THE DEVELOPMENT OF AND GAPS IN THE THEORY OF PRODUCTS OF INITIALLY M-COMPACT SPACES

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1 Vlume 6, 1981 Pages THE DEVELOPMENT OF AND GAPS IN THE THEORY OF PRODUCTS OF INITIALLY M-COMPACT SPACES by R. M. Stephensn, Jr. Tplgy Prceedings Web: Mail: Tplgy Prceedings Department f Mathematics & Statistics Auburn University, Alabama 36849, USA tplg@auburn.edu ISSN: COPYRIGHT c by Tplgy Prceedings. All rights reserved.

2 TOPOLOGY PROCEEDINGS Vlume THE DEVELOPMENT OF AND GAPS IN THE THEORY OF PRODUCTS OF INITIALLY M-COMPACT SPACES R. M. Stephensn, Jr. Accrding t A. Tychnff's therem, btained in 1930, every prduct f cmpact spaces is cmpact. Since that time, tplgists and ther matematicians have been cnsidering prperties similar t, but weaker than, cmpactness, and they have been attempting t determine the extent t which analgues f Tychnff's therem hld fr these prperties. One such prperty in which there has been cnsiderable interest is initial m-cmpactness. Beginping in 1938 when E. Cech asked if cuntable cmpactness is prductive, r perhaps earlier when P. Alexandrff and P. Uryshn published [AU], it became apparent that sme questins cncerning prducts f linearly cmpact" spaces might turn ut t be bth very interesting and difficult t answer. In this talk we shall fcus ur attentin n initially m-cmpact prduct spaces fr varius infinite cardinal numbers m. The cardinality f a set X will be dented by lxi, and fr a filter base J n a tplgical space X, the set f adherent pints f J, n{f: F E J}, will be dented by ad J. The successr f a cardinal number m will be dented by m+. We shall write c fr 2 K, and MA (CH, GCH) will dente Martin's Axim (the Cntinuum r Generalized Cntinuum Hypthesis).

3 100 Stephensn A tplgical space is called initially m-cmpact (where m is an infinite cardinal number) if any f the fllwing equivalent cnditins hlds: ad J t- ~; (i) fr every filter base J n X, (ii) fr every pen cver V f X, if I JI < m then if IV/ < m then V has a finite subcver; r (iii) fr every subset A f X, if IAI ~ m then A has a cmplete accumulatin pint, i.e., a pint every neighbrhd f which cntains IAI pints f A. Initially K-cmpact spaces are called cuntably cmpact. These ideas are due t Alexandrff and Uryshn. That initial m-cmpactness culd be quite different frm cmpactness was discvered in the early 50's. Examples published in 1952 by H. Terasaka [T2] and in 1953 by J. Nvak [N3] shwed that, in general, cuntable cmpactness was nt prductive and, in fact, that ne culd cnstruct tw cuntably cmpact cmpletely regular Hausdrff spaces whse prduct is nt even pseudcmpact (by pseudcmpact ne means a space n which every cntinuus real valued functin is bunded). During the last 20 years the thery f cuntably cmpact spaces and prduct spaces has been extensively develped. While there has als been much activity in the area f initially m-cmpact spaces, fr m ~ K ' the thery f the latter is, at this time, much less cmplete than the thery f cuntably cmpact spaces. In bth areas a crnmn trend, since the discveries f Nvak and Terasaka,

4 TOPOLOGY PROCEEDINGS Vlume has been the search fr well behaved prperties, nly slightly strnger than initial m-cmpactness, which frce certain prduct spaces t be initially m-cmpact. We shall utline much f the prgress that has ccurred since 1953 and indicate sme f the gaps in the thery and pen prblems n which further wrk is needed. In the late 50 l s Z. Frlik [Fl], [F2] and I. Glicksberg [G] independently cnsidered cnditins smewhat strnger than initial m-cmpactness which culd be used t prduce initially m-cmpact prduct spaces. Several very useful therems were btained. Therem 1. Glicksberg [G]. Let X be a prduct f n mre than m cmpletely regular Hausdrff spaces each f which is initially m-cmpact. Then X is initially m-cmpact if either (i) all but ne f the factrs are lcally cmpact r (ii) all but ne f the factrs have character < m. Frlikls results cncerned the case m = K and the family F f cmpletely regular Hausdrff spaces X such that every infinite subset f X has an infinite subset whse clsure in X is cmpact. Therem 2. Frlik [F2]. Every prduct f cuntably many members f F is a member f F. Anther nice therem cncerning a prperty which frces certain prduct spaces t be cuntably cmpact was btained by A. H. Stne in 1966.

5 102 Stephensn Therem 3. Stne [SS2]. Every prduct f at mst K l sequentially cmpact spaces is cuntably cmpact. Next, N. Nble [Nl] studied the family F' f Tl-spaces X such that every infinite subset f X has an infinite subset whse clsure in X is cmpact. He succeeded in imprv ~ ing Frlik's and Glicksberg's therems by btaining, amng ther results, the fllwing. Therem 4. Nble [Nl]. Fr each space X E F' and cuntably cmpact Tl-space Y, the space X x Y is cuntably cmpact. Fr cmpletely regular Hausdrff spaces, Therem 4 was als used by T. Isiwata [I]. Therem 5. Nble [N2]. Every prduct f at mst m+ spaces, each f which is initially m-cmpact and f character < m, is initially m-cmpact. Abut the same time three ther authrs, S. L. Gulden, W. M. Fleishman, and J. H. Westn, defined a tplgical space X t be m-bunded prvided that fr every subset A f X, if IAI ~ m then there is a cmpact subset K f X with A c K. Althugh this cncept appeared t be much strnger than initial m-cmpactness, it was different frm thse used by Glicksberg, and ne culd easily shw that it was prductive and, therefre, wuld prduce initially m-cmpact prduct spaces. In 1969 I defined a Tl-space X t be strngly initially m-cmpact prvided that fr every filter base] n X, if

6 TOPOLOGY PROCEEDINGS Vlume IJI ~ m then there exists a cmpact subset K f X such that JIK is a filter base. Fr Tl-spaces, this cncept generalized m-bundedness, and fr m = H ' it generalized sequential cmpactness and was equivalent with membership in F'. One culd prve that strng initial m-cmpactness is finitely prductive, and the prduct f a strngly initially m-cmpact space and an initially m-cmpact space is initially m-cmpact. In additin, I was able t use the cncept, fr the case m = H ' t strengthen Therem 3. Therem 6. Stephensn [551]. Every prduct f at mst H l spaces in F' is cuntably c~mpact. The best prperty f this type was discvered and later refined by J. E. Vaughan in the early and mid 70's. He defined a space X t be (l)m prvided that fr every filter base J n X, if IJI ~ m, then there exist a cmpact set K c X and a filter base y n X such that \yl < m, and y is finer than bth J and the filter base f all pen sets cntaining K [VI]. Later [V4] he defined a space X t be TI m-cmpact prvided that fr every filter base J n X, if IJI < m then there exists a finer ttal filter base n X with I I < m. (A filter base n X is called ttal if every finer filter base has an adherent pint- this cncept is due t B. J. Pettis--see [P], [V2].) A regular, TI m-cmpact space is (l)m. Vaughan's cncept might be cnsidered "best" in that it made pssible a simultaneus generalizatin f mst f the therems abve, namely, by Therems 7 and 8.

7 104 Stephensn Therem 7. Vaughan [VI]. Let X each X is {l)m. a (i) If IAI (ii) If IAI < m then X is {l)m. < m+ then X is initially rn-cmpact. Therem 8. Stephensn-Vaughan [SV]. If a space X is {l)m and Y is an initially m-cmpact space~ then X x Y is initially m-cmpact. Analgus therems were btained fr his secnd definitin. Therem 9. Vaughan [V4]. Let X each X is TI m-cmpact. a (i) If IAI < m then X is TI m-cmpact. (ii) If IAI < m+ then X is initially m-cmpact. Therem 10. Vaughan [V4]. If X is TI m-cmpact and Y is an initially m-cmpact space~ then X x Y is initially m-cmpact. T verify that Therems 7 and 8 (r 9 and 10) include mst f the previus nes, it suffices t nte that ne has (in sme cases, require T ) the fllwing implicatins. l lcallr cmpact } m-bunded ~ initially m-cmpact ~ strngly initially m-cmpact f character < m } initia~ly m-cmpact ~ (I) ~ m TI m-cmpact 4 initially m-cmpact

8 TOPOLOGY PROCEEDINGS Vlume A number f interesting examples, sme quite simple, shw that mst f these cncepts are distinct. We briefly describe them. Example 11. Let X be the set f rdinal numbers < m+, with the rder tplgy, and let P = x n, where n is a cardinal number. Then P is m-bunded but nt cmpact. P is lcally cmpact if and nly if n < K ' and P is f character < m if and nly if n < m. Example 12. The fllwing space, C, similar t nes due t H. H. Crsn, I. Glicksberg, L. S. Pntryagin, and J. Kister, is m-bunded but nt initially m+-cmpact. Fr each rdinal number a < m+ chse a cmpact Hausdrff space X with Ix I > 2 and fix a pint p EX. Let C be a a - a a the set f all pints x in the prduct space TI{X } such a that I{a: x ~ Pall < m. In case each X is a tplgical a a grup with identity Pa' then C is a tplgical grup. Example 13. Frlik [Fl], [F2]. If x E SN\N let K SN\{x}. Then each KEF but is nt K -bunded r x x 0 sequentially cmpact. Mrever, there exist sets A and B e such that IAI = c and IBI = 2, but rr{k : x E A} F' and x TI{K : x E B} fails t be cuntably cmpact. In 1975 S. H. x Hechler prved [HI], under MA, the space TI{K : x E C} is x in F if Ici < c and is cuntably cmpact if Ici ~ c. Example 14. M. E. Rudin [R]. If CH hlds, then there exists a separable, nncmpact, sequentially cmpact space. (In Trans. AQer. l1ath. Sc. 155 (1971) , Franklin and Rajagpalan btain such a space within ZFC.)

9 106 Stephensn The next, very nice cnstructin, as well as a tplgical grup versin f it [SSl], was discvered by Victr Saks while he was a graduate student wrking under W. W. Cmfrt. It shws that fr every m ~ K ' if m has the discrete tplgy, then there exists an initially m-cmpact space m c P c Bm such that Ipi < 2 m. Fr the case m = K, - 0 the result that such a space exists is due t Frlik [Fl]. Example 15. Saks [SSl]. Let m be an infinite cardinal number. Fr a subset A f a cmpact space, dente by A' a set btained as fllws: fr each filter base J n A such that IJI ~ m and IFI ~ m fr every F E J, chse ne adherent pint PJ f J, and let A' be the set f all such adherent pints. Nw in any Hausdrff cmpactificatin f the discrete space m, define P = m, P = (U{p : a < b})' b a fr each rdinal number 0 < b < m+, and, finally, P = U{P b : b < m+}. Then, as pinted ut by Saks, (i) P is initially m-cmpact, (ii) P is nt m-bunded if the cmpactificatin has cardinality > 2 m, and (iii) P can be btained as a tplgical grup by a similar cnstructin. Saks and I als nted in [SSl] that fr m regular and fr a cmpactificatin f cardinality> 2 m, P fails t be strngly initially m-cmpact~ In 1974 [SV], Vaughan and I prved, mrever, that if the cmpactificatin is Bm, then P fails t be (l)m' whether m is regular r nt. Example 16. Eric van Duwen [vd3]. Let m be an uncuntable regular cardinal number and < its usual well rder. Call a subset C f m a cub set if sup C = m and

10 TOPOLOGY PROCEEDINGS Vlume if sup B E C U {m} fr every subset B f C. Next, viewing m as a discrete space, define V t be the fllwing sub~ space f Sm: V U{I: I c m and Inc = ~ fr sme cub set C}. The space V is then lcally cmpact and initially m-cmpact (hence strngly initially m-cmpact) but nt m-bunded r f character < m. Example 17. Vaughan [VS]. If V is as in Example 16 and n is a cardinal number, then the space X V n is (i) nt strngly initially m-cmpact if n ~ K ' (ii) nt TI m-cmpact if n >2 m and (iii) initially m-cmpact if GCH. Befre discussing sme pen questins cncerning the prductivity f initial m-cmpactness, let us cnsider tw psitive therems f a different srt than thse discussed earlier. The first is a general reductin therem which, fr the case m = K ' strengthened A. H. Stne's reductin therem in [SS2]. m > K. Therem 18. 5aks [51]. Let X = IT{X : a E A} and a Then X is initially m-cmpact if and nly if IT{X : b E B} is initially m-cmpact fr each B c A such b m that IBI < 22. The secnd is a surprising result I btained and later imprved which cncerns nly singular cardinals m, i.e., m such that m can be expressed as a sum f fewer, smaller cardinals. In set thery sme f the mst unexpected (and

11 108 Stephensn interesting) results have turned ut t be thse cncerning singular cardinals--the same has been the case in this area. Therem 19. Stephensn [SV]. Let m be a singular cardinal number and suppse that 2 n < m fr every n < m. Then initial m-cmpactness is prductive. Thus, in cntrast with the situatin fr m = K ' it fllws frm GCH that initial m-cmpactness is prductive fr every singular cardinal number m. Even withut the assumptin f GCH, a standard technique shws that Therem 19 applies t cfinally many cardinals (given a, define m. m a, m i + = 2 1, and m = sup{m : i E w}). l i Obtaining answers t sme f the prblems belw wuld significantly help the develpment f the thery f initially m-cmpact spaces and prduct spaces. Prblem 20. Let m be a singular cardinal number. Is every TI m-cmpact space strngly initially m-cmpact (m-bunded)? Is every strngly initially m-cmpact space m-bunded? P~blem 21. Let m be a regular uncuntable cardinal number. Is every prduct f strngly initially m-cmpact (TI m-cmpact) spaces initially m-cmpact? Prblem 22. Using Example 13, Vaughan [SV] prved that, fr m = K ' it cannt be determined within ZFC if the bunds in the cnclusin f Therem 7 can be imprved. What can be said fr m > K?

12 TOPOLOGY PROCEEDINGS Vlume Prblem 23. Can the bund in Saks' Therem 18 be imprved within ZFC? Saks [Sl] shwed that fr m = ~, if MA then it is the best pssible. Several, but nt all, f Prblems have been raised in the literature--see [SSl], [SV], [Sl], and [V5]. Befre stating sme additinal nes, I wuld like t state fur imprtant results btained by Eric van Duwen during the last 2 r 3 years. These results have filled majr gaps in the thery f initially m-cmpact prduct spaces. Several f them are based n a very nice technique van Duwen has devised fr cnstructing spaces X ' Xl such that each Xi is a unin f (i) the nnunifrm ultrafilters in Sm and (ii) a space similar t the type described in Example 12. Therem 24. van Duwen [vd2]. If c > ~w and MA~ then there are tw initially ~w-cmpact nrmal spaces whse prduct is nt initially ~w-cmpact. Therem 25. van Duwen [vd2]. If MA~ then there exist tw nrmal spaces~ each initially rn-cmpact fr every rn < c~ whse prduct is nt even cuntably cmpact. Therem 26. van Duwen [vd2]. Let rn be an uncuntable regular cardinal number and assume GCH. There exist tw nrmal initially m-cmpact spaces whse prduct is nt initially m-cmpact. Therem 27. van Duwen [vdl]. Assume MA. (i) There exist tw cuntably cmpact tplgical grups whse prduct

13 110 Stephensn is nt cuntably cmpact. (ii) Assume MA and the negatin f the Cntinuum Hypthesis. There are tw initially Kl-cmpact tplgical grups whse prduct is nt cuntably cmpact. Prblem 28. In 1966, W. W. Cmfrt and K. Rss published a prf that every prduct f pseudcmpact grups is pseudcmpact [CR], and abut that time Cmfrt raised the related questin f whether r nt every prduct f cuntably cmpact grups is cuntably cmpact. What ther results besides Therem 27 can be btained cncerning initially m-cmpact grups and their prducts? Prblem 29. While Therem 19 implies that initial m-cmpactness is prductive fr cfinally many singular cardinals m, it fllws frm Therems 19 and 24 that it cannt be determined within ZFC if initial Kw-cmpactness is prductive. Fr which singular cardinals m > c is an analgus result true? See 7.2 f [vd2]. Prblem 30. Can the assumptin f GCH in Therem 26 be deleted? In 1966 A. H. Stne asked [882] if every prduct f sequentially cmpact spaces is cuntably cmpact. M. Rajagpalan and R. Grant Wds prved in 1977 that under CH, Stne's questin has a negative answer, and J. E. Vaughan prved in 1978 that under an additinal axim, there exist sequentially cmpact, perfectly nrmal spaces whse prduct is nt cuntably cmpact.

14 TOPOLOGY PROCEEDINGS Vlume III Prblem 31. Within ZFC, is every prduct f sequentially cmpact spaces cuntably cmpact? Prblem 32. Within ZFC, des there exist a first cuntable, cuntably cmpact space that is nt K-bunded? Like Cech ',s questin in 1938, the prblems listed abve (sme f which als have been raised elsewhere) are likely t turn ut t be difficult but very wrthy f effrts t slve them. Bibligraphy [AU] P. Alexandrff and P. Uryshn, Memire sur les espaces tplgiques cmpacts, Ver. Akad. Wetensch. Amsterdam 14 (1929), [C] W. W. Cmfrt, Ultrafilters: Sme ld and sme new results, Bull. Amer. Math. Sc. 83 (1977), [CH] and A. W. Hager, The prjectin mapping and ther cntinuus functins n a prduct space, Math. Scand. 28 (1971), [CR] W. W. Cmfrt and K. A. Rss, Pseudcmpactness and unifrm cntinuity in tplgical grups, Pacific J. Math. 16 (1966), 483~496. [vol] E. van Duwen, The prduct f tw cuntably cmpact tplgical grups, Trans. Amer. Math. Sc. 262 (1980), [vd2], The prduct f tw nrmal initially K-cmpact spaces, Trans. Amer. Math. Sc. (t appear). [vd3], Cmpactness-like prperties and nnnrmality f the space f nnstatinary ultrafilters (t appear). [Fl] z. Frlik, The tplgical prduct f cuntably cmpact spaces, Czechslvak Math. J. 10 (85) (1960), [F2], The tplgical prduct f tw pseudcmpact spaces, Czechslvak Math. J. 10 (85) (1960),

15 112 Stephensn [GS] J. Ginsburg and V. Saks, Sme applicatins f ultrafilters in tplgy, Pacific J. Math. 57 (1975), n. 2, [G] I. G1icksberg, Stne-Cech cmpactificatins f prducts, Trans. Amer. Math. Sc. 90 (1959), [GFW] S. L. Gulden, W. M. Fleishman, and J. H. Westn, Linearly rdered tplgical spaces, Prc. Amer. Math. Sc. 24 (1970), [HI] S. H. Hechler, On sme weakly cmpact spaces and their prducts, General Tplgy and its Appl. 5 (1975), [H2], Tw R-clsed spaces revisited, Prc. Amer. Math. Sc. 56 (1976), [I] T. Isiwata, Sme classes f cuntably cmpact spaces, Czechslvak Math. J. 14 (89) (1964), [N1] N. Nble, Cuntably cmpact and pseudcmpact prducts, Czechslvak Math. J. 19 (94) (1969), [N2], Prducts with clsed prjectins. II, Trans. Amer. Math. Sc. 160 (1971), [N3] J. Nvak, On the cartesian prduct f tw cmpact spaces, Fund. Math. 40 (1953), [P] B. J. Pettis, Cluster sets f nets, Prc. Amer. Math. Sc. 22 (1969), [R] M. E. Rudin, A technique fr cnstructing examples, Prc. Amer. Math. Sc. 16 (1965), [Sl] V. Saks, Ultrafilter invariants in tplgical spaces, Trans. Amer. Math. Sc. 241' (1978), [S2] A. H. Stne, Hereditarily cmpact spaces, Amer. J. Math. 82 (1960), [SSl] V. Saks and R. M. Stephensn, Jr., Prducts f m-cmpact spaces, Prc. AIDer. Math. Sc. 28 (1971), [SS2] C. T. Scarbrugh and A. H. Stne, Prducts f nearly cmpact spaces, Trans. Amer. Math. Sc. 124 (1966),

16 TOPOLOGY PROCEEDINGS Vlume [SV] R. M. Stephensn, Jr. and J. E. Vaughan, Prducts f initially m-cmpact spaces, Trans. Amer. Math. Sc (1974), [Tl] H. Taman, A nte n the pseudcmpactness f the prduct f tw spaces, Mem. CIl. Sci., Univ. Kyt Sere A Math. 33 (1960/61), [T2] H. Terasaka, On cartesian prduct f cmpact spaces, Osaka J. Math. 4 (1952), [Vl] J. E. Vaughan, Prduct spaces with cmpactness-like prperties, Duke Math. J. 39 (1972), [V2], Ttal nets'and filters, Tplgy Prc. Memphis State Univ. Cnf., Marcel Dekker Lecture Ntes 24 (1976), [V3], Sme examples cncerning a-bunded spaces, Set-Theretic Tplgy, 1977, Academic Press, [V4], Prducts f tplgical spaces, General Tplgy and its Appl. 8 (1978), [V5], Pwers f spaces f nn-statinary ultrafilters, Fund. Math. (t appear). University f Suth Carlina Clumbia, Suth Carlina 29208

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