数理解析研究所講究録
第 1922 巻 2014 年 1-7

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Definable slices in -minimal structures$0$

Tomohiro Kawakami
Department of Mathematics, Wakayama University

AbstractLetbe a definably compact definable group and $X$ a definable
set. We prove that there exists a definable slice at every point of $X$
and $X$ is covered by finitely many definable tubes.
$G$

$G$

$G$

1Introduction

In this paper we consider definable slices in an -minimal expansion$(R,$ $+,$
,of a real closed field $R$ . It is known that there exist uncountably many -minimal expansions of the field of real numbers([ll]).
Definable set and definable maps are studied in [2], [3], and see also[12]. Everything is considered in $\mathcal{N}=(R,$ $+,$ $<$ ,and definable maps are
assumed to be continuous unless otherwise stated.
In this paper we prove the existence of a slice in the definable category.

$\mathcal{N}=$$0$$\cdot,$$<$$\mathbb{R}$$0$$\cdot,$Theorem 1.1. Let be a definably compact definable group and $X$ a definable set.
(1) For every point $x\in X$ , there exists a definable slice at(2) $X$ is covered by finitely many definabletubes.
$G$

$G$

$S$

$G$

Theorem 1.1 is a generalization of [5].

2010 Mathematics Subject Classification. $14P10,$ $57S99,$ $03C64.$

Key Words and Phrases. Definable slices, -minimal, real closed fields.
$0$

$x.$

2

Preliminaries

2

Let be a topological group, $X$ a space and $x\in X.$ A slice at is a subset
of $X$ containing such that $G_{x}S=S$ and the map : $G\cross c_{x}Sarrow X$ definedinvariant open neighborhood
imbedding onto aby $\phi([g, s])=gs$ is a$X$
$GS$
$GS$ of $G(x)$ inis called a tube. It is known that there exists a, andslice when is a compact Lie group and $X$ is a completely regular space([4], [8], [9]).
is definable (in ) if it is defined by a formula
A subset $X$ of(with parameters). Namely, there exist a formulasuch that $X$,...,and elements
is true in }.
For any $-\infty\leqq a<b\leqq\infty$ , an open interval $(a, b)_{R}$ means $\{x\in R|a<$$b\in R$ with $a<b$ , a closed interval $[a, b]_{R}$ means $\{x\in R|a\leqq$$x<b\}$ , for any$x\leqq b\}$ . We call-minimal (order-minimal) if every definable subset ofis a finite union of points and open intervals.
A real closed field $(R, +, \cdot, <)$ is an -minimal structure and every definable set is a semialgebraic set [13], and a definable map is a semialgebraic
map [13]. In particular, the semialgebraic category is a special case of adefinable one.
is theThe topology of is the interval topology and the topology ofis a Hausdorﬀ space.
product topology. Note thatis algeraic over } areof real nubmers,
The field{Archimedean real closed fields.
is a, namelyThe Puiseux series
non-Archimedean real closed field.
$G$

$G$

$S$

$x$

$\phi$

$x$

$G$

$G$

$G$

$G$

$G$

$\mathcal{N}$$R^{n}$$\phi(x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{m})$$b_{m}$$b_{1}$

$R$

$\in$

$=$

$\{(a_{1)}\ldots, a_{n})$$\in$$\mathcal{N}$$R^{n}|\phi(a_{1}, \ldots, a_{n}, b_{1}, \ldots, b_{m})$$a,$$\mathcal{N}$

$R$

$0$

$0$

$R^{n}$$R$$R^{n}$$\mathbb{R}$$\mathbb{R}_{al_{9}}=$$x\in \mathbb{R}|x$$\sum_{i=k}^{\infty}a_{i}X^{\frac{i}{q}},$$\mathbb{R}[X]^{\wedge}$$\mathbb{Q}$$k\in \mathbb{Z},$ $q\in \mathbb{N},$ $a_{i}\in \mathbb{R}$Fact 2.1. (1) The characteristic of a real closed field is O.
many non-isomorphic real, there exist(2) For any cardinality
closed fields whose cardinality arefunction, the interme(3) In a general real closed field, even for adiate value theorem, existence theorem of maximum and minimum, Rolle’stheorem, the mean value theorem do not hold. Even for afunction f in
one varianble, the result that $f’>0$ implies is increasing does not hold.
$2^{\kappa}$$\kappa\geqq\aleph_{0}$$\kappa.$$C^{\infty}$$C^{\infty}$$f$Definition 2.2. Let(1) A continuous map
is definable.

$X\subset R^{n},$$(\subset R^{n}\cross R^{m})$$f$:

be definable sets.
is a definable map if the graph of$Y\subset R^{m}$$Xarrow Y$$f$

3(2) A definable map : $Xarrow Y$ is a definable homeomorphism if thereexists a definable map $f’$ : $Yarrow X$ such that $fof’=id_{Y},$ $f’ of=id_{X}.$$f$

Definition 2.3. A group is a definable group ifgroup operations $G\cross Garrow G,$ $Garrow G$ are definable.
$G$

$G$

is definable and theLet
be a definable group. A pair $(X, \phi)$ consisting a definable set $X$
and a action : $G\cross Xarrow X$ is a definable set if is definable. We
simply write $X$ instead of $(X, \phi)$ .
$G$

$G$

$G$

$\phi$

Definition 2.4. Let $X,$
(1) A definable map$X,$ $g\in G,$

$Y$
$f$

$\phi$

be definable sets.
: $Xarrow Y$ is a definable
$G$

$G$

map if for any $x\in$$f(gx)=gf(x)$ .

(2) A definable
map $f:Xarrow Y$ is a definable homeomorphism if
there exists a definable map : $Yarrow X$ such that $foh=id_{Y},$ $hof=id_{X}.$$G$$G$$G$$h$Definition 2.5. (1) A definable set $X$
if for any definable map: $(a, b)_{R}$$f$$\lim_{xarrow a+0}f(x)$$\subset$$arrow$

is definably compact, there exist the limits$R^{n}$$X$, $\lim_{xarrow b-0}f(x)$ in $X.$

(2) A definable set $X\subset R^{n}$ is definably connected if there exist nodefinable open subsets
of $X$ such that $X=U\cup V,$ $U\cap V=\emptyset,$ $U\neq$$U,$ $V$$\emptyset,$$V\neq\emptyset.$

A compact (resp. A connected) definable set is definably compact (resp.
definably connected). But a definably compact (resp. a definably connected)
definable set is not always compact (resp. connected). For example, if $R=$
, then $[0, 1]_{\mathbb{R}_{alg}}=\{x\in \mathbb{R}_{alg}|0\leqq x\leqq 1\}$ is definably compact anddefinably connected, but it is neither compact nor connected.
$\mathbb{R}_{al_{9}}$Theorem 2.6 ([10]). For a definable set
and only if $X$ is closed and bounded.

$X\subset R^{n},$$X$is definably compact ifThe following is a definable version of the fact that the image of a compact(resp. a connected) set by a continuous map is compact (resp. connected).
: $Xarrow Y$ a
Proposition 2.7. Let $X\subset R^{n},$ $Y\subset R^{m}$ be definable set,definable map. If $X$ is definably compact (resp. definably connected), then$f(X)$ is definably compact (resp. definably connected).
$f$

4Theorem 2.8. (1) (The intermediate value theorem) For a definable function
on a definably connected set $X$ , if $b\in X,$ $f(a)\neq f(b)$ then takes allvalues between $f(a)$ and $f(b)$ .
(2) (Existence theorem of maximum and minimum) Every definable function on a definably compact definable set attains maximum and minimum.
(3) (Rolle’s theorem) Let : $[a, b]_{R}arrow R$ be a definable junction such thatis diﬀerentiable on $(a, b)_{R}$ and $f(a)=f(b)$ . Then there exists between aand with $f’(c)=0.$(4) (The mean value theorem) Let : $[a, b]_{R}arrow R$ be a definable junction
which is diﬀerentiable on $(a, b)_{R}$ . Then there exists between and with
$f$

$f$

$a,$

$f$

$c$

$f$

$c$

$f$

$a$

$c$

$c$

$f’(c)= \frac{f(b)-f(a)}{b-a}.$(5) Let : $(a, b)_{R}arrow R$ be a diﬀerentiable definable function.
on $(a, b)_{R}$ , then is increasing.
$f$

If $f’>0$$f$

Example 2.9. (1) Letis not defined([14])(2) Letbenable in , and$\mathcal{N}$

be$(\mathbb{R}_{atg}, +, \cdot, <)$. Then

$f:\mathbb{R}_{alg}arrow \mathbb{R}_{alg},$$f(x)=2^{x}$$\circ$$\mathcal{N}$$(\mathbb{R}, +, \cdot, <)$$\mathcal{N}$$h:\mathbb{R}arrow \mathbb{R},$$\mathcal{N}de.fi$

3

is defined but not$h(x)=\sin x$ is defined but not definable in. Then$f$:$\mathbb{R}arrow \mathbb{R},$$f(x)=2^{x}$

Idea of proof of Theorem 1.1

sets are equivalent if they are
We say that two homogeneous definablehomeomorphic. Let $(G/H)$ be the equivalence class of $G/H.$
definably
The set of equivalence classes of homogeneous definable sets has an defined$(X)\geqq(Y)$ if there exists a definable
map $Xarrow Y$ . Then the reflexivity
and the transitivity hold and the anti-symmetry is true.
$G$

$G$

$G$

$G$

be a definably compact definable group. ThenTheorem 3.1 ([6]). Let
every definable set has only finitely many orbit types.
$G$

$G$

be a definably compact definable group, $X$ a deTheorem 3.2 ([6]). Letfinable Gset with transitive action and $x\in X$ . Then the map : $G/G_{x}arrow X$
defined by $f(gG_{x})=gx$ is a definable $Ghomeomorphi_{\mathcal{S}}m.$
$G$

$f$

The following is a fundamental facts of -minimal structures.
$0$

Theorem 3.3. (1) (Monotonicity theorem (e.g. 3.1.2, 3.1.6. [2])). Let: $(a, b)_{R}arrow R$ be a function with the definable graph. Then there exist$f$5finitely many points $a=a_{0}<a_{1}<\cdots<a_{k}=b$ such that on each subinterval, the function is either constant, or strictly monotone and continuous. Moreover for any $c\in(a, b)_{R}$ , the limits $\lim_{xarrow c+0}f(x)$ , $\lim_{xarrow c-0}f(x)$exist in $R\cup\{\infty\}\cup\{-\infty\}.$(2) (Cell decomposition theorem ($e.g$ . 3.2.11. [2])). For any definable
subsets , . . . ,of , there exists a cell decomposition of partitioning
each , . . .
Letbe a definable subset ofand : $Aarrow R$ a function with thedefinable graph. Then there exists a cell decomposition of partitioning$f|B$ : $Barrow R$ is continuous.
such that each $B\subset A,$
be a definable(3) (Triangulation theorem $(e.g.$ $8.2.9.$ $[2])$ ). Let,...,
set and let
be definable subsets of S. Then has a triangulationcompatible with , . . . ,in(4) (Piecewise trivialization theorem $(e.g.$ $8.2.9.$ $[2])$). Let : $Sarrow A$ be adefinable map between definable sets and A. Then there is a finite partition,...,of into definable sets such that each $f|f^{-1}(A_{i}):f^{-1}(A_{i})arrow A_{i}$is definably trivial.
(5) (Existence of definable quotients ($e.g$ . 10.2.18 [2])). Let be a definset. Then the orbit space
ably compact definable group and $X$ a definable$X/G$ exists as a definable set and the orbit map: $Xarrow X/G$ is surjective,definable and definably proper.
$(a_{j}, a_{j+1})_{R}$$A_{k}$$A_{1}$$R^{n}$$R^{n}$$A_{k}.$$A_{1}$$A$$R^{n}$$f$$\mathcal{D}$$A$$R^{n}$$B\in \mathcal{D},$$S\subset R^{n}$$S_{1},$$S_{2}$$R^{n}$$S$$S_{k}$$S_{k}.$$S_{1}$$f$$S$$A_{1}$$A$$A_{k}$

$G$

$G$

$\pi$

set andbe a definably compact definable group, $X$ a definable$x\in X.$ A
definable slice at is a definable subset of $X$ containing suchthat $G_{x}S=S$ and the map : $G\cross G_{x}Sarrow X$ defined by $\phi([g, s])=gs$ is adefinable $G$ imbedding onto a invariant definable open neighborhood $GS$of $G(x)$ in $X$ , and $GS$ is called a definable tube. Remark that $G\cross G_{x}S$
exists a definable set because
is definably compact and Theorem 3.3, andthe natural action $G\cross G\cross c_{x}Sarrow G\cross G_{x}S,$ $(g, [g’, x])\mapsto[gg’, x]$ induced
by $G\cross G\cross Sarrow G\cross S,$ $(g, (g’, x))\mapsto(gg’, x)$ is definable.

Let$G$$G$$S$$x$$x$$\phi$$G$$G$$G_{x}$$G$

Proposition 3.4 (e.g. II. 4.2 [1]). Let
be a compact Lie group, $X$ aset,aof $X$ and $x\in S.$ Then the following three conditions areequivalent.
invariant(1) There exists aimbedding : $G\cross c_{x}^{A}arrow X$ onto a$G(x)$$\phi([e,A])=S$space.
open neighborhood of, where is awith(2) is a slice at(3) $GS$ is a invariant open neighborhood of $G(x)$ and there exists a
retraction $f:GSarrow G(x)$ such that $f^{-1}(x)=S.$
$G$

$G$

$S$

$\mathcal{S}ubset$

$G$

$G$

$\phi$$A$$S$$G_{x}$

$x.$

$G$

$G$

6
By a way similar to the proof of Proposition 3.1, we have the following
proposition.
Proposition 3.5 ([6]). Let
be a definably compact definable group, $X$ adefinable set, a definable subset of $X$ and $x\in S.$ Then the following
three conditions are equivalent.
imbedding : $G\cross G_{x}Aarrow X$ onto a(1) There exists a definable
invariant definable open neighborhood of $G(x)$ with $\phi([e, A])=S$ , where isa definableset.
(2) is a definable slice at(3) $GS$ is a $G$ invariant definable open neighborhood of $G(x)$ and thereexists a definable
retraction $f:GSarrow G(x)$ such that $f^{-1}(x)=S.$
$G$

$S$

$G$

$G$

$G$

$\phi$$A$$G_{x}$

$S$

$x.$

$G$

By a way similar to the proof of 2.5 [7], we have the following theorem.

be a definably compact definable $group_{f}X$ a deTheorem 3.6 ([6]). Letfinable set, a definable set and : $Xarrow Y$ a invariant surjectiveof $Y$ into definabledefinable map. Then there exists a finite partition
sets such that each $f|f^{-1}(C_{i})$ : $f^{-1}(C_{i})arrow C_{i}$ is definably trivial.
$G$

$G$

$Y$

$G$

$f$

$\{C_{i}\}_{i}$$G$a definable closed
Proposition 3.7 ([6]). Let $X$ be a definable set andsubset of X. Suppose that is a definable strong deformation retract of $X.$
Then for any definable open neighborhood of in $X$ , there exist a definable
closed neighborhood $N$ of in and a definable map : $Xarrow U$ such that$\rho|N=id$ and $\rho(X-N)\subset U-N.$
$A$

$A$

$U$

$A$

$U$

$A$

$\rho$

Idea of Proof of Theorem 1.1.
We prove the condition (3) in Proposition 3.5.
To do so, we use finiteness of orbit types, piecewise trivialization of re1
strictions of orbit map, triangulation, Proposition 3.7.

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