PRIMITIVE σ-spaces. by Rick E. Ruth

Transcription

1 Volume 8, 1983 Pages PRIMITIVE σ-spaces by Rick E. Ruth Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume PRIMITIVE ti -SPACES RickE. Ruth A iterestig result due to Fletcher ad Lidgre [FL ] is that spaces havig primitive bases are 8-spaces. 2 Wicke explored this relatioship further by discussig the use of 8-space cocepts i base of coutable order theory [Will ad by givig a fuctioal characterizatio of primitive base [Wi ]. His work led him to ask for a otrivial 2 property to add to 8-space i order to obtai primitive base. I sectio 1 we provide a aswer to Wicke's questio by defiig a property P ad showig that a space has a primitive base if ad oly if it is a 8-space havig property P. The author wishes to thak J. Chaber for poitig out that this property is ot ew ad that it was defied ad studied i [Ch ] as it relates to a-spaces. Further 2 ramificatios (as it relates to spaces havig a primitive base) icludig additioal diagoal theorems are discussed here. We tur i sectio 2 to a exploratio of the class of a-scattered spaces as a special subclass of spaces havig this property. Uless otherwise stated, all spaces are regular TO except i defiitios, where oly those separatio properties stated are assumed. 1. Primitive (T -Spaces We begi with some defiitios.

3 ~ rt =r 6-refiablet1 1 e-refiable BCD-spaces.. 11 ~ Sc-spaces + SIC-diagoal 1 11 Closed sets Closed sets SIC SIC Primitive 11 base spaces ~ ~ Primitive 1 q-spaces + Primitive diagoal Primitive Primitive a-spaces Developable spaces -c lr w~-spaces + G8-diagrr e-spaces C Do-sp~es First coutable :4c spaces.. we-spaces L ~ q-spaces + It e-diagoal a~spaces w.-s 0"\ Diagram 1

4 TOPOLOGY PROCEEDINGS volume Defiitio (Hodel, [Ho]). A space (X,]) is a e-space if there is a fuctio g: Nx X ~ ] havig the followig properties. a) x c EN g (, x) for each x E X. b) If {p,x } ~ g(,y ) ad Y E g(,p) for each E N, the p is a cluster poit of ( x : EN). If i b) we just require that (x : EN) has a cluster poit, we call X a we-space. b Fletcher ad Lidgre [FL 2 ] characterize e-spaces by replacig b) with the followig. l ) If for each E N, Y E g(,p) ad p E g(,y ), the {g(,y ): E N} is a base at p. If i b l ) we just require that ENg(,y ) a-diagoal [Wi ]. l {p}, X is said to have a The cocept of primitive base was itroduced by Worrell ad Wicke i We state below characterizatios of this cocept ad of the associated cocepts of primitive q-space ad primitive diagoal foud i [Wi 2.3]. These may serve l, as defiitios. 1.2 Theorem [Wi 2.3]. For each E N~ let W.be a l, well-ordered collectio of ope sets i X such that W covers X. For each x E X, let F(X,W ) deote the first elemet of W that aotais x. a) If for each x E X, {F(X,W ): E N} is a base at x, the X has a primitive base. b) If Y E F(x,W) for all E N implies (y : E N> clusters, the X is a primitive g-space. c) If for each x E X, ENF(x,W) = {x}, the X has a primitive diagoal.

5 318 Ruth From this characterizatio it is ot hard to see that the class of all e-spaces cotais the class of spaces havig a primitive base. (For a more comprehesive view of some of the relatioships existig amog the spaces metioed here, the reader is referred to Diagram 1 i the Itroductio.) It is the a atural questio to ask what otrivial property a e-space must possess i order to have a primitive base. We aswer this questio after the followig defiitio ad observatios. 1.3 Defiitio. Suppose a space X has a sequece (y : EN> of well-ordered ope covers. a) If F(X,,q) = F(X,,q) for each E N implies (x : E N> clusters to x, the X is called a primitive a-space [Ch 2 ]. b) If F(x,~) = F(x,~ ) for each E N implies I I (x : E N> clusters, the X is called a primitive wa-space. c) If for each x E X, EN{Y: F(X'~) = F(Y'~)} = {x}, the X is called a primitive ai-space [see Ch 2 ]. 1.4 Questios. a) Is every ai-space a primitive ai-space? b) Is every wa-space a primitive wa-space? Chaber poses a) i [Ch 2 ]i we pose b) i view of the cocept of wa-space foud i [FL ] ad for the sake of coml pleteess. ai-space. Clearly, a) Every T l primitive a-space is a primitive a-space. b) Every space havig a primitive base is a primitive

6 TOPOLOGY PROCEEDINGS Volume ai-space. c) Every primitive q-space is a primitive wa-space. d) Every space with a primitive diagoal is a primitive 1.5 Theorem. A space X has a primitive base if ad ozy if it is a e-space ad a primitive a-space. Proof. We eed oly show sufficiecy. Assume X is well-ordered. Let ( Y : EN> be a sequece witessig the fact that X is a primitive a-space, ad let g be a fuctio witessig that X is a e-space. For each E N ad for each x E X, let H(x,) = F(X,y ) g(,x'), where x' is the first y i X such that F(y,y ) = F(X'~) ad x E g(,y). Thus, {H(x,): x E X} is a well-ordered cover of X such that for each x E X, the first elemet cotaiig x is H(x,). Suppose H(x,) = F(x,y ) g(,x ) for each E N. The F(X,y ) F(X,y ) for each E N. Thus (x : E N> clusters to x. {H(x,): ~ N} is a base at x. Sice X is a e-space, it follows that 1.6 Theorem. A primitive a-space which is azso a we-space is a primitive q-space. 1.7 Theorem. A primitive a-space with a e-diagoaz has a primitive diagoaz. The proofs of the above two theorems are obtaied by makig the appropriate chages i the proof of 1.5. The followig examples show that we caot improve upo Theorem 1.5 by usig DO-space or we-space i place of e-space. (See Diagram 1 i the Itroductio.)

7 320 Ruth 1.8 Examples. a) Heath [He, 4.1] has a example of a bow-tie-type space which is a DO-space (i.e., compact sets have coutable character) but which is ot a we-space. This space is a primitive a-space. b) Let X [0,1] U [2,3]. Defie a topology o X as follows. If x E [0,1], the a eighborhood U' of x is a usual eighborhood Utogether with {y + 2: y E U\{x}}. If x E [2,3], the {x} is ope. This space is a primitive a-space ad a we-space, but ot a a-space. 1.9 Questios. a) Is every primitive wa-space which is also a a-space a primitive a-space? b) Does every primitive q-space with a a-diagoal have a primitive base? If we try improvig 1.5 by replacig primitive a-space with primitive wa-space, the a) is a atural questio. This is related to 6) sice every primitive q-space is a primitive wa-space ad a wa-space, ad every wa-space with a a-diagoal is a a-space. Thus, should our questio a) be aswered i the affirmative, so will b); should b) be aswered i the egative, so will a). Questio b) was posed by Wicke i [Will. A stregtheig of the cocept of a-space alog with the associated cocept of wa-space was itroduced by Fletcher ad Lidgre i [FLll. These cocepts are called ~-space ad w~-space respectively. We add a defiitio to theirs ad obtai results aalogous to 1.5, 1.6 ad 1.7 for primitive wa-spaces. Hece, primitive a-space ca be

8 TOPOLOGY PROCEEDINGS Volume weakeed if a-space is stregtheed ad still obtai primitive base Defiitio [FL ]. A space (X,]) is a ~-space l provided there is a fuctio g: N x X + ] such that if for each E N, {p,x } c g(,y ) ad {y : EN> has a cluster - poit, the p is a cluster poit of {x : E N> (or equiva letly, if p E g(,y ) for E Nad {y : E N> has a cluster poit, the {g(,y ): E N} is a base at p). If we just require that {x : E N> has a cluster poit, we call X a w~-space Defiitio. A space (X,]) is said to have a ~-diagoal if ad oly if there is a fuctio g: N x X + ] such that for each x E X, if for some p E X ad each E N, Clearly, ~-spaces have ~-diagoals ad spaces with ~-diagoals have a-diagoals Theorem. A primitive wcr-space whiah is also a ~-space has a primitive base Theorem. A primitive wcr-space which is also a w~-space is a primitive q-space Theorem. A primitive wa-space with a ~-diagoal has a primitive diagoaz. The proofs of the above are all aalogous to the proof of 1.5.

9 322 Ruth 1.15 ExampLes. a) 001' the ~et of all ordials less tha the first ucoutable ordial, with the order topology has a base of coutable order (hece, a primitive base) but is ot a ~-space. Thus, the coverse of ~heorem 1.12 is false (cf. 1.5). b) The Michael lie (i.e., the set of real umbers R with the topo~ogy geerated by the usual topology for the reals together with the sigletos of the irratioal poits) shows that a primitive a-space which is also a ~-space eed ot have a base of coutable order. Clearly, if we demad of primitive wa (resp. w~)-spaces that (x : E N> clusters to y E ENF(X'~) (ENg('Y))' the we obtai results such as the followig Theorem [FL Corollary to 2.2]. A we-space with 2, a e-diagoaz is a e-space. This raises the followig questios Questio. Does every regular we (primitive wa, w~)-space have a we (primitive wa, w~)-fuctio which satisfies the stroger coditio above? This type of questio (cf below) arises agai cocerig the cocept of mootoic S-space i view of Theorem 1.19 ad the remark which follows it. This theorem illustrates the "completeess" required of e-spaces i order to obtai spaces havig a base of coutable order.

10 TOPOLOGY PROCEEDINGS Volume Defiitio (Chaber [ChI' 1.6]). A space X is called a mootoic a-space if ad oly if for all x E X there is a decreasig sequece < B (x): E N> of bases at x such that if, for all E N, B E B (x ), B +1 c B, - ad EN B ~~, the <x : EN> has a cluster poit Theorem [Wi l, 5.1]. Let X be a regular T mool toic a-space. The the folzowig are equivalet: a) X has a base of coutable order. b) X is a e-space. If i the defiitio of mootoic a-space oe required the stroger coditio that <x : E N> has a cluster poit y E ENB' the it is easier to argue tha the above that a e-space which is this stroger mootoic a-space has a base of coutable order. Thus, we ask the followig Questio. Does every regular mootoic a-space satisfy the stroger coditio above? 2. q -Scattere4 Spaces Spaces which are a-scattered (icludig a~(closed ad scattered) ad scattered spaces) are special cases of primitive a-spaces. This might be expected i light of the followig which are foud all together i [WW]. a) Every T first coutable scattered space has a l mootoically complete base of coutable order. b) Every a-(closed ad scattered) T l first coutable space has a base of coutable order. c) Every a-scattered first coutable space has a primitive base.

11 324 Ruth It is clear the that the property of beig a-scattered has a rather strog aspect of primitivity. This must, i fact, be stroger tha primitive a-space sice we have c) above i cotrast to Theorem 1.5 where e-space appears i place of first coutable. Oe of our iterests here is to provide fuctioal characterizatios of these various scattered cocepts i view of our characterizatio of,primitive a-space ad later i view of the fuctioal characterizatio of primitive base of [Wi ]. Usig well-orderig we may i fact formulate 2 these cocepts by meas of the followig. 2.1 Defiitio. Let ( : EN) be a sequece of wellordered ope covers of a space X. For each x E X ad E N, let F(x,) be the first elemet of that cotais x. a) F is called s-primitive a if ad oly if for each x E X, there is x for all m > x E N such that {y: F(y,m) = F(x,m)} = {x} b) F is called rak 1 s-primitive a if ad oly if there is E Nsuch that for each x E X, {y: F(y,m) = F(x,m)} {x} for all m >. c) F is called azosed s-primitive a if ad oly if F is s-primitive a ad for each E N, {x: {y: F(y,) = F(x,)} = {x}} is closed. 2.2 Defiitio. A space X is scattered if ad oly if every subset A c X has a isolated poit (i.e., there is a ae A such that {a} is ope i A). It is a-scattered if it is the uio of a coutable collectio of scattered

12 TOPOLOGY PROCEEDINGS Volume subspaces; if each of these scattered subsets is closed, the space is called o-{closed ad scattered). 2.3 Theorem. A space X is scattered if ad oly if it has a sequece ( : E N> of well-ordered ope covers such that F is rak 1 s-primitive o. Proof. Suppose X is scattered ad is well-ordered. Let X be the first isolated poit of X. There is a ope o set U such that U X = {x O }. Havig chose x ad U a o a a for all a < a, let x be the first isolated poit of a X\{x : a < S} provided this is oempty. a The there is a ope set U such that U X\{x : a < a} {x }. Evetually a a a S X\{x : a < S} = ~; thus we have covered X. a Now we have a well-ordered ope cover Uof X such that for each x EX, {y: F(y,U) =F(x,U)} ={x}. Defiig =U for all E Ngives us a rak 1 s-primitive a fuctio with respect to ( : EN>. Suppose X has a rak 1 s-primitive a fuctio with respect to some sequece ( : E N> of well-ordered ope covers ad that E Nwitesses that it is rak 1. Let A c X be oempty, G E be the first elemet of which meets A, ad x E G A. The F(x,) = G. If Y E G A, the also F(y,) G. Thus x = y. Hece, x is a isolated poit of A. The proofs of the followig two theorems are similar to the proof of 2.3 ad are left to the reader.

13 326 Ruth 2.4 Theorem. A space X is a-{closed ad scattered) if ad oly if it has a sequece < : E N> of well-ordered ope covers such that F is closed s-primitive a. 2.5 Theorem. A space X is a-scattered if ad oly if it has a sequece < : E N> of well-ordered ope covers such that F is s-primitive a. From the above characterizatios, we ca ow see the precise relatioship of primitive a-spaces to those which are a-scattered. These characterizatios, however, are extrisic i that the fuctio F is defied by a sequece of well-orderigs. I order to elimiate the explicit wellorderigs we provide purely fuctioal characterizatios by applyig Wicke's cocept of a iitial fuctio foud i 2.6 Defiitio (Wicke, [Wi 2 ]).. A fuctio h: N x X ~ J is called iitial if ad oly if for j E Nad A ~ X, if A ~ ~, there exists a E A such that for all b E A, either h (j, a) = h (j,b) or b ~ h (j, a). This gives rise to the followig characterizatio of primitive base. 2.7 Theorem [Wi 2.2]. A space has a pripitive base 2, if ad oly if it has a iitial fuctio which is first coutable {i.e., for all p E X, {h(j,p): j E N} is a base at p).

14 TOPOLOGY PROCEEDINGS Volume I the same maer, we obtai characterizatios of scattered, a-(closed ad scattered), a-scattered ad primitive a. We state here oly the characterizatio of primitive a as the others will be apparet i light of Defiitio Theorem. A space X is a primitive a-space if ad oly if it has a iitial fuctio h such that h(,x) h(,x ) for all E N implies (x : EN> clusters to x. Proof. Suppose X is a primitive a-space, ad let ( U : EN> witess this. It is ot hard to show that the fuctio h defied by h(,x) = F(X,U ) is a iitial fuc tio satisfyig the coditio above. To show sufficiecy, suppose X has a iitial fuctio satisfyig the coditio above. Followig the proof of [Wi 2, Theorem 2.2], we obtai a sequece (W j : j E N> of well-ordered ope covers such that F(x,W.) = h(j,x). J Refereces [Ch ] J. Chaber, O poit-coutable collectios ad mool toic properties, Fud. Math. 94 (1977), [Ch ] 2, Primitive geerazizatios of a-spaces, Colloquia Mathematica Societatis Jaos Bolyai 23 (1978), P. Fletcher ad W. F. Lidgre, O w~-spaces, wa-spaces ad ~#-spaces, Pacific J. Math. 71 (1977), [FL ], 8-spaces, Ge. Top. Appl. 9 (1978), [He] R. W. Heath, O certai first coutabze spaces, Topology Semiar, Wiscosi 1965, Aals of Math. Studies 60 (1966),

15 328 Ruth [Ho] [Will [Wi ] 2 [WW) R. E. Hodel, Spaces defied by sequeces of ope covers which guaratee that certai sequeces have cluster poits, Duke Math. J. 39 (1972), H. H. Wicke, Usig 8-space cocepts i base of coutable order theory, Topology Proc. 3 (1978), , A fuctioal characterizatio of primitive base, Proc. Amer. Math. Soc. 72 (1978), ad J. M. Worrell, Jr., Spaces which are scattered with respect to collectios of sets, Topology Proc. 2 (1977), Shippesburg Uiversity Shippesburg, Pesylvaia 17257