An Open Mapping Theorem for O-Minimal StructuresAuthor(s): Joseph JohnsSource: The Journal of Symbolic Logic, Vol. 66, No. 4 (Dec., 2001), pp. 1817-1820Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2694977 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 66. Number 4, Dec. 2001

AN OPEN MAPPING THEOREM FOR O-MINIMAL STRUCTURES

JOSEPH JOHNS

Introduction. We fix an arbitrary o-minimal structure (R, <,...), where (R, <) is a dense linearly ordered set without end points. In this paper "definable" means "definable with parameters from R". We equip R with the interval topology and Rn with the induced product topology. The main result of this paper is the following.

THEOREM. Let V C Rn be a definable open set and suppose that f: V > Rn is a continuous infective definable map. Then f is open, that is, f ( U) is open whenever U is an open subset of V.

Woerheide [6] proved the above theorem for o-minimal expansions of a real closed field using ideas of homology. The case of an arbitrary o-minimal structure remained an open problem, see [4] and [1]. In this paper we will give an elementary proof of the general case.

Basic definitions and notation. A box B C Rn is a Cartesian product of n definable open intervals: B = (a,, b1) x ... x (an, bn) for some ai, bi E R U {-oo, +oo}, with ai < bi. Given A C R , cl(A) denotes the closure of A, int(A) denotes the interior of A, bd(A) := cl(A) - int(A) denotes the boundary of A, and AA := cl(A) - A denotes the frontier of A. Finally, we let z: Rn R- I denote the projection map onto the first n - 1 coordinates.

Background material. Without mention we will use notions and facts discussed in [5] and [3]. We will also make use of the following result, which appears in [2].

FACT 1. Let K C R' be a definable closed and bounded set and suppose that f: K - R' is a continuous definable map. Then f (K) is also closed and bounded.

Organization. The paper is divided into two sections. Section 2 contains the proof of the theorem. Section 1 contains some simple preliminary observations. The reader may wish to skip the first section and refer to it only as needed.

?1. Some preliminary results. Local dimension. Let S C Rn be a definable set and let b E Rn. Then it is easy

to see that there is a number d c -oo, 0, dimS} such that dim(S n U) = d for all sufficiently small definable neighbourhoods U of b, that is, for all definable

Received October 15, 1999; revised September 24, 2000. Supported by NSERC grant OGP0009070 as well as an NSERC Undergraduate Student Research

Award.

? 2001, Association for Symbolic Logic

0022-4812/01/6604-0019/$1 .40

1817

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1818 JOSEPH JOHNS

neighbourhoods of b contained in some fixed definable neighbourhood of b. We call this number d the local dimension of S at b and denote it by dimb (S).

We will say that a definable set A C Rn has constant local dimension d if dimb(A) = d for each b E cl(A).

PROPOSITION 1. Let A C Rn be a definable set of constant local dimension d and suppose that f: A > R' is a continuous invective definable map. Then f (A) also has constant local dimension d.

PROOF. First note that if U is any definable open set with U n A 7 0, then dim(U n A) = d. The result now follows easily from the fact that definable bijections preserve dimension. -

PROPOSITION 2. Let B C Rn be a box and suppose that W C B is a nonempty definable open set with B - cl(W) :& 0. Then dim(B n aW) = n - 1.

PROOF. By induction on n. The case n = 1 is obvious. Assume that n > 1 and the result holds for lower values of n.

Let 9 be a cell decomposition that partitions B, B - cl( W), B n aW and W. Let B = B' x I, where B' C Rn-1 and I C R. Put V := z(W), so that V is a nonempty definable open subset of B'.

Suppose first that there is an open cell D EE c = {zc(C): C E 9} such that (D x I) n (B - cl(W)) :8 0 and (D x I) n W :8 0. Then it is easy to see that (D x I) n (B n aW) :8 0. Hence in this case there is an (n - 1)-dimensional cell in 9 lying above D which is contained in B n aW.

We may therefore assume that for each open D c 7r(g) with D C B', we have (D x I) n aW = 0 and either D x I C B - cl(W) or D x I C W. It is then easy to see that B - cl(W) 7 0 implies B' - cl(V) 7 0. By induction, we then have dim(B' n V) = n-2. - Now, since V is open and 7(9) partitions V, it follows that each b EE aV is in OD for some open D E 7c(9) such that D C V. Hence there is an open D E 7c(9) such that D C V and dim(B' n aV n oD) = n -2. Let E be an (n - 2)-dimensional cell contained in B'V n a V n D. Since D x I C W, it follows that E x I C aOW n B. ]

PROPOSITION 3. Let D C Rn be a cell of dimension n - 1. Then there is a coordinate permutation R n R n such that b(D) is a cell which is the graph of a function.

PROOF. Let p: Rn > R- I denote the projection map that maps D homeo- morphically onto an open cell in R 1 . Let us say that

p(xi,, Xn) (XI,... Xi-,, Xi+l,. *., Xn) = (XI, )2)

and let z7: Rn R denote the projection map onto the ith coordinate. Then, putting h :=r o (p [ D)1 p(D) > R, we see that

D - {(Y1, h( 1, X2), X2) E Rn (x1, 2) c p(D)},

which is the graph of the function h with some coordinates permuted. -

PROPOSITION 4. Let A1 and A2 be disjoint definable sets in R . Suppose that D C 0A1 0n aA2 is a cell of dimension n - 1. Then there is a box B containing some point of D such that B C A1 U D U A2.

PROOF. Using Proposition 3, we can assume without loss of generality that D is the graph of a function. Let 9 be a cell decomposition which partitions A1, A2, and D, and let E E 9 be a cell of dimension n - 1 contained in D. Then E is

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AN OPEN MAPPING THEOREM FOR O-MINIMAL STRUCTURES 1819

itself the graph of a function. Let C1, C2 be the open cells of 9 which lie directly above and below E. Then V := C1 U E U C2 is an open set containing E and since E COA1OnA2,itfollowsthatVnAi 0,i =1,2. SinceEnAi= 0and partitions A1, A2, it follows that Ci C Ai, provided we label things appropriately. The result follows easily. -

?2. Proof of the Theorem. For this section, we fix V and f as in the statement of the Theorem. To establish the Theorem we must show that f ( U) is open whenever U is an open subset of V. However, since any open set is the union of some boxes, it suffices consider the case where U is a bounded box with cl( U) C V. Fix such a box U for the remainder of this section. We will prove that f (U) is open by showing that it does not contain any of its boundary points. In other words, we will show that

H := f (U) n bd(f (U))

is empty. Before giving the proof of the theorem we make two observations about H, which appear below as Lemmas 1 and 2.

LEMMA 1. H is an open subset of bd(f (U)). PROOF. This follows from two facts:

(i) af (U) is a closed subset of Rn. (ii) H = bd(f (U)) - af (U).

Statement (ii) is easily verified, so we prove (i). By Fact 1, f (cl( U)) is a closed set and since it contains f ( U), we have

f (U) U af (U) = cl(f(U)) C f (cl(U)) = f (U) U f (aU).

We have the reverse inclusion by continuity of f, and so we see that

af(U) = f (aU),

which is closed by Fact 1. -

LEMMA 2. If H is nonempty, then H has dimension n - 1.

PROOF. For this proof we aim to use Proposition 2. Let b E H. By Lemma 1, there is a box B containing b such that B n bd(f( U)) C H. Since Proposition 1 implies f (U) is of constant local dimension n, we have W := B n int(f (U)) #8 0. Now suppose for a contradiction that B - cl(JW) = 0. Then B C cl(f (U)) and since B n bd(f(U)) C H C f(U), it follows that B C f(U). But then b E B implies b E int(f( U)), contradicting the fact that b c H C bd(f( U)). This proves that B -- cl( W) #8 0. Hence by Proposition 2, dim(B n aW) = n - 1. On the other hand it is easy to check that B n aW C B n bd(f (U)) C H.

Proof of the Theorem. Suppose for a contradiction that H is nonempty. Then Lemma 2 implies dim H = n - 1 and since f is invective, we have

dimf-'(H)= n - 1.

Let D be a cell of dimension n - 1 contained in f - (H). Using Proposition 3, it is not hard to see that there is a box B C U such that B - D = B1 U B2, where B1 and B2 are some disjoint definable open sets, with B n D C aB1 n aB2. Obviously B n D has dimension n - 1, and since f is continuous and invective it follows that H' := f (B n D) is contained in a(f (B1)) n a(f (B2)) and has dimension

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1820 JOSEPH JOHNS

n - 1. By Proposition 4 there is a box B' containing some point b E H' such that B' C f (B1) U H' U f (B2) C E (U). But then b E B' implies b E int(f (U)), a contradiction, since b E H C bd(f ( U)). We therefore conclude that H must be empty. Hence f (U) is open and since U C V was fixed arbitrarily, this proves that f is open. A

Acknowledgments. I would like to thank Patrick Speissegger for acting as faculty supervisor, as well as Edward Bierstone for his support.

REFERENCES

[1] A. NESIN, A. PILLAY, and V. RAZENJ, Groups of dimension two and three definable over o-minimal structures, Annals of Pure and Applied Logic, vol. 53 (1991), pp. 279-296.

[2] Y PETERZIL and C. STEINHORN, Definable compactness and definable subgroups of o-minimalgroups, Journal of the London Mathematical Society, vol. 59 (1999), pp. 769-786.

[3] A. PILLAY and C. STEINHORN, Definable sets in ordered structures. I, Transactions of the American Mathematical Society, vol. 295 (1986), pp. 565-592.

[4] L. VAN DEN DRIES, o-Minimal structures, Logic: from foundations to applications (W Hodges et al., editor), Oxford University Press, 1996.

[5] , Tame topology and o-minimal structures, London Mathematical Society Lecture Note

Series, no. 248, Cambridge University Press, 1998.

[6] A. WOERHEIDE, Topology of definable sets in o-minimal structures, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1996.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CHICAGO,

5734 S. UNIVERSITY AVENUE, CHICAGO ILLINOIS 60637, U.S.A.

E-mail: johnsgmath.uchicago.edu

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