular, in the definitions of O -[ µ , λ ]- compactness and o f O ✷ -[ µ, λ ]-compactness, w e get an equiv alen t notion if we consider only those Z ∈ H . Similarly , in Prop ositions 2.4 a nd 4.3 w e can equiv alently consider H -indexed sequences, rat her t han S µ ( λ )- indexed sequences, that is, w e can replace ev erywhere Z ∈ S µ ( λ ) by Z ∈ H , still obtaining the results. Moreo v er, we can say that an ultrafilter D o ve r H cov ers λ if and only if, for ev ery α ∈ λ , it happ ens that [ α ) H = { Z ∈ H | α ∈ Z } ∈ D . With this definition, w e ha ve that Theorems 3.2 and 4.4, to o , hold, if Z ∈ S µ ( λ ) is ev erywhere replaced by Z ∈ H . Moreo v er, let f : S µ ( λ ) → H b e defined in suc h a wa y tha t Z ⊆ f ( Z ). If D is ov er S µ ( λ ) a nd co ve rs λ , then f ( D ) is o v er (a subset of ) H , and f ( D ), to o, co v ers λ . The a b o ve observ ations give us the p ossibilit y of pro ving Theorem 4 .6 with the impro v ed v alue κ = cf S µ ( λ ) in Condition (5). R e m ark 4.8 . In order t o g et results like Theorem 4.6, it is actually nec- essary to deal with O ✷ -[ µ, λ ]-compactness, rather than with O -[ µ, λ ]- compactness. Indeed, [GiSa, Example 4.4 ] constructed a T yc honoff space X suc h that all p ow ers of X ar e pseudo compact but for no ul- trafilter D uniform ov er ω , X is D -pseudo compact. By Remark 2.3, GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 15 all pow ers of X are O -[ ω , ω ]- compact. The condition that, for no ul- trafilter D unifo rm ov er ω , X is D -pseudo compact is easily seen t o b e equiv alent to the prop ert y t ha t for no ultr a filter D ov er S ω ( ω ) and co v ering ω , X is D - pseudo compact. The equiv a lence can b e pro ve d di- rectly; otherwise, notice that, for µ = λ a regular cardinal, Condition (4) in Theorem 4.6 coincides with Condition (5) in [Li4, Corollary 5.5 ], hence the resp ectiv e Conditio ns (1 ) are equiv alen t. Since, for no ultr a filter D o v er S ω ( ω ) and co ve ring ω , X is D - pseudo compact, we get, by Theorem 4.6, that not ev ery p o w er of X is O ✷ -[ ω , ω ]-compact, but, as w e remark ed, ev ery p o w er o f X is O - [ ω , ω ]-compact, th us the tw o notions are distinct, in general. Indeed, b y Remark 4.7, w e ha v e that X δ is not O ✷ -[ ω , ω ]-compact for δ = 2 2 ω . In particular, Conditions (4) and (5 ) in Theorem 4 .6 ar e in gen- eral not equiv alen t to t he o ther conditions, if we replace O ✷ -[ µ, λ ]- compactness with O -[ µ, λ ]-compactness. Indeed, as is the case for pseudo compactness, w e can show that the O -[ µ , λ ]-compactness of a pro duct dep ends only o n the O -[ µ, λ ]- compactness o f a ll subpro ducts of some small num ber of factors. Th us, w e hav e an analogue for O -[ µ, λ ]-compactness of the equiv alence (4) ⇔ (5) in Theorem 4.6. Lemma 4.9. If X an d Y ar e top o l o gic al sp ac es, f : X → Y is a c ontinuous and surje ctive function, and X is O - [ µ, λ ] -c o m p act then also Y is O - [ µ, λ ] -c omp act. Theorem 4.10. Supp ose that ( X j ) j ∈ J is a se quenc e of top o l o gic al sp ac es. Then the pr o duct Q j ∈ J X j is O - [ µ, λ ] -c o m p act if and only if any sub- pr o duct of ≤ κ factors is O - [ µ, λ ] -c omp act, w her e κ = λ <µ . Inde e d, the r e s ult c an b e impr ove d to κ = cf S µ ( λ ) . Pr o of. The only-if part is immediate f rom Lemma 4.9. T o prov e the con v erse, giv en ( C α ) α ∈ λ as in the definition of O -[ µ, λ ]- compactness, w e might assume, without loss of generalit y , that the O Z ’s are members of the canonical base o f Q j ∈ J X j , that is, eac h O Z has the f orm Q j ∈ J Q j , where eac h Q j is an op en set of X j , and Q j = X j , for all j ∈ J \ J Z , f or some finite J Z ⊆ J . If J ′ = S Z ∈ S µ ( λ ) J Z , and π : Q j ∈ J X j → Q j ∈ J ′ X j is the canonical pro jection, then, by assumption, Q j ∈ J ′ X j is O -[ µ, λ ]-compact, since | J ′ | ≤ κ , hence T α ∈ λ π ( C α ) 6 = ∅ , and this clearly implies T α ∈ λ C α 6 = ∅ . By arguments similar to those in Remark 4.7, we can impro ve the v alue of κ to cf S µ ( λ ).  16 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS F o r sak e of simplicit y , in the statemen t o f Theorem 4.6 w e hav e dealt with a single top ological space X . Ho w ev er, a vers ion of the theorem holds for families of top olo gical spaces. Theorem 4.11. F or every fam ily T of top olo gic al sp ac es, and λ , µ infinite c ar dinals, the fol lowing ar e e quivalent. (1) Ther e exists some ( µ, λ ) -r e gular ultr afilter D (which c an b e taken over S µ ( λ ) ) such that, for every X ∈ T , we have that X is D - pseudo c omp act. (2) Every pr o duct o f a n y numb er of memb ers of T (a l lowing r ep e- titions) is O ✷ - [ µ, λ ] -c omp act. (3) Every pr o duct of memb ers of T (al lowing r ep etitions) w ith at most δ factors is O ✷ - [ µ, λ ] -c omp act, wher e δ = min { 2 2 κ , sup {| T | , ν }} , for ν = sup X ∈ T ( w ( X )) κ and κ = λ <µ (inde e d, this c an b e im- pr ove d to κ = cf S µ ( λ ) ). Corollary 4.12. F or µ , λ , µ ′ and λ ′ infinite c ar dinals, the fol lowing ar e e quivalent. (a) Every ( µ, λ ) -r e gular ultr afilter is ( µ ′ , λ ′ ) -r e gular. (b) F o r every famil y T of top ol o gic al sp ac e s , if eve ry pr o duct of any numb er of memb ers of T (al low ing r ep etitions) is O ✷ - [ µ, λ ] -c omp act, then every pr o duct of any numb er of mem b ers of T ( a l lowing r ep eti- tions) is O ✷ - [ µ ′ , λ ′ ] -c omp act. (c) F o r every top olo gic al sp ac e X , if every p ower of X is O ✷ - [ µ, λ ] - c omp act, then every p ower of X is O ✷ - [ µ ′ , λ ′ ] -c omp act. (d) Same as (c), r estricte d to T ychonoff sp ac es. Pr o of. (a) ⇒ (b) Supp ose that t he assumption in (b) holds. By Theo- rem 4.11 (2) ⇒ (1) , there exists some ( µ, λ )-regular ultrafilter D suc h that, fo r ev ery X ∈ T , we ha v e that X is D - pseudo compact. By (a), D is ( µ ′ , λ ′ )-regular. Hence , by Theorem 4.11 ( 1 ) ⇒ (2), ev ery pro duct of an y n umber of mem b ers of T is O ✷ -[ µ ′ , λ ′ ]-compact. (b) ⇒ (c) and (c) ⇒ (d) are trivial. (d) ⇒ (a) Garcia-F erreira [G a] constructs, for ev ery ultrafilter D , sa y o v er I , a T yc honoff space P RK ( D ) suc h that, for ev ery ultrafilter E , sa y o ve r J , the space P RK ( D ) is E -pseudo compact if and only if E = f ( D ) fo r some function f : I → J , that is if and only if E ≤ RK D in t he R udin-Keisler order. Let D b e a ( µ, λ )-regular ultra filter, sa y ov er I . By ab o ve , X = P RK ( D ) is D -pseudo compact, hence, b y Theorem 4 .6 (2) ⇒ (4), ev ery p ow er of X is O ✷ -[ µ, λ ]-compact. By (d), eve ry p o w er of X is O ✷ -[ µ ′ , λ ′ ]-compact and, by Theorem 4.6 (2) ⇒ (4) , X is E -pseudocompact, for some ( µ ′ , λ ′ )-regular ultrafilter E GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 17 o v er some set J . By the ab ov e-mentioned result from [Ga], E = f ( D ), for some function f : I → J . By a trivial prop erty of the R udin-Keisler order, D is ( µ ′ , λ ′ )-regular, th us (a) is prov ed.  Man y results are know n ab out cardina ls fo r whic h Condition (a) in Corollary 4.12 holds. See [Li3] for a surv ey . Corollary 4.12 can b e applied in eac h of these cases. R e m ark 4.13 . As w e men tioned in Remark 2.6, [ µ, λ ]-compactness is equiv alen t to [ κ, κ ]-compactnes s for ev ery κ suc h that µ ≤ κ ≤ λ . W e no w sho w that the analogous result fails, in general, for O -[ µ, λ ]- compactness. Under some set-theoretical a ssumption, [Ka] constructed an ultra- filter D uniform ov er ω 1 and an ultrafilter D ′ o v er ω suc h that, for ev ery ultrafilter E , it happ ens that E ≤ RK D if a nd only if E is Rudin-Keisler equiv alent either to D or to D ′ . By the results from [Ga] mentioned in t he pro of of Corollary 4.12, the space P RK ( D ) is b oth D - pseudo compact and D ′ -pseudo compact, hence b oth O -[ ω , ω ]- compact and O -[ ω 1 , ω 1 ]-compact, since ev ery uniform ultrafilter ov er some cardinal λ is ( λ , λ )-regular (see, e. g., [Li3]). Indeed, b y Corol- lary 4.5, all p o we rs of P RK ( D ) are ev en b oth O ✷ -[ ω , ω ]-compact and O ✷ -[ ω 1 , ω 1 ]-compact. Ho w ev er, [Ga] prov ed that P RK ( D ) is not ev en ω 1 -pseudo compact. Since, b y [R e, Theorem 2(c)], ev ery O -[ ω , λ ]-compact T yc honoff space is λ -pseudo compact, w e ha v e that P RK ( D ) is not O - [ ω , ω 1 ]-compact ( O -[ ω , λ ]-compact spaces are called w eakly-initially compact in [Re]). 5. The abstract framework In this final section we mention that our results actually hold in the g eneral framew ork in tro duced in [Li4]. In [Li4] eac h compactness prop erty is defined r elat ive to some fa mily F of subsets of a top ological space X . By taking F to b e either the set of all singletons of X , or the set of all nonempty op en sets of X , this generalized approach pro vides a unified treatmen t o f definitions and results ab out [ µ, λ ]- compactness and related compactness no tions, on one side, and a b out O -[ µ, λ ]-compactness and related pseudo compactness-lik e notions, on the other side. In the case of [ µ, λ ]-compactness, as we shall p o in t after each single result, mo st of the theorems w e get a r e kno wn; in the case when F = O w e usually get back the results obtained in the previous sections. 18 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS Definition 5.1. The definitions o f F -[ µ, λ ]-compactness and o f F - D - compactness can b e obtained, respectiv ely , from the definitions of O - [ µ, λ ]-compactness (Definition 2.1) and o f D -pseudo compactness (Def- inition 3.4), b y replacing the fa mily O off all nonempt y op en sets with another sp ecified family F of subsets of X . In more detail, let X be a top olog ical space, a nd let F b e any f amily of subsets of X . Let λ and µ b e infinite cardinals. W e sa y that X is F -[ µ, λ ]- c omp act if a nd only if, for ev ery seque nce ( C α ) α ∈ λ of closed sets of X , if, for ev ery Z ⊆ λ with | Z | < µ , there exists F ∈ F suc h that T α ∈ Z C α ⊇ F , then T α ∈ λ C α 6 = ∅ . Let D b e an ultr a filter ov er some set Z . W e sa y tha t X is F - D - c omp act if and only if ev ery seque nce ( F z ) z ∈ Z of mem b ers of F has some D -limit p oint in X . When, in the preced ing definitions, F = O , the family of a ll the nonempt y op en sets of X , w e get bac k D efinitions 2.1 and 3 .4. When F is tak en to b e the family of all singletons of X , we g et ba ck the more familiar notions of, re sp ectiv ely , [ µ, λ ]-compactness and of D - compactness. See [Li4] for mo r e information. In pa rticular, no t ice that, for µ = λ a regular cardinal, [Li4] prov ides a v ery refined theory of F - [ λ, λ ]-compactness . In the par t icular case µ = λ regular, the results presen ted in [Li4] are usually stronger than the results presen ted here for F -[ µ, λ ]-compactness . Notice also t ha t, b y Remark 4.13, the theory of F -[ µ, λ ]-compactness, in g eneral, cannot b e “reduced” to the theory of F -[ κ, κ ]-compactness. On the contrary , it is a v ery useful fact that [ µ, λ ]-compactness can b e studied in terms of [ κ, κ ]-compactness, for µ ≤ κ ≤ λ (Remark 2.6). Notice that if X is realized a s a Tyc honoff pro duct Q j ∈ J X j , then O ✷ -[ µ, λ ]-compactness, as in tro duced in Definition 4 .1, is the same as F -[ µ, λ ]-compactness of Q j ∈ J X j , when we tak e F t o b e the f a mily o f all op en sets in ✷ j ∈ J X j , t hat is, the op en sets in the b ox to p ology . If F is a f a mily of subsets of some top ological space, w e denote b y W F (r esp., W ≤ κ F ), the family of all subsets of X whic h can b e obtained as the union of t he mem b ers of some subfamily of F (resp., some subfamily of cardinalit y ≤ κ ). Theorem 5.2. Supp o s e that X is a top olo gic a l sp ac e, F is a family of subsets of X , and λ and µ ar e infinite c a r dinals. Then the fo l lowing ar e e quivalent. (1) X is F - [ µ, λ ] -c om p act. GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 19 (2) F or every se quenc e ( P α ) α ∈ λ of subsets of X , if, for every Z ⊆ λ with | Z | < µ , ther e exists some F Z ∈ F such that T α ∈ Z P α ⊇ F Z , then T α ∈ λ P α 6 = ∅ . (3) F or every se quenc e ( Q α ) α ∈ λ of sets in W F (e q uiva lently, in W F ≤ κ , whe r e κ = λ <µ ), if, for every Z ⊆ λ with | Z | < µ , ther e exists some F Z ∈ F such that T α ∈ Z Q α ⊇ F Z , then T α ∈ λ Q α 6 = ∅ . The value of κ c an b e impr ove d to cf S µ ( λ ) . (4) F or every se quenc e { F Z | Z ∈ S µ ( λ ) } o f memb ers of F , it happ ens that T α ∈ λ S { F Z | Z ∈ S µ ( λ ) , α ∈ Z } 6 = ∅ . (5) F or every s e quenc e { F Z | Z ∈ S µ ( λ ) } of memb ers of F , the fol lowing holds. If, fo r every fi nite subset W of λ , we put Q W = S { F Z | Z ∈ S µ ( λ ) and Z ⊇ W } , then T { Q W | W is a finite subset of λ } 6 = ∅ . (6) F or every λ -indexe d op e n c over ( Q α ) α ∈ λ of X , ther e exists Z ⊆ λ , with | Z | < µ , such that F ∩ S α ∈ Z Q α 6 = ∅ , for every F ∈ F . (7) F or e v ery se quenc e { F Z | Z ∈ S µ ( λ ) } of ele ments of F , ther e exists an ultr afilter D over S µ ( λ ) wh i c h c overs λ an d such that { F Z | Z ∈ S µ ( λ ) } has some D -limit p oint i n X . Pr o of. Same as the pro ofs o f Pro p osition 2.4, of the last remark in Definition 2.1 and of Theorem 3.2. See also Remark 4.7.  Prop osition 2 .4 and Theorem 3.2 can b e obtained as the particular case of Theorem 5.2, when F = O is the family of the nonempt y op en sets o f X . Prop osition 4 .3 and Theorem 4.4 can b e obtained as the particular case of Theorem 5.2, when X is the top olo gical space Q j ∈ J X j (with the T yc honoff top ology), and F is the fa mily of the nonempt y op en sets o f ✷ j ∈ J X j (with the b ox top o logy). Th us, Theorem 5.2 prov ides a generalization of all the ab ov e results. As w e men tioned in Remark 3.3, in the particular case when F is the family S of all singletons, the implication (1) ⇒ (7) in Theorem 5.2 is pro v ed in [Ca1 , Ca2]. Again when F = S , the equiv a lence of (1) and (2) in Theorem 5.2 has b een prov ed in [G´ a], with different no tation. See also [V a, Lemma 5(b)]. Theorem 5.3. Supp ose that X is a top olo g ic al sp ac e , F is a family of subsets o f X , and D is an ultr afilter over some se t I . Then the fol lowing ar e e quivalent. (1) X is F - D -c omp act. (2) F or every se quenc e { F i | i ∈ I } of memb ers of F , if, for Z ∈ D , we put C Z = S i ∈ Z F i , then we have that T Z ∈ D C Z 6 = ∅ . 20 GENERALIZA TIONS OF PSEUDOCOMP ACTNESS (3) Whenever ( C Z ) Z ∈ D is a se quenc e of clo se d sets of X with the pr op erty that, for every i ∈ I , ther e exists some F ∈ F such that T i ∈ Z C Z ⊇ F , then T Z ∈ D C Z 6 = ∅ . (4) F or every op en c over ( O Z ) Z ∈ D of X , ther e is some i ∈ I such that F ∩ S i ∈ Z O Z 6 = ∅ , for every F ∈ F . Pr o of. Similar to the pro of of Theorem 3.8.  Theorem 3 .8 could b e obtained as t he particular case F = O of Theorem 5.3. The pa r t icular case of Theorem 5.3 when F is the set of all singletons of X might b e new, so w e state it explicitly . Corollary 5.4. Supp ose that X is a top olo gic al sp ac e, and D is an ultr afilter ove r some set I . Then the fol lo w ing ar e e quivalent. (1) X is D - c omp act. (2) F or every se quenc e { x i | i ∈ I } of elem ents of X , if, for Z ∈ D , we put C Z = { x i | i ∈ Z } , then we have that T Z ∈ D C Z 6 = ∅ . (3) Whenever ( C Z ) Z ∈ D is a se quenc e of clo se d sets of X with the pr op erty that, for every i ∈ I , T i ∈ Z C Z 6 = ∅ , then T Z ∈ D C Z 6 = ∅ . (4) F or every op en c over ( O Z ) Z ∈ D of X , ther e is some i ∈ I such that ( O Z ) i ∈ Z is a c over of X . Theorem 5.5. Supp ose that λ and µ ar e infini te c ar dinals, T is a family of top olo gic al sp ac e s, and, for every X ∈ T , F X is a family of subsets of X . T o e v e ry pr o duct Q j ∈ J X j , wher e e ac h X j b elongs to T , asso ciate the family F = { Q j ∈ J F j | F X j ∈ F j , for ev e ry j ∈ J } . Then the fol lowing ar e e quivalent. (1) Ther e exis ts some ultr afilter D over S µ ( λ ) wh i c h c overs λ , an d such that, for every X ∈ T , we have that X is F X - D -c omp act. (2) Ther e e x ists so me ( µ, λ ) -r e gular ultr afilter D (over any set) such that, f o r every X ∈ T , we h ave that X is F X - D -c omp act. (3) Ther e exists some ( µ , λ ) -r e gular ultr afilter D such that, for ev- ery set J , every pr o duct Q j ∈ J X j of memb ers of T (al lowing r e p etitions) is F - D -c omp act. (4) F or every set J , ev ery pr o duct Q j ∈ J X j of memb ers of T (al- lowing r e p etitions), is F - [ µ, λ ] -c om p act. (5) L et δ = min { 2 2 κ , sup {| T | , sup X ∈ T |F X | κ } , wher e κ = λ <µ (in- de e d, this c an b e impr ove d to κ = cf S µ ( λ ) ). F o r every set J with | J | ≤ δ , every pr o duct Q j ∈ J X j of memb ers of T ( a l lowing r e p etitions) is F - [ µ, λ ] -c omp act. GENERALIZA TIONS OF PSEUDOCOMP ACTNESS 21 Pr o of. Same as the pro ofs of Corollar y 3.7 and of Theorem 4.6, using [Li4, F act 6 .1 and Prop osition 5 .1 (b) with ν = | J + | ] and Theorem 5 .2 (7) ⇔ (1). F or (5), see also Remark 4.7.  Theorem 5.5 is more general than Theorems 4.6 and 4.11. In the particular case when F is the family S of all singletons, Theorem 5.5 is essen tially [Ca2, Theorem 3.4] (in some cases, our ev a lua tion of δ migh t b e sligh tly sharp er). Corollaries 3.7 and 4.5 are immediate con- sequence s o f Theorem 5.5 (2) ⇒ (4), by t a king, for ev ery j ∈ J , F j to b e the family of all nonempt y op en sets of X j . The following easy prop osition, generalizing Lemma 4.9, describ es the b eha vior of F - D - compactness with resp ect to quotien ts. Prop osition 5.6. Supp ose that X and Y ar e top olo gic al sp ac es, and f : X → Y is a c ontinuous function. Supp ose that F is a family of subsets of X , and supp ose that G is a family of s ubsets of Y , s uch that for eve ry G ∈ G ther e is F ∈ F such that F ⊆ f − 1 ( G ) . Then the fol lowing hold. (1) If X is F - [ µ, λ ] -c omp a c t then Y is G - [ µ, λ ] -c omp act. 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Dip ar timento Ma tema ticæ, Viale della Ricer ca Scientifica, II Uni- versit ` a di Roma (Tor Verga t a), I-00133 R OME IT AL Y URL : http: //www .mat.u niroma2.it/~lipparin