Volume 1, 1976 Pges 253 260 http://topology.uburn.edu/tp/ A NOTE ON PREPARACOMPACTNE by J. C. mith Topology Proceedings Web: http://topology.uburn.edu/tp/ Mil: Topology Proceedings Deprtment of Mthemtics & ttistics Auburn University, Albm 36849, UA E-mil: topolog@uburn.edu IN: 0146-4124 COPYRIGHT c by Topology Proceedings. All rights reserved.
TOPOLOGY PROCEEDING Volume 1 1976 253 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 1 1 ] 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:
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 4. 2. 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
TOPOLOGY PROCEEDING Volume 1 1976 255 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.
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 2.4. 3. ~ 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
TOPOLOGY PROCEEDING Volume 1 1976 257 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
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].
TOPOLOGY PROCEEDING Volume 1 1976 259 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), 105-108. 2. H. R. Bennett nd D. J. Lutzer, A note on wek e-refinbility, Gen. Top. Anl. 2 (1972), 49-54. 3. J. Boone, On irreducible spces, Bull. Aust. Mth. oc. 12 (1975), 143-148. 4., On irreducible spces II, Pcific J. Mth 62 (1976), No.2, 351-357. 5. R. C. Briggs, Preprcompctness nd ~ -preprcompctness in q-spces, Colloq. Mth. (1973), 227-235.
260 mith 6. U. Christin, Concerning certin miniml cover refinble spces, Fund. Mth. 76 (1972), 213-222. 7. E. Michel, A note on prcompct spces, Proc. Amer. Mth. oc. 4 (1953), 831-838. 8. J. mith, Properties of wek 8-refinble spces, Proc. Amer. Mth. oc. 53 (1975), 511-517. 9., A remrk on irreducible spces, Proc. Amer. Mth. oc. 57 (1976), 133-139. 10. J. M. Worrell, Jr. nd H. H. Wicke, Chrcteriztions of developble topologicl spces, Cnd. J. Mth. 17 (1965), 820-830. 11. P. Zenor, ome continuous seprtion xioms, Fund. Mth. 90 (1975/76), No.2, 143-158. Virgini Polytechnic Institute nd tte University Blcksburg, Virgini 24061