A Note on the Theorem of Baturov

BULL. AUSTRAL. MATH. SOC. 54A25, 54C60, 54E18 VOL. 76 (2007) [219-225]

A NOTE ON THE THEOREM OF BATUROV

MARIA MUNOZ

D.P. Baturov proved in 'Subspaces of function spaces' Vestnik Moskov University

Series I (1987) that Lindelof degree equals extent for subspaces of CP(X) when X is a Lindelof S-space. We prove that if the Lindelof degree of the subspace is "big enough" the equality is true for a topological space X not necessarily Lindelof E.

1. INTRODUCTION

Baturov proved in [5] that Lindelof degree equals extent for subspaces of CP(X) when X is a Lindelof S-space. This classical theorem answers a question of A. V. Arkhangel'skii. Later, Buzyakova established in [8] a connection between the D-property and Baturov's theorem. A topological space X is a D-space if, and only if, given a neighbourhood assignment {N(x) : x G X}, that is, x € IntiV(x) for each x e X, there is a closed discrete subset D of X such that X = u{iV(x) : x £ D}, [10], see also [3, 4, 12, 6].

Buzyakova showed that CP(X) is hereditarily D whenever X a compact space, see [8]. Gruenhage describes the theorem of Buzyakova as an "explanation" of Baturov's result, whose proof is quite technical. Recently, Gruenhage has proved the same result for X a Lindelof E-space, [13]. The theorem of Baturov is established in terms of Lindelof E-spaces. The notion of Lindelof E-space was introduced in [17] (in different terminology K-countably determined space, [18]). We use the following characterisation of Lindelof E-spaces (see [18, pp. 95- 96]). DEFINITION 1.1: A topological space X is said to be Lindelof E-space if it is the image of some subset of NN under an upper semicontinuous compact-valued map. The class of Lindelof E-spaces is the minimal class which contains all second countable spaces and all compact spaces and is closed with respect to finite products, closed subspaces and continuous images. The goal of this paper is to give a different point of view of the theorem in the way that the restrictive hypothesis is not supported by the space X or even by the subspace

of CP(X). Our result is better in the sense that X is not necessarily a Lindelof E-space.

Received 8th January, 2007 Copyright Clearance Centre, Inc. Serial-fee code: 0004-9727/07 SA2.00+0.00.

219

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To achieve our aim, we use the invariant cardinal function index of K-determination. This cardinal function is close to the ideas of Nagami [17] and the cardinal functions defined by Hodel [14]. It has turned out to be an useful tool to establish relation- ships between different cardinal functions of a topological space [9]. The index of K- determination, £E(X) of a topological space X, measures how the space X is determined by its compact subsets via upper semicontinuous compact valued maps defined on metric spaces. Now we can enunciate the theorem of Baturov in these new terms. We shall prove

that for a subspace Y oiCp(X), the Lindelof number of Y is equal to the extent of Y, if the Lindelof number of the subspace is greater than or equal to the index of AT-determination of-Y.

2. NOTATION AND RESULTS All our topological spaces X, Y are assumed to be Tychonoff. Every Tychonoff space has a compactification. We denote by bX a compactification of X. Our basic references are [15, 2, 11, 16]. A cardinal number m is the set of all ordinals which precede it. In particular, m is a set of cardinality m. m+ is the smallest cardinal after m. A cardinal function is a function from the class of all topological spaces (or some precisely defined subclass) into the class of all infinite cardinals such that (j>{X) = (Y) whenever X and Y are homeomorphic. The requirement that cardinal functions take on only infinite cardinals as values simplifies statements of theorems. We assume that to all the cardinal functions we add the countable cardinal LJ. The weight of X, w(X), is the minimal infinite cardinality of a base for the topology of X. The Lindelof degree of X, denoted i(X), is defined as the smallest infinite cardinal m such that every open cover of X has a subcollection of cardinality ^ m which covers X. The density of X, d(X), is the minimal cardinal of an everywhere dense set in X. The extent of X e(X) is the supremum of the cardinalities of discrete closed sets in X. The hereditary density of X, hd(X), is sup{d(F) : Y C X}. The tightness of a point x in a topological space X, t(x, X), is the smallest infinite cardinal number m with the property that if x € A, then there exists AQ C A such that |>lo| ^ w and x G AQ. The tightness of a topological space X, t(X) is the supremum of all numbers T(X, X) for x 6 X. It is obvious that for every subspace Y of X, t(Y) ^ t(X). We say that a topological property V is hereditary for a space X if every subspace of X has the property V. A multi-valued map : X —> 2Y is said to be upper semicontinuous if it is compact valued and upper semicontinuous, that is, for every x G X the set (x) is compact, non- empty and for every open set V CY with (x) C V there is an open neighbourhood U of x such that (U) C V. In terms of nets, if : X —• 2Y is an upper semicontinuous map, (xj)j£j is a convergent net to x in X and (yj)jej is a net in Y such that Vj € ~~{xj) for every j € J, then we have that (y,-)i€J has a cluster point y which belongs to (x).~~

More about upper semicontinuous maps can be found in [7]. DEFINITION 2.1: Let X be a topological space. The number of K-determination of X £E(X), is the smallest infinite cardinal number m for which there are a metric space (M, d) of weight m and an upper semicontinuous map : M —> 2X such that x=\jMY When £E(X) is countable we simply get Lindelof E-spaces. The behavior of this invariant cardinal with respect to the usual operations and more properties can be found in [9]. We include in the proposition that follows some properties of €£ to make this paper self-contained. All of them can be found in [9].

~~PROPOSITION 2.2. Let X be a topological space. Then:~~

~~(ii) Let Xi be a topological space for i = 1,..., r, then~~

~~i = l,...,r}. t=l~~

(iii) t{Cp(X)) X PROOF: (i) Let (M,d) be a metric space with w{M) < 0£{X) and : M ->• 2 an upper semicontinuous map such that X = u{(t) : t € M}. Let O = {Oi : i 6 /} be an open cover of X. The set 4>{t) is compact for every t € T, hence there exists a finite set of indices i\,... t^ such that {t) C U{Oj« : j = 1,..., r^}. Since cj> is upper semicontinuous, for the open set O(t) := U{Ojt : j = 1,.. .nt} there exist an open set U{t) in M such that (£/(*)) c O(t). Now the" family, U = {U(t) : t € M} is an open cover of M. Since iu(M) = £(M), there exists a subcover {U(t) : t € 5} of M with |5| ^ IL(X). Thus,

X = U *(tf(*)) c U °W = U U{°.j = i = 1. • • •, "f}, tes tes tes and the proof is finished. (ii) There exist a metric space Mi with w(Mi) ^ £E(Mj) and an upper semicontinuous map fa : Mi -> 2Xi such that Xi = (J &(£) for each i = 1,..., r. The space

r M := n -Mi is a metric space with w(M) ^ sup{iu(Mj) : i = 1,... ,r} and the multi- i=l n * t i=1 valued map := \ x ... x T defined on M which takes values in 2 is an upper r semicontinuous map such that J~[ A"< is covered by the union (J 4>(t). «=i teM (iii) This is a consequence of a theorem of Arkhangel'skii, [1] see ([2, Theorem n II.l.l]) which affirms that if i{X ) ^ m for every n e N then t(Cp{X)) ^ m, where m is

an infinite cardinal number. Now (iii) is obvious because using 2.2 and 2.2 we have that £(Xn) ^ ffipf") sj £E(X) for every n € N. D We recall that a continuous map / : X —• Y is perfect if / is a closed map and all fibres f~l(y) are compact subsets of X. PROPOSITION 2.3. Let X be a topological space. Tien, there exist a metrisable space M with w(M) < tZ(X), a topological space Y with £Y,{Y) < £Z(X) and continuous functions f : Y —> X and p : Y ->• M such that f is onto and p is a perfect map. X PROOF: Let M be a metric space with w(M) ^ £Y,(X) and $ : M —>• 2 an upper semicontinuous map. We define Y := {(t,x) : x € {t)} C M x 6X. Now the set

y is closed in M x 6A\ Let (*,-, Xj)jey be a convergent net to (t, x) in M x kY. We shall prove that (i, x) € Y, that is x 6 0(t). Since {tj,Xj)j€j converges to (t,x), this implies that (tj)j^j converges to t in M and (xj)jgj converges to a; in bX. For each open neighbourhood U in Wf of x there exists ju such that if j ^ jV then Xj € {/. Since 0 is

upper semicontinuous, the net (XJ)J€J has a cluster point x* in X such that x* € (f>(t). If we assume that x ^ x* then there exist open sets V and V* in Wf such that x € V, x* e K* and V n V* = 0. But now there exist an open set U* in X such that x* G t/* and {7* C V* D A\ and j ^ jV such that Xj € £/* C V*, but also x_, e V, which is a contradiction, so x = x*, that is x € X and x £ {t). On the other hand, we have that ffi(Af x bX) ^ max{ffi(Af),ffi(6X)}, by 2.2 (ii). £E(6X) is countable since bX is a compact set and M is a metrisable space. Hence £E(M) ^ w(M) ^ £S(X). The index of /^-determination does not increase under closure [9], hence ffi(r) ^ £E(M x bX) < ffi(X). The maps / : y —> X and p : Y -*• M given by f(t, x) := x and p(t, x) := t are continuous. The space X is covered by the union L){/(t,x) = x, (t,x) € y}, hence / is onto. On the other hand, p is a perfect map. In fact, the set p"1 (t) = {t} x {t) is compact and we have by Kuratowski's theorem [11, Theorem 3.1.16] that p : M x bX ->• M given by p(t, x) = t is closed. Since y is closed, the map p is closed too and the proof is over. D Although in- the proof of the theorem that follows we follow the general argument used in the proof of the theorem of Baturov, we include all the steps and changes because of the technical arguments used in it.

THEOREM 2.4. Let X be a topological space and T C CP(X) be a subspace such that £(T) > tZ{X) then £(T) = e(T). PROOF: The inequality e(T) < £(T) is always true. To prove the other inequality, we may assume that X is a topological space such that there exists a continuous perfect map p defined on X which takes values in a metrisable space M with w(M) < i£(X). In fact, by proposition 2.3, there exist a metrisable space M with w(M) ^ £T:(X), a topological space Y with £L(Y) ^ £E(X), and maps / : Y -¥ X continuous onto and

~~p : Y -¥ M continuous and perfect. Since X is a continuous image of Y, the map~~

~~/ : CP{X) -¥ CP(Y) given by J{h) := h o / for every h G CP(X) is a homeomorphic~~

embedding. Let T C CP{X) be a subspace such that £(T) > iT,{X), then l(J{T)) > PL{Y). This inequality follows since 1{J{TJ) = £(T) > ffipQ ^ ffi(V). To prove that ^(T) ^ e(T) we shall prove that for every infinite cardinal m > £E(X), if e(T) > m, then e(T) > m. We suppose £(T) > m. There is an open cover 7 of T in which there is no subcover of cardinality less or equal to m. We can assume that the elements of the cover 7 have the form

~~U{xh,. ..,xin;Okl,.. .,Okn) = {/i G CP{X) : h(x{j) G Okj, j = l,...,n},~~

where x^ are points in X and Okj G O for j = l,...,n, where O = {Oi : i£ N} n is a standard countable base of R. For every n G N we define the family O := {Oix n x Oi2 x • • • x Oin : Oj. gO,i = l,...,n}. The cardinality of O is countable, so we can n consider O = {On,m : m € N}. We denote f/n,m(x) = U{xiit.. .,xin;Okl,. ..,Okn) n where x = (i*,,. ..,xin) and On>m = Okl x • • • x Okn e O . We define An

~~if, and only if, j4n,m ^ 0. The family 7 can be represented as 7 = u{7n,m : (n, m) £ P},~~

where 7n,ro = {C/n>m(x) : x € An)m}. For every map h G CP(X) and n G N we denote by n n n n h the map from X to R such that h (xi,..., xn) = (/i(ii),..., h(xn)) and by p" the n n map from X to M" such that p (x\,..., xn) = (p(ii),... ,p(xn))- With this notation, n we have that for each h G T there exist (n, m) G P and x G j4nim such that h (x) G On,m- The fact that there is no subfamily of 7 of cardinality less or equal to tn which covers

T can be written as if for every (n, m) G P we choose Sn,m C An,m and |Bn,m| ^ m, then n there is g G T such that g {Bn>m) n On>m = 0 for every (n,m) G P. + We construct by transfinite induction a set F = {ha : a < m } C T which is closed and discrete in T. Choose ho G T arbitrarily, and suppose that we have the functions {hp G T : /3 < a} where a < m+. For each n G N the space Rnr x Mn is a metrisable space and io(Rnr x Mn) ^ w{M) < ^S(X), [11, Theorem 2.3.13]. The hereditarily density degree of Rnr x Mn coincides with the weight, see [15, Theorem 8.1], hence hd{RnT x M") ^ ffi(X).

~~For each finite collection /3i,...,0r < a and n G N, we consider the map h^i~~

= /$,A... A/i^Ap" which is the diagonal product of the maps /i^,..., /ijjr and, p" defined on Xn and taking values in Rnr x Mn. Now we have that hjjx,...* is a perfect map since p is perfect, [11, Theorem 3.7.11].

~~For each {n,m) G P we can choose in An~~

~~such that ^, ^(Sft™ A) is dense in ^, ^(i4n>m). We'define i5«m =~~

~~Pi,.. .,0r < a}. It is obvious that |B"m| ^ max{m,^E(A')} = tn. We assume that we~~

have B*m for every (n, m) G P and hence we obtain that there exists ha G T such that + ftS(Bn,m)nOB,m = 0 for every (n, m) G P. The construction of the set F = {ha : a < m } is finished. We show that F is discrete and closed in T. Assume the contrary. Then there is n a point g G T which is a limit for F. For some (n,m) G P and xg G X , we have n g G £/n,m(x9), that is g {xg) G On>m. By proposition 2.2 (iii), t(Cp(X)) < £Z{X), and hence t(T) ^ m. Let ao be the smallest a' < m+ such that g is a limit point of the set

~~{ha : a < a'}, and we put~~

~~P={ha€F: hl(xg) G On,m, a < a0}.~~

n l 1 1 : h e p is not The set (g )~ (On,m) n njC ")" ^"^)) } empty because xg belongs to n l n n 1 the intersection. We define the set N = n{{h )- (h {xg)) : heP}\ (^ )- (On,m)- n We distinguish two situations. If N is empty, we put p (xg) = t. The map p" : Xn -i- Mn is perfect,- hence (p")"1^) is compact. Since the set (^"H^n.m) is open,

~~there is a finite set {hpt,..., hpr} C P such that l * = nU1(hl)- (hl(xg))n(P»)-\t) c (sT'tOvn).~~

I Now, * = (/i^ Pr)- {h}l(xa),...,h%r{xg),t). The map /i^ Pr is closed and the set n l 1 (g )~ (On,m) is open such that $ C (fl")" ^,™)- Since An,m n $ ^ 0, there exists a point y G /^...^(S^J such that (/ft ^{y) c (s")-1^,.,™), [11, Theorem 1 1.4.12]. Thus, there is a point x G S^'™ 0r D (s")" ^,™)- But then, by construction,

~~the function ha, for a > max{/?j : i = 1,.. .,r}, does not belong to the neighbourhood~~

~~Un,m{x) and g G Un,m(x). We define a* = max{/?j : t = 1,...,r}. Now, a* < a0, and~~

g is a limit point for the set {ha : a < a*}. But ao is the smallest cardinal with that property and we get a contradiction. n n On the other hand, if N is not empty, let i' G N. Then g {x') ^ g (xg), since n n n g {xg) G On,m. But h (x') = h (xg) for every h G P. This implies that 5 does not belong to the closure of P, which is a contradiction. We have proved that F is discrete and closed in T and the proof is finished. D The theorem of Baturov follows as corollary of the main theorem. COROLLARY 2.5. (Theorem of Baturov, [5]) Let X be a Lindelof E-space and

~~T C CP(X) a subspace ofCp{X), then e(T) = £(T).~~

REFERENCES [1] A.V. Arkhangel'skii, 'On some topological spaces that arise in functional analysis', Rus- sian Math. Surveys 31 (1976), 14-30. [2] A.V. Arkhangel'skii, Topological function spaces, Mathematics and its Applications (So- viet Series) 78 (Kluwer Academic Publishers Group, Dordrecht, 1989). Translated from the Russian by R.A.M.Hoksbergen.

[3] A.V. Arkhangel'skil and R. Buzyakova, 'Addition theorems and D-spaces', Comment. Math. Univ. Car. 43 (2002), 653-663. [4] A.V. Arkhangel'sku, 'D-spaces and finite unions', Proc. Amer. Math. Soc. 132 (2004), 2163-2170. [5] D.P. Baturov, 'Subspaces of function spaces', Vestnik Moskov. Univ. Ser. I Mat. Mekh. 4 (1987), 66-69. [6] C.R. Borges and A.C. Wehrly, 'A study on D-spaces', Topology Proc. 16 (1991), 7-15. [7] B. Cascales and L. Oncina, 'Compactoid filters and USCO maps', J. Math. Anal. Appl. 282 (2003), 826-845. [8] R. Buzyakova, 'Hereditarily D-property of function spaces over compact a', Proc. Amer. Math. Soc. 132 (2004), 3433-3439. [9] B. Cascales, M. Munoz, and J. Orihuela, 'Index of if-determination of topological spaces', (preprint). [10] E.K. van Douwen and W. Pfeffer, 'Some properties of the Sorgenfrey line and related spaces', Pacific J. Math. 81 (1979), 371-377. [11] R. Engelking, General topology (PWN-Polish Scientific Publishers, Warsaw, 1977). Trans- lated from the Polish by the author, Monografie Matematyczne, Tom 60. [Mathematical Monographs, Vol. 60]. [12] W. Fleissner and A. Stanley, 'D-spaces', Topology Appl. 114 (2001), 261-271. [13] G. Gruenhage, 'A note on D-spaces', Topology Appl. 153 (2006), 2229-2240. [14] R. Hodel, 'On a Theorem of Arkhangel'skfi concerning Lindelof p-spaces', Canad. J. Math. 27 (1975), 459-468. [15] R. Hodel, 'Cardinal Functions I', in Handbook of Set-Theoretic Topology (Elsevier Science Publishers, 1984), pp. 1-61. [16] J.L. Kelley, General topology (Springer-Verlag, New York, 1975). Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics 27. [17] K. Nagami, 'E-spaces', Fund. Math. 65 (1969), 169-192. [18] A. Rogers and E. Jayne, Analytic sets (Academic Press, London, 1980). Departamento de Matematica Aplicada y Estadfstica Universidad Polite'cnica de Cartagena (Murcia) Spain e-mail: [email protected]