Sum theorems for the strong small transfinite dimension

Abstract

We state some sum theorems for the strong small transfinitedimension in differentclasses of topological spaces. Introduction P. Borst introduced in [1] the strong small transfinitedimension, sind. There, HipQtatprlTHEOREM ([1], corollary of proposition IV.5). Let X be a normal space which has a locally finite closed cover {Fs}_ye g such that for each Fs itis sind (Fs)* A . Then sind(X)±A. We could say that thisis a locally finite sum weak theorem, because in it there is no relation between the strong small transfinitedimensions of the closed sets Fs and that of the whole space X. We'll connect them obtaining the locally finitesum theorem for sind, in its classical formulation, in the class of strongly hereditarily normal spaces. In addition, we establish an open sum theorem in the class of regular spaces. Preliminaries We shall use the notation and definitionsin [1], [3], [4] and [5]. For every ordinal t, we have £= A(£)+ ≪(£)where A(|) is a limit ordinal and ≪(£)is a finiteordinal. We take the extra symbol A ,satisfying A > t,and A + % = ];+ A = A for each ordinal number £. In order to define the strong small transfinitedimension (due to P. Borst, [1]) 1991 A.M.S. Subject Classification: 54F45 Received March 7, 1994. Revised March 22, 1995. Firstauther has been supported by the DGICYT grant PB93-0454-C02-02 94 Eduardo Cuchillo-IbAnez and Juan Tarres we define, for a subspace Y of a topological space X, the sets: Pn(y) = u{£//£/open in Y and Ind [ClY(U)] < n] where neiVu{0}, and Ao[^] = ^For every ordinal number £ we obtain inductively: a^[Y] = y kjP(Y) and P4(Y) = n<£, '.BlKlflM) We simplify by denoting A JX] = A* and PAX) = P? for every ordinal t, DEFINITION. Let X be a topological space, then: sind(Z) = -1 iff X = 0 sind(X)<£ iffA§= 0 smd (X) = ^ iffsind (X) < £ and sind {X)<£, does not hold in other case, we say that X has not sind or sind (X) = A Locally finite sum theorem Recall that a topological space X is called strongly hereditarily normal (see [3], definition 2.1.2) if X is a T, -space and for every pair A, B of separated sets in X there exist open sets U, V <z X such that A<zU,BczV,U nV = 0 and U and Vcan be represented as the union of a point-finitefamily of FCT-setsin X. Theorem 1. Let X be a strongly hereditarily normal space. If <^={C',},e/is a locally finiteclosed cover of X such that for each i e / sind(C,) ^ £,then sind(X)<£. Proof. We'll obtain sind (X) < ^ by proving At = 0, thatis to say, X=uP,. So, we'll see that for each point xeX there exists an ordinal number 7]0,with 7]0< % ,such that xeP^. Let's take a point xo^ X and let V be an open neighbourhood of x0 such that intersects to a finitenumber of elements of the closed cover %: C/,,...,C/n.We'll show the existence of an ordinal Tfo<£, for which xo^P^o≫ by induction on n, the number of elements of % whose intersection with V is not empty, i.If n=\, V only cuts C,,,whereupon xoeVcCi,. Sum Theorems for SIND 95 As sind(C/,)<£, As[C/,] = 0. Since V is open in Ci{, from the corollary of lemma 3 of [2] we obtain AdV] = AdCil]nV = $. Anew due to the mentioned result, 0 = AdV] = A^nV. Hence, xo<£A$ = XUPV. Thus, there exists an ordinal f]Q< £ such that xo e P^o. ii. When n=2, ^only cuts two members, say C,-,and C,-2,of the cover %. We'll have V a C,-,u C,-2and suppose that because if, for example, iog Cix-d2, after considering the open neighbourhood of *o = Vn(X-Ci2), we'll be situated in the section i. Hence, as sind(C,-,)££and sind(C,-2)<<^, there exist ordinals a,, a2 < £ such that ^ePai(Ci)nPa2(C/2) Let's define the ordinal number p = max{a,,a2} < £. We are going to prove, by transfinite induction on p, that Xo e u P^. ii.l.Ifp = O,a. a2 = 0 too. In this case *oePo(C,) so there exists JJ\,open neighbourhood of xo in C,,, such that Ind[C7CVlU/I)]=Ind[tf1]£0. Analogously, x0 e P0(C-2) and there exists U2, open neighbourhood of jt0in d2, such that Ind [ClCi2{U2)]=lnd [(72]<0. Take open subsets W and W2 of X with [/i = lVinCf, and U2 = W2nCh.