REMARKS ON CLOSURE-PRESERVING SUM THEOREMS

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1 Volume 13, 1988 Pages REMARKS ON CLOSURE-PRESERVING SUM THEOREMS by J. C. Smith ad R. Telgársky 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 REMARKS ON CLOSURE-PRESERVING SUM THEOREMS J. C. Smith ad R. Telgarsky 1. Itroductio I 1975 Jui1a ad Potoczy [6J ad Katuta [2J idepedetly showed that, if a space X has a c1osurepreservig cover by com~act sets, the X is metacompact. Thus bega a study of spaces which have closure-preservig covers by "ice" (fiite, coutab1y compact, Lidelof, etc.) sets. The reader is referred to [1,2,4,5,6,8,11,12, 13,16,17,18,19,24J for these developmets as well as their relatioship with wiig strategies i some topological games. I 2 we show that if a space X has a closurepreservig cover by owhere dese sets, the X eed ot be 1 st category. However, if these sets are also coutably compact, the X must be 1 st category. The c1osurepreservig property of the cover is stregtheed i a atural way i 3, ad a somewhat more geeral result is obtaied for the ideal of closed subsets of a space. The desired result for owhere dese sets follows as a special app ~cat~o. Fially i 4 closure-preservig sum theorems are obtaied for the dimesio fuctios dim ad Id for ormal ad totally ormal spaces. lthe authors would like to thak the referee for his/her commets o this paper.

3 , 126 Smith ad Telgarsky Defiitio 1.1. (i) A family J of subsets of a space X is closure-preservig (c-p) if U{P: F E J'} = 01]' for every J' ~ J. (ii) A c-p family J is special if there exists a poit fiite ope collectio U such that for F E J, X - F U{u E U: U F = ~}. I this case we say that U geerates J. Defiitio 1.2. A family J of closed subsets of a space X is a ideal of closed sets if, (i) for every fiite J' ~ J, ~ J' E J ad (ii) if H is a closed set ad H C J E J, the H E J. We will deote the fa~ly of coutable uios of members of J by aj. Remark. As oted i [1], if lj is a poit fiite ope cover which geerates a c-p J-cover, the the family C = {X[Ux]: x E X ad lux I = } cosists of J-small sets, ad C is a discrete collectio of relative closed sets i X - X - l, where X = {x EX: I IJ I ~ } x Defiitio 1.3. A space X is called weakly a-refiable if every ope cover of X has a ope refiemet U~=l ~i satisfyig (i) {Gi}~=l is poit fiite, where G i = U i ad (ii) for each x E X, there exists some (x) such that o < ord(x'~(x» < 00

4 TOPOLOGY PROCEEDINGS Volume A cover U~=l i satisfyig (i) ad (ii) above is called a weak a-cover. It is well kow that the class of weakly e-refiable spac~s lies strictly betwee the classes of e-refiable ad weakly e-refiable spaces (see [8J). All spaces will be assumed to be T Closure-preservig covers by owhere dese sets I this paper we cosider spaces which have a closurepreservig cover by owhere dese sets. Note that if Q = {r l,r, } is the set of ratioals ad F 2 the {F.: i ~ {ri}i=l' < w} is a closure-preservig cover of Q by fiite sets. Clearly, Q is a 1 st category space. The followig example shows that if a space X has a closure-preservig cover by owhere dese sets, the X eed ot be of the 1 st category. Example. Let L = {O,l}wl where {O,l} is the two poit discrete space. The topology o Y is the coutable box topology; that is, a basic ope set i Y is a coutable itersectio of basic ope sets i the usual product topology.. Defie X {y E Y: Ia < U].: y (a) ~ 0 I < wi} ~ow if Fa = {x E X: x(b) o for each B ~ a}, it is easy to check that {Fa: a < WI} is a closure-preservig cover of X by discrete closed sets. Furthermore each Fa is owhere dese. We assert that X is ot 1 st category. Ideed, let {G : < w} be a sequece of ope dese sub sets of X. Let B be a basic ope set such that B ego' O O ad defie

5 128 Smith ad Telgarsky Co = {a < wi: x(a) = y(a) for all x,y E B }. O Now by iductio choose a basic ope set B ~ G B - l ad let e {a < WI: x(a) = y(a) for all x,y E B }. Sice C ~w C is.coutable, the~e exists a poit z E X such that z E Q B ~ Q G ad z(a) ~ 0 for w w a > supc. Therefore X is ot 1 st category. The followig result is a easy cosequece of Zor's Lemma so the proof is omitted. Lemma 2.1. Let J = {F : a E A} be ay o-empty a famizy of o-empty sets. pairwise disjoit subfamizy of J. The there exists a maximal Defiitio 2.2. Let ~ e E C X ad J a family of subsets of X. (1) m(e) = a maximal pairwise disjoit subfamily of {F E: F E J}. (2) M* (E) = U m(e). The followig uses a techique due to Telgarsky ad Yajima [16: Theorem 3.3]. Lemma 2.3. Let C = {C : a E A} be a czosure-pre CL servig cover of X by coutably compact sets. {H}~=l is ay decreasig sequece of czosed subse~s oo X satisfyig H + 1 ~ X - M*(H ) for each " the. lh = <p. = Proof Suppose there ~xists a sequece of {H}ri=l closed sets satisfyig the above coditio, If but of

6 TOPOLOGY PROCEEDINGS Volume O H -J The there exists some C E [ such that =l T. ao C H ~ for each. Furthermore (C H) ~ m(h )i ao ao for otherwise, C H 1 =. Therefore for each, there ao + exists some C E [ such that (C H ) E m(r ) ad a a C (C H ) ~. Now choose x E C (C H) so a ao a ao that x E M*(H ) ad hece x e H + l. The {x}~=l is a sequece of distict poits i C ad must cluster at ao y E C Now Y E {x : > I} C U OO lc ad y E ~oo lh a O - - = a = as well. But this is a cotradictio, sice Y E (C (IH) E ftj(h ) implies that y ~ H + l. Therefore a it must be the case that O lh =~. = It' Remark. It should be oted that the above proof oly used the fact that every coutable subfamily of [was closure-preservig. Theorem 2.4. Let X be a regular space with a closurepreservig cover Cby coutably compact owhere dese sets. The X is 1 st category. Proof. Sice M*(X) is closed ad owhere dese i X, by LeITma 2.1 above, there exi?ts a maximal pairwise disjoit family H(X) of regular closed subsets of X such that U H(X) c X - M*(X}. Furthermore, it is easy to see that U H(x) is dese i X. Defie O = {it(h): H E H(X)} ad GO = Uyo Note that GO is also dese i X. Now by iductio we costruct the sequece ( O,GO' l,g1',g ) where ~+l = {it(h): H E H(G) where G E ~}' G + l = U Y + l ad

7 130 Smith ad Telgarsky = a maximal pairwise disjoit family of regular closed subsets of X such that U#(G) is dese i G - M* (G) As before it is easy to see that G is ope ad dese i X. Fially, if H = X ad H E #(H ) 1 by Lemma 2.3, O + l st =OH =~. Therefore =OG = ~ ad hece X 1S category Corollary 2.5. Let X be a regular ooutably oompaot spaoe. If J is a olosure-preservig oover of X, the some member of J has o-empty iterior. 3. Special c-p covers I [1] the authors studied spaces havig c-p J-covers iduced by a poit fiite ope cover. Here we get aalogous results to Theorem 2.4 above by omittig the regularity ad coutably compact coditios. The otio of B(D,A)-refiability has bee show to play a importat role i the study of weakly e-refiable spaces itroduced i [7J. Defiitio 3.1. A space X is B(D,A)-refiable provided every ope cover lj = {U : y E r} of X has a "t refiemet C= U{Ca = fe(b,y):y E r}: a < A} where satisfies (i) E(a,y) C U for each y E r, a < A, y (ii) {Lt : a < A} partitios X, a (iii) for each a < A, Ca is a closed discrete collectio i X - U{U C ; 1..1 < a}, ad 1..1 (iv) for each a < A, u{uf : 1..1 < S} is closed i x

8 TOPOLOGY PROCEEDINGS Volume I [7J the author has show that every space which is B(O,w)-refiable is weakly e-refiable ad that every weakly a-refiable space is B(O, (w)2)-refiable. Throughout this paper A will deote a coutable ordial. Defiitio 3.2. Let J be a ideal of closed subsets of a space X. We say that J is discretely additive if (i) J i~ closed uder discrete uios, ad (ii) whe F E a] ad 0 C J such that 0 is discrete i X - F, the H = F U {u ij} E oj. Theorem 3.3. If a space X has a B(O,A) - J cover for some coutable ordial A ad J is discretely additive~ the X E a]. Proof y E r}, be a B(O,A) - Let C = U{C S : S < A}, where Cs = {E(S,y): J cover of X such that J is discretely additive. The proof is by iductio o A. It is easy for A = 1; sice J is closed uder discrete uios, so X E J. Let S < A ad assume true for all y < S. S a O + 1, defie F = U{UC : II ~ ao} so that F E a]. ll Now Cs is discrete i X - F ad Cs ~ J, ad hece H S F U {UCa} E aj. If B is a limit ordial, let a i < B such that a~ < a.+ 1 ad sup{a.} = S. Now for each i, ~ ~ ~ H. = U{C II < a.} E a], so H Q = U~ lh. E a]. Therefore ~ ll - ~ ~ ~= ~ X E a]. We ow cosider the ideal of closed owhere dese subsets of a space X. If

9 132 Smith ad Te1garsky Lemma 3.4. Let J be the ideal of closed owhere dese subsets of a space X. The J is discretely ad~itive. Proof Clearly, J is closed uder discrete uios. Let F E 01 ad ij = {D : a. E A} c J such that ij is discrete a. i X - F. Let W be ay o-empty ope subset of X such that W C H F U {UiJ} If U = it(f), the W ~ U; otherwise, W Do. ~ ~ for some a. E A. Thus Do. fails to be owhere dese i X. Therefore, U{(D - U): a. E A} E J; a. ad hece H E a]. Corollary 3.5. owhere. dese sets. Let X be a space with B(D,A)-cover by The X is 1 st category. Proof. Let J be the ideal of closed owhere dese subsets of X. The J is discretely additive by Lemma 3.4, ad hece X E oj by Theorem 3.3. Corollary 3.6. If a space X has a special c-p cover by owhere dese sets~ the X is 1 st category. 4. Applicatios to Dimesio Theory I [24] the author obtaied the followig result which also ca be foud usig a techique similar to that of Theorem 2.4 above. Theorem 4.1. [24]. Let X be a ormal space. If X has a c-p cover C by coutably compact sets such that dim(c) < for C E C~ the dim(x) ~.

10 TOPOLOGY PROCEEDINGS Volume To obtai the special closure-preservig sum theorem for coverig dimesio as a applicatio of Theorem 3.6 above, we eed the followig result foud i [24]. Lemma 4.2. Let X be a ormal space ad F a closed subset of X with dim(f) ~. For each fiite ope cover u= {U : i i < m} of X, there exists a ope refiemet V ={Vi: i < m} of U, - - (i) Vi ~ for each i < m, ad ad a ope set G ::) F such that U i (ii) ord(x, V) < + 1 for each x E - G. Lemma 4.3. Let X be a ormal space ad F a closed subset of X with dim(f) ~. Let 0 = {D : a E A} be a a family of closed subsets of X such that (i) dim(d ) ~ for each a E A, ad a (ii) 0 is discrete i X-F. The dim(f U {U l)}) ~ Il. Proof. Let U = {U.: i < m} be a fiite ope (i X) J. cover of H F U {Ul)}. By Lemma 4.2 there exists a ope refiemet V {Vi: i < m} of Uad a ope set G ~ F such that ord(x, V> < + 1 for all x E G. Now {D - G: a E A} a is a discrete closed collectio i X with dim(da. - G) < - for each a E A. Therefore dim(u o - G) < ad hece V - has a partial ope refiemet W= {W.: i < m} coverig ~ {U l) - G), such that W. C V. ad ord(x,w) < + 1 for ~ - J. x E (UO- G). Defie W~ = W.U(V. - G) for i < m. It is ~ ~ ~ easy to show that W* = {W~: i < m} is a ope refiemet ~ of lj coverig Had ord(x,w*) < + 1 for x E H. Therefore dim(h) <.

11 134 Smith ad Telgarsky We ow obtai the desired c-p sum theorem for coverig dimesio. Theorem 4.4. Let X be a ormal space which has a B(D,A) cover Csuch that dim(e) < for each E E C. The dim(x) <. Proof. Let J be the ideal of closed subsets of X such that dim(j) < for J E J. Sice coverig dimesio satisfies the coutable sum theorem for ormal spaces, J = oj. Therefore J is discretely additive by Theorem 4.3 above. Now from Theorem 3.3 it follows that X E oj = J. Hece dim(x) ~. Corollary 4.5. Let X be a ormal space. If X has a special c-p cover C, such that dim(c) < for each C E C, the dim (X) <. For totally ormal spaces similar results hold for large iductive dimesio usig the lemma below. The proofs are straightforward ad left for the reader. Yajima [18] has obtaied a a-closure-preservig sum theorem for Id. Lemma 4.6. Let X be totally ormal ad F a closed subset of X such that Id(F) ~. If Id{K) < for each czosed subset K of X such that K F = ~, the Id(K) ~. Theorem 4.7. Let X be a totazly ormal space. If X has a special c-p cover Csuch that Id(C) ~ for C E C, the Id(X) <.

12 TOPOLOGY PROCEEDINGS Voll.lll1e Refereces [lj H. Juila J. C. Smith, ad R. Telgarsky, Closurepreservig covers by small sets~ Top. Appl., 23 (1986) [2] Y. Katuta, O spaces which admit closure-preservig covers by compact sets~ Proc. Japa Acad., 50 (1974) [3J A. R. Pears, Dimesio theory of geeral spaces, Cambridge Uiv. Press (1975). [4] H. B. Potoczy, A oparacompact space which admits a closure-preservig cover of compact sets~ Amer. Math. Soc., 32 (1972) Proc. [5] H. B. Potoczy, Closure-preservig families of compact sets~ Ge. Topology Appl. 3 (1973) [6] H. Potoczy ad H. Juila, Closure-preservig families a d metacompactess~ 53 (1975) Proc. Amer. Math. Soc., [7] J. C. Smith, Irreducible spaces ad property bl~ Topology Proceedigs, Vol. 5 (1980) [8] J. C. Smith, Properties of weak 6-refiable spaaes~ Proc. Arer. Math. Soc., 53 (1975) [9] J. C. Smith ad L. L. Krajewski, Expadability ad aolleatiowise ormality~ Tras. Amer. Math. Soc., 160 (1~71) [10J J. C. Smith ad R. Te1garsky, Closure-preservig aovers of cr-produats~ Proc. of Japa Acad., Vol. 63, No. 4 (1987) [11] H. Tamao, A aharaaterizatio of paraaompaatess, Fud. Math., 72 (1971) [12] R. Te1garsky, C-scattered ad paracompact spaces~ Fud. Math., 73 (1971) [+3] R. Te1garsky, Closure-preservig covers~ Fud. Math., 85 (1974) [14] R. Telgarsky, Spaces defied by topological games~ Fud. Math., 88 (1975)

13 136 Smith ad Telgarsky [15J R. Telgarsky, Spaces defied by topological games, II, Fud. Math., 116 (1983) [16J R. Telgarsky ad Y. Yajima, O order locally fiite ad closure-preservig covers, Fud. Math., 109 (1980) [17J Y. Yajima, O spaces which have a closure-preservig cover by fiite sets, Pacific J. Math., 69 (1977).---,~-._.-, ---" [18J Y. Yajima, O order star-fiite ad closure-preservig covers, Proc. Japa Acad., 55 (1979) 19-21~ [19J Y. Yajima, A ote o order locally fiite ad closurepreservig covers, Bull. Acad. Po1o. Sci. Sere Math., 27 (1979 ) [20 ] Y. Yajima, Topological games ad products I, Fud. Math., 113 (1981) [21] Y. Yajima, Topological games ad products II, Fud. Math., 117 (1983a) [22 J Y. Yajima, Topological games ad products III, Fud. Math., 117 (1983b) [23J Y. Yajima, Notes o topological games, Fud. Math., 121 (1984) [24 J Y. Yajima, Topological games ad applicatios, (to appear). Virgiia Polytechic Istitute ad State Uiversity Blacksburg, Virgiia Florida Atlatic Uiversity Boca Rato, Florida Uiversity of Texas at El Paso El Paso, Texas 79968