cell decomposition and dimension functions in first

CELL DECOMPOSITION AND DIMENSION FUNCTIONS IN FIRST-ORDER TOPOLOGICAL STRUCTURES LARRY MATHEWS [Received 2 February 1993—Revised 26 July 1993 and 1 December 1993]

0. Introduction The notion of a cell and that of a cell decomposition has been a central one in the study of certain first-order theories. A cell is a particular kind of definable set. The notion of a cell was first explicitly considered in [8], in the context of the theory of real closed fields. Collins defined a class of cells in this context, and showed that every definable subset of a real closed field is a finite disjoint union of cells. This property is often referred to as a cell decomposition for definable sets. The definable subsets of a real closed field are precisely the semi-algebraic sets (see [6]), and therefore this notion of a cell is of considerable importance in real algebraic geometry. Real closed fields are the principal examples of o-minimal structures. These have been extensively studied by model theorists during the last decade, and in this connection it was readily noticed that the notion of a cell was appropriate to the study of definable sets in this more general context. It was shown by van den Dries in [14] that o-minimal expansions of the real ordered field admit a cell decomposition for definable sets. However, Knight, Pillay and Steinhorn proved more generally in [19] that any o-minimal structure whatever admits a cell decomposition for definable sets. The notion of cell for o-minimal structures in [19] makes sense for linearly ordered structures, but a related notion was subsequently considered in a different context, that of the first-order theory of the p-adic numbers. Denef introduced a notion of cell appropriate to Qp in [10], and showed that any definable subset of Qp in the language of rings is a finite disjoint union of cells. By analogy with the real case, definable subsets of p-adically closed fields (structures elementarily equivalent to Qp) are often referred to as p-adic semi-algebraic sets. By contrast, however, a satisfactory analogue of an o-minimal structure in the p-adic setting has yet to be developed, much less an analogue of the cell decomposition for o-minimal structures in [19]. The structure of p-adic semi-algebraic sets was studied extensively by van den Dries and Scowcroft in [30]. They proved a result concerning the structure of definable subsets of Qp which, though weaker than the cell decomposition result in [10], is the basis for the work in this paper. We give a definition of cell (Definition 6.2) which makes sense in the context of any first-order structure in which there exists a definable topology, a first-order topological structure in the terminology of [24]. We also define in this context the Cell Decomposition

This research forms part of my doctoral thesis. I would like to thank my supervisor Angus Macintyre for his invaluable assistance. Thanks also to Dugald Macpherson and Alex Wilkie for helpful discussions concerning this work. 1991 Mathematics Subject Classification: 03C60, 03C45. Proc. London Math. Soc. (3) 70 (1995) 1-32.

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Property for a first-order topological structure (Definition 6.3), which is essentially that every definable set is a finite disjoint union of cells. Now o-minimal structures and p-adically closed fields are both examples of first-order topological structures, the definable topology in each case being the order and valuation topology respectively. Furthermore, the cells mentioned above are cells in our sense, and therefore o-minimal structures and /?-adically closed fields are examples of structures satisfying the Cell Decomposition Property. For the case of Qp, this is essentially Corollary 3.1 of [30], and in this paper we take this result and its proof as a basis for a unified theory of cell decomposition for these structures. Another context in which the notions and results of this paper apply is the theory of cell decomposition for definable subsets of real closed rings given in [21]. For another context in which the idea of a cell appears, see the cell decomposition theorem for the so-called C-minimal structures proved by Haskell and Macpherson in [17]. The cells defined there are cells in our sense, but the theory developed in this paper does not apply to their context. The principal result of this paper is Theorem 7.1, which gives sufficient conditions for a first-order topological structure to satisfy the Cell Decomposition Property. These conditions are satisfied by the examples of first-order topological structures discussed above. The first part of the paper is devoted to preliminaries to the proof of Theorem 7.1. The remainder of the paper develops a theory of dimension for definable subsets of structures satisfying the Cell Decomposition Property and the Exchange Principle (see Definition 4.6). The principal result is Theorem 8.8, which states that a number of notions of dimension for definable sets are equivalent. This theorem generalises known results concerning dimension for semi-algebraic sets (see, for example, [6]), o-minimal structures, and for p-adic semi-algebraic sets in [30]. A number of properties of this notion of dimension are observed. We prove in particular that it is a dimension function in the terminology of [15]. We now describe briefly the contents of the paper. In § 1 we discuss some notation and terminology that we employ. In § 2 we introduce the notion of a first-order topological structure, and also the notion of a t-minimal structure, which will also be of central importance in this paper. In § 3 we recall some basic facts concerning the model theory of p-adically closed fields and real closed rings, which will be needed subsequently. Section 4 introduces some further notions, that of a topological system, of a theory being model-theoretically bounded, and of a theory satisfying the Exchange Principle. We recall the fact (Corollary 4.8) that real closed fields, p-adically closed fields and real closed rings are t-minimal, model-theoretically bounded and satisfy the Exchange Principle. Section 5 introduces further fundamental notions, that of a theory having finite Skolem functions (Definition 5.1), and that of a structure having the Local Continuity Property. The latter is a first-order version of the Implicit Function Theorem, and is utilised to prove a result concerning the behaviour of definable functions (Theorem 5.6), which is the main preliminary lemma required for the proof of the main theorem. In § 6 we define a cell (Definition 6.2) and the Cell Decomposition Property (Definition 6.3). In § 7 we prove the main theorem (Theorem 7.1) giving sufficient conditions for the Cell Decomposition Property to hold. In § 8 we introduce various notions of dimension for definable sets in structures satisfying the Cell Decomposition Property and the Exchange Principle, and prove (Theorem 8.8) that they are all equivalent. Finally, in § 9 we

CELL DECOMPOSITION AND DIMENSION FUNCTIONS

3

deduce a number of elementary properties of this notion of dimension for definable sets. 1. Notation We employ standard model-theoretic notation for the most part. Throughout L denotes a first-order language, and M, Jf denote L-structures. The complete L-theory of an L-structure M is denoted by T\\{M). We denote L-formulas by , if/ and X- We often consider a formula with its free variables partitioned into special and parameter variables, in which case we write it as \p(x;y) or (j>(x ;y). We also sometimes display the special and parameter variables explicitly, and write *l*(xi,-,xn;yl,...,ym) or (x;yu...,yn). If x = (xu ..., xn) is a tuple of variables, then we may write l(x) = n. Let M be an L-structure, and let b = (bu ..., bn) be a tuple of elements of M. Suppose also that A^Ji. Then we may write A U {b} to denote A U {bu ..., bn} {x) be an L-formula, and let k e Z + . Then (3 ^kx)(x ; b)M denotes the definable set

In addition, if P is a relation symbol of L, and if / is a function symbol of L, then PM, fM denote the interpretations of P and f in M respectively. However, we often follow the convention of simply identifying function symbols, terms and relation symbols with their interpretations in a given L-structure. We also employ standard notation for types. Let M be an L-structure, and let

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Al)-formula «//(x) such that \\tM = X. Suppose that p is such that for every N> M, p has only finitely many realisations in M. Then p is said to be an algebraic type. In this case, an easy compactness argument shows that there exists i//ep such that for every Jf>M, «// is finite, and hence Let M be an L-structure, and let A 4) = {y G M: there exists ij/(v) e L(y4) such that

We often employ the convention of working in an L-structure M which is saturated with respect to any set of parameters under consideration. We therefore say that M is sufficiently saturated to mean that if A g M is any set of parameters under consideration, then M is \A\*-saturated. We occasionally consider a polynomial / in several variables over a domain D, with a distinguished variable Y. In this case we may write / a s f(Xu ..., Xn, Y) e £>[*,,..., * „ , Y] or just f(X,Y)eD[X,Y]. Then deg y /(A\ Y) denotes the degree of the polynomial / i n the variable Y. We also employ some standard notation from general topology. Let ^ be a topological space, and let Yc.X. Then int(K) denotes the set of interior points of V, bd( V) denotes the set of boundary points of Y, and Y denotes the closure of Yin X.

2. First-order topological structures and t-minimal structures This paper is concerned with the study of t-minimal structures, which were introduced in [22]. We recall their definition and give a brief indication of the motivation for considering them. We recall first the definition of a first-order topological structure, introduced by Pillay in [24]. DEFINITION 2.1. Let M be an L-structure. Let (v ;y) G L, where l(y) = n, for some n G Z + . Then (M, ) is a first-order topological theory if T = T\\(M), where {Ji, ) is a first-order topological structure. We often suppress mention of the distinguished formula (f> defining the topology. If M is a first-order topological structure, then for every n e Z + , we consider Ji" as a topological structure with the product topology induced by the topology on Ji. Note that the standard topological operations are 'definable' in M, in the sense that if J f g J t is definable, then X, \n\(X), bd{X) etc., are all definable. We make use of this fact without explicit mention. We will impose some very minor restrictions on the underlying topology of a first-order topological structure. DEFINITION 2.2. Let {Ji, (\>) be a first-order topological structure. Then {Ji, $) is proper if (a) the topology determined by is Tu that is, for all a, b e M such that a ¥• b, there exists a basic open ball U 9) j u s t says, \r is the least element of the definable set given by \p{x ; y)\ Therefore in particular Th(RCF) has finite Skolem functions. However, more is true in this case. The theory Th(RCF) has full definable Skolem functions, which was observed by van den Dries in [13]. In fact, applying a theorem of Pillay and Steinhorn in [26] that an o-minimal group is divisible and abelian, one can conclude that any o-minimal group has definable Skolem functions. It is also the case that Th(pCF) has finite Skolem functions, though this is far less apparent than in the case of an o-minimal theory. This was proved by Denef in [9]. However, it is also the case that Th(pCF) has definable Skolem functions, which was also proved by van den Dries in [13]. The case of real closed rings differs slightly. Since Th(RCR) is a theory of linear order, it has finite Skolem functions by the above remark. It is worth pointing out that Th(RCR) does not have definable Skolem functions in the language of ordered rings with divisibility. However, it was pointed out by van den Dries in [13] that only a minor modification to the theory is required for definable Skolem functions. Suppose that D t Th(RCR). Then it follows directly from Theorem 3.3(b) that D is a valuation ring, and it is also easily shown (see [7]) that the unique maximal ideal in D, namely M = {x ED: D !=-»(* 11)}, the set of non-invertible elements in D, is a convex subset of D. Hence, in particular, we have that for every x e M, \x\ < q, for every q e +, since

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Q c D \ M . Note also that M is a 0-definable subset of D. It is an easy consequence of quantifier elimination that if y e Z)\{0} is 0-definable, then in fact f(y) = 0, for some / e Q[X], and therefore y cannot be infinitesimal. Hence the definable set A/\{0} cannot be Skolemised in the language L ord of ordered rings with divisibility. In order to overcome this, we add a new constant symbol c to the language, and add the axiom

to Th(RCR). It was proved by van den Dries in [13] that Th(RCR) has definable Skolem functions in this extended language. For a very different proof of the existence of definable Skolem functions for Th(RCF), Th(/?CF) and Th(RCR), see [29]. We also need to introduce a property which holds for the algebraic examples of structures that we consider, which is essentially a weak first-order version of the Implicit Function Theorem. The Implicit Function Theorem (see [1, p. 84]) holds for IR and Qp, since they are both complete fields with respect to an absolute value. The following weak version of this for real closed fields is taken from [12]. 5.2. Let K be a real closed field. Let x0 G Km, and let y0 G K, for some m el+. Let F = F(X, Y) (where X = (Xu ..., Xm)) be a polynomial such that F(x0, yQ) = 0 and F'(xQ, y0) ¥" 0, where F' denotes the derivative of F with respect to Y. Then there exists an open neighbourhood U of x0, there exists an open neighbourhood V of yQ, and there exists a definable function : U —> V which is differentiable on U, such that for every x e U, and for every y e V, THEOREM

F(x,y) = 0 &(x) = y. This result is an immediate consequence of the Implicit Function Theorem for U and the completeness of the theory of real closed fields. A similar result also holds for p-adically closed fields, for the same reason. In what follows M tTh(RCF), Th(pCF), or Th(RCR). Let px(X, Y), ...,pk(X, Y) e M[X]}..., Xn) Y] be finitely many non-trivial polynomials over M, containing a distinguished variable Y. We assume that this set of polynomials is closed under derivatives with respect to Y, except for constant polynomials. Let K = 2*=i degY Pi(X, Y). For every / with 1 =s/ ^K, let [ * EAT: Mt(3=jy)^pi(x,y) = Note that V) is definable, for every j with U / ^ / C . Recall that Th(M) has finite Skolem functions, and hence for every; with 1 =s/ *s K, if Y} ¥" 0 , then there exist definable functions Fi M such that for every x G Yjt k

M b A \ / n.(x F- (x}}

= 0A

A

F- (X~}

T£

F- (X}

We refer to these as the definable root functions of p\(X, Y), ...,pk(X, Y). Then M has the Local Continuity Property if, for every j with l^j^K, given x G int(>y), there exists an open ball B c Yj, with x E B, such that FjA,..., Fu may all be chosen to be continuous on B. It is a routine exercise to check that the Local Continuity Property holds for

CELL DECOMPOSITION AND DIMENSION FUNCTIONS

13

any model of Th(RCF) or Th(pCF), by virtue of Theorem 5.2 and the corresponding result for /?-adically closed fields. It is also clear that any real closed ring also satisfies the Local Continuity Property, since by Theorem 3.3(a), a real closed ring is a convex subring of a real closed field. We now point out that as a consequence of this, definable unary functions are piecewise continuous, that is, not discontinuous on any open neighbourhood. The proof of this is essentially derived from a result in [30]. First, we introduce the following temporary definition. DEFINITION 5.3. Let M be a first-order topological structure. Then M is regular, if, given S g= Mn definable, and a partition

of S into finitely many definable subsets, if S has non-empty interior in M", then 5, has non-empty interior in Jt, for some i with 1 =s / ss k. It follows from what we do later that if M is o-minimal, then M is regular. This was noted in [21], and is an elementary consequence of the dimension theory for definable sets in o-minimal structures. In addition, it is easily seen from quantifier elimination that our three main examples of t-minimal structures have this property also. We state this result in terms of the general terminology introduced earlier. LEMMA 5.4. Let M be an L-structure, where L is an extension of the language of rings {+, -, •, 0, 1} with extra constant and relation symbols only. Suppose also that M is a topological system such that Th(M) has quantifier elimination. Then M is regular.

Proof. Suppose that we have a partition m

s = U s, /=i

of the definable set S

In addition, let y0 = {x G Jt: M^{^y)[pi{x, y) = 0], for some i with 1 ^ / ^ k}. Let {/>,,}] =syssxM«,«, be definable root functions of P\{X, Y), ...,pk(X, Y) as above. Note that these functions are all A -definable, and that x = \j(Yjnx) is a partition of X into finitely many /1-definable subsets. For every j with O^j-^K, putting Sj = YjC\X, partition 5, further into the sets Sj^ - {x £ Sj\ f is continuous at x},

SJ2 = Sj\Sj},

and then into the sets int(5>2), 5>2\int(Sy2). Hence for every ; with O^j^K, 5, can be partitioned into finitely many A -definable sets, which are either open or have no interior in Mn. Let Z be the union of the open definable sets in this partition. By Corollary 5.5, M is regular, and hence X\Z has no interior in M". Let x e Z. Then x $ Yo, since Vo has no interior in AT. Let xe Yjt where / is such that l^j^K. Suppose that / is discontinuous at x. Then by the construction, / is discontinuous on C, where C c Z is the open definable set in the above partition such that x e C. By the Local Continuity Property, let x e B, where B l) = dcl^(v4), let / : Z—>M be an A -definable function such that if (xu...,xn) e Z, then f((ax,...,an)) = an+1 and for all xu...,xneM, (*!,..., xn,f(xi,..., *„)) e X By the CDP and the inductive hypothesis, we may suppose that / is continuous on Z. Then W = G(f) is an ^-definable cell, {ax,..., fln+1) e W, IVgA', and the projection map ;r£ + 1: W-^VV^ is a homeomorphism, with Wk = Zk open in Mk, as required. LEMMA 8.14. Suppose that the underlying topology on M is Hausdorff. Let X Z is a homeomorphism, where Z = n"k[Y\, and that Z is open in Mk. Suppose that X, Y are /4-definable, where A • • • >

x

x

x

n ) X (

a

\ >

•••>

a

k>

x

k + \> •••}

x

j>

•••>

X

n )

an

where x( \y •••> n) is L(y4)-formula defining X. Then for some j with k + 1 ss/s£/?, (/^(fli,..., fl* \Xj)M is infinite. This will suffice, since by compactness we may then find bj e M such that u

..., ak, bj), bj $ ac\M{A U {au ..., ak}).

Let c = (cx,..., cn) E A', where Cj = flj,..., ck = ak, c, = bj. Then clearly rk(cM) ^ k + 1, and therefore rk(A r )^A: + l. By Lemma 8.11, we have that dim(A') s* A: + 1, as required. Suppose then that for every ; with k + l = s / =£n, iftj(au ...,ak ;xj)M is finite. Then, by the CDP, let T^Z be ,4-definable and open such that (ait..., ak) e T, and for every j with /c + l = s / ^ n , there exists m{j) e Z + such that for all *!,..., xk G J Xj).

For every y G M, and for every / with k + 1 = s / ^ n , if ^t= 0,(fli,..., fl/t, y), then (flj,..., ak, y) is a generic point of 0, over A. Fix / with A: + 1 ^ / ^ n, and y e Mas above. By Lemma 8.13, let S c Oj(xu ..., xk> Xj)M be an A -definable cell such that (au ..., ak, y) G S, the projection function 7r*+1: 5-^5^. is a homeomorphism, and Sk is open in ^*. Hence there exists an ^-definable open set R^T^Z, with (fl 1} ..., ak) G R, and there exists an ^-definable function g;: R-+M such that for all z\,..., ^ e ^ , if (zi,..., z j £ R then

Carrying out the above argument m(j) times, altering the relevant definition at each stage, we may show the following, which we leave as an exercise. (*) Without loss of generality there exist m(j) A -definable functions «A/(i> •••> *l>j,muy Z^>M such that for all Z\,..., zk G M, if (z\,..., zk) G Z then for all i, k with l^i^k^m(j), if!jt,(zu .-, zk)* iffj,k(zi, •••> zk), for every i with 1 ^ i ^ m(j), Mtipjizi,..., zk, ijfjAzu - , Zk)), and

If this is done for each co-ordinate in turn, then we may assume that for every ; with k + 1 s£;s£rt, there exist m(j) ^-definable functions 0- ;1 ,..., ^j,mu): Z-+-M satisfying (*). Furthermore, by the CDP we may suppose that each of these functions is continuous on Z. Now fix / with k + l^j^n. Since the underlying topology on M is Hausdorff, there exist open balls BjU ..., #;>m(;) in M such that for every / with ls=/=£ra(;), «//y>J(ai,..., ak) e Bj>h and for all i, k with

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1 ^ / # k =s m(j), Bjj Pi Bjk = 0 . For every j with k + 1 =s/ =s n, and for every i with 1 =s / s= m(y), let ZjA — ^jJ[BjA\ Then ZjA is a definable open set containing (au ...,ak), with Z,-,,