A NOTE ON PREPARACOMPACTNESS

Transcription

1 Volume 1, 1976 Pges A NOTE ON PREPARACOMPACTNE by J. C. mith Topology Proceedings Web: Mil: Topology Proceedings Deprtment of Mthemtics & ttistics Auburn University, Albm 36849, UA E-mil: topolog@uburn.edu IN: COPYRIGHT c by Topology Proceedings. All rights reserved.

2 TOPOLOGY PROCEEDING Volume A NOTE ON PREPARACOMPACTNE J. C. mith 1. Introduction In 1973 R. C. Briggs [5] introduced two properties, preprcompctness (ppc) nd ~ -preprcompctness (~-ppc) nd compred them with the properties of prcompctness nd collectionwise normlity in vrious q-spces. The purpose of this pper is to show tht most of the results obtined in [5] cn be generlized, hence closing the somewht lrge gp between these properties. Definition 1.1 A T spce X is pj?eprcompct (resp. ~- 2 preprcompctj if ech open cover of X hs n open refinement JC = {Hex: ex E A} such tht, if B c A is infinite (resp. uncountble) nd if Ps nd qs E H for ech E B with Pex ~ Ps nd qex # qs for ex #, then the set Q = {q: E B} hs limit point whenever P = {P: E B} hs limit point. The notions of 0-ppc nd 0- ~ -ppc should be cler. Collections stisfying the bove property will be clled ppc( ~ -ppcj collect-ions. ince neither of the bove properties implies prcompctness, even in the presence of collectionwise normlity, the specil setting of q-spces is chosen for their study. Definition 1.2 A spce X is clled q-spce if ech point p E X hs sequence of neighborhoods {N such tht if i }:=1 y. E N. for ech i with y. # y. for i # j, then the set {y.}~ ] 1 1= hs limit point. In [5] Briggs obtined the following. Theorem 1.3 Let X be pegulp q-spce. Then the following re equivlent:

3 254 mith (1) X is prcompct. (2) X is ~ -ppc nd subprcompct. (3) X is X -ppc nd metcompct. ince the notion of e-refinbility of J. Worrell nd H. Wicke [10] is generliztion of both subprcompctness nd metcompctness, it is nturl to sk whether the bove result cn be generlized ccordingly. In 2 of this pper we ctully obtin much stronger result using the notion of irreducible spces [6]. Theorems involving the properties of oe-refinbility [1] nd wek 88-refinbility [9] re obtined in 3, nd in 4 it is shown tht every - ~ -ppc, norml q-spce is collectionwise norml. Exmples nd open questions re lso included in Irreducible q-spces Definition 2.1 An open cover of topologicl spce X is clled miniml provided no proper subcollection of covers X. A spce X is clled irreducible if every open cover of X hs miniml open refinement. The following lemms re esy to verify nd hence the proofs re omitted. Lemm 2.2 Let = {G : E A} be n open cover of n irreducible spce X. Then hs miniml refinement Lemm 2.3 A cover U") = {W : E A} is minim l cover of X iff there exists discrete collection of non-empty closed sets {F: E A} such tht F ~ w for ech E A. Theorem 2.4 Let X be q-spce nd let = {G : E A} be ~ -ppc collection of open subsets of x. If there exists discrete collection {D : E B} of non-empty subsets of X such

4 TOPOLOGY PROCEEDING Volume th t DB <:= G for ech E B <:= A~ then {G B: E B} is ei ther countble or loclly finite. uppose B is uncountble nd {G : E B} is not loclly finite t p E X. ince X is q-spce, there exists countble subcollection {G. }:=l of nd sequence of points 1 (i) for ech i, Pi E G. 1 (ii) p. t p. nd G. t G. for i t j, 1 J 1 J (iii) {p. }~ 1 hs limit point in X. 1 1= Now let qs E D for ech E B nd define P = qs for ll ~ {i: i=1,2, }. Then P {P: E B} hs limit point while Q = {q: E B} does not. This contrdicts the fct tht is n ~ -ppc collection. Hence {G : E B} is loclly finite. Remrk: If ~ -ppc is replced by ppc in the bove theorem then {G : E B} is loclly finite in ech cse. Theorem 2.5 Let X be regulr q-spce. Then X is prcompct iff X is ~ -ppc nd irreducible. The necessity is cler. Let X be ~ -ppc nd irreducible nd let GU be ny open cover of X. Then GtL hs n open ~ -ppc refinement = {G : E A}. ince X is irreducible hs n open refinement J( which covers X minimlly. By Lemm 2.2 bove we my ssume tht X = {H : E B} where H ~ G for I3 ech B E B C A. By Lemm 2.3 there exists discrete collection of non-empty closed sets {D : E B} such tht D ~ H for ech I3 E B. Therefore, {G : E B} is -loclly finite open refine ment of ~, nd hence X is prcompct by Theorem 1 of [7]. Corollry 2.6 Let X be q-spce. Then X is ppcompct iff X is ppc nd irreducible. The proof follows immeditely from the remrk fter Theorem 2.4 bove.

5 256 mith Corollry 2.7 Let X be regulr q-spee. Then the follo~ing re equivlent: (1) X is preompet. (2) X is X-ppe nd 8-refinble. (3) X is X -ppe nd ~ek 8-refinb leo In [9] the uthor hs shown tht 8-refinble nd wek 8-refinble spces re irreducible. Remrk: It should be noted t this point tht the bove results (ssuming regulrity) remin true when ~-ppc is replced by - ~ -ppc by Theorem ~ e-refinble pces In [1] Aull proved tht Xl-compct 88-refinble spces re Linde16f nd in [8] the uthor obtined n nlogous result for wek 88-refinble spces. Definition 3.1 A spce X is clled o8-refinble if every open cover X hs refinement = U ~ 1. stisfying, 1= 1 (i) ech is n open cover of X. (ii) for ech x E X there exists n integer n(x) such tht ord(x, n(x)) 2- ~o Definition 3.2 A spce X is clled wek 88-refinble if every open cover of X hs refinement = U :=1 i stisfying, (i) ech i is collection of open subsets of X. (ii) for ech x E X there exists n integer n(x) such tht o < ord (x, ~ n (x)) < X 0 (iii) {Gi U{G: G E i}}:=l is point finite. Even though o8-refinble spces need not be irreducible it is nturl to sk whether similr results to those in 2 cn be obtined since such spces re generliztions of 8-refinble spces. Here we provide such results using the notion of mximl

6 TOPOLOGY PROCEEDING Volume distinguished sets, due to Aull [1]. Let Gl.L be n open cov'er of topologicl spce X. Definition 3.3 A set M is distinguished with respect to Gl.L if for ech pir x, y E M with x ~ y, then x E U E (~ => Y t U. Lemm 3.4 For every subset M of spce X nd every open (in X) cover Gl.L of M." there exists mximl distinguished set wi th respect to Gl.L which is discrete in U{U: U E GlL}. Theorem 3.5 Let X be regulr q-spce. Then X is prcompct iff X is ~ -ppc nd 88-refinble. Let X be ~ -ppc nd o8-refinble nd let GlL be n open cover of X. Then GlL hs n ~ -ppc refinement = {G : E A}. ince X is o8-refinble, hs refinement U ~ lw, stisfying, 1= 1 (i) ech W. {W(,i): E A} is n open cover of X, 1 (ii) for ech x E X, there exists n integer n(x) such tht ord(x' n(x)) 2. ~o As before we my ssume W(,i) C G for ech E A nd ech i. Now let H {x: ord(x, ) < ~ } so tht X = U OO lh. Let M n n - 0 n= n n be mximl distinguished set of H with respect to for n n ech n. By Lemm 3.4 the collection of singletons of points of ech M is discrete collection in X. By Theorem 2.4 bove n H n is covered by -loclly finite subcollection of ~n for ech n. ThereforeGlL hs -loclly finite open refinement, nd hence X is prcompct. The nlogous result for wek 88-refinble spces is lso true. The proof is modifiction of the one bove nd hence is omitted. Theorem 3.6 Let X be regulr q-spce. Then X is prcompct iff X is.~ -ppc nd wek 86"-refinble. 4. Norml-q-spces

7 258 mith In [5] Briggs obtined the following result using somewht involved rgument. We now generlize this result using theorem of Zenor [11]. Theorem 4.1 (Briggs) Let X be norml q-spce. If X is ~-ppc~ then X is collectionwise norml. Theorem 4.2 (Zenor) A spce X is collectionwise norml iff fo! ech discrete collection {F: E A} of closed sets~ there exists sequence of collections {V{,i): E A}~=l of open subsets of X stisfying~ (i) {V(,i)}~=l covers F for ech E A~ (ii) F n [U s~v(,i)]- ~ for ech E A nd ech i. Theorem 4.:3 Let X be norml q-spce. If X is - ~ -ppc~ then X is collectionwise norml. Let {F: E A} be n uncountble discrete collection of closed subsets of X. ince X is norml there exists for ech E A n open set G contining F such tht IT n [U ~ F] IJr IJ =~. We my ssume tht 0 ~ A. Then let GO = X - [U EAF]' nd = {G : E A} U {GO}. ince X is -~-ppc, hs re finement U c:' IX. where X. = {H(,i): E A} hs the ~-ppc 1= 1 1 property nd H(,i) ~ G for ech E A nd ec~ i. Let JCi = {H{,i): H{,i) n F t ~} for ech i. Then by Theorem 2.4, ech JC~ 1 is either countble or loclly finite so tht {H{,i): E A}:=l stisfies the conditions of Theorem 4.2 bove. Therefore X is collectionwise norml. Briggs [5] used severl exmples to demonstrte the necessity of specil setting (q-spces) in order to study the reltionships between preprcompct spces nd other more common generliztions of prcompctness. These exmples re summrized here for the benefit of the reder. For more detils see [5].

8 TOPOLOGY PROCEEDING Volume Exmple I: A countbly compct, first countble, norml q-spce which is ppc nd collectionwise norml but not prcompct. Exmple II: A first countble, collectionwise norml q- spce which is not ~ -ppc. Exmple III: A norml, metcompct, ppc spce which is not collectionwise norml. Exmple IV: A regulr, loclly countbly compct q-spce which i s ~ -ppc nd -ppc but not ppc. Exmple V: A regulr, countbly compct, q-spce which is ppc but not norml. Exmple VI: A metcompct, first countble, Lindelof q- spce which is ~ -ppc but not regulr. everl interesting open questions remin: (1) Is every regulr, first countble, ppc spce norml? (2) Is Theorem 3.5 true for wek e-refinble spces? (3) In wht setting, other thn q-spces, re the bove results true? (4) When re ppc spces expndble? (5) When re ~ -ppc spces countbly prcompct? References 1. C. Aull, A generliztion of theorem of Aquro, Bull. Aust. Mth. oc. 9 (1973), H. R. Bennett nd D. J. Lutzer, A note on wek e-refinbility, Gen. Top. Anl. 2 (1972), J. Boone, On irreducible spces, Bull. Aust. Mth. oc. 12 (1975), , On irreducible spces II, Pcific J. Mth 62 (1976), No.2, R. C. Briggs, Preprcompctness nd ~ -preprcompctness in q-spces, Colloq. Mth. (1973),

9 260 mith 6. U. Christin, Concerning certin miniml cover refinble spces, Fund. Mth. 76 (1972), E. Michel, A note on prcompct spces, Proc. Amer. Mth. oc. 4 (1953), J. mith, Properties of wek 8-refinble spces, Proc. Amer. Mth. oc. 53 (1975), , A remrk on irreducible spces, Proc. Amer. Mth. oc. 57 (1976), J. M. Worrell, Jr. nd H. H. Wicke, Chrcteriztions of developble topologicl spces, Cnd. J. Mth. 17 (1965), P. Zenor, ome continuous seprtion xioms, Fund. Mth. 90 (1975/76), No.2, Virgini Polytechnic Institute nd tte University Blcksburg, Virgini 24061