Definable Tietze extension property in o-minimal expansions of ordered groups

Abstract

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots )$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $f:A \rightarrow M$ defined on a definable closed subset of $M^n$ has a definable continuous extension $F:M^n \rightarrow M$.

1 Introduction

In this paper, we investigate the Tietze extension property for functions definable in an o-minimal expansion of an ordered Abelian group. For the basics of o-minimality, good references are [2, 6, 7]. In this context, the Tietze extension property is defined as follows:

Definition 1.1

Consider an expansion ${\mathcal {M}}=(M,<,\ldots )$ of a dense linear order without endpoints. The structure ${\mathcal {M}}$ enjoys the definable Tietze extension property if, for any positive integer n, any definable closed subset A of $M^n$ and any continuous definable function $f:A \rightarrow M$, there exists a definable continuous extension $F:M^n \rightarrow M$ of f.

The definable Tietze extension property is a convenient tool for the geometric study of o-minimal structures. We prove the following theorem in this paper.

Theorem 1.2

Consider an o-minimal expansion ${\mathcal {M}}=(M,<,+,0,\ldots )$ of an ordered group. The following are equivalent:

• 1. There exists a definable bijection between a bounded interval and an unbounded interval.

• 2. The structure ${\mathcal {M}}$ enjoys the definable Tietze extension property.

We make a comment on the theorem. Miller and Starchenko studied the asymptotic behavior of o-minimal expansions of an ordered group ${\mathcal {M}}=(M,<,+, \ldots )$ in [4]. They introduced the notion of linear boundedness. An o-minimal structure is called linearly bounded if, for any definable function $f:M \rightarrow M$, there exists a definable automorphism $\lambda :M \rightarrow M$ with $|f(x)| \le \lambda (x)$ for all sufficiently large $x \in M$. Their main theorem is that there exists a definable binary operation $\cdot$ such that $(M,<,+,\cdot )$ is an ordered real closed field when the structure is not linearly bounded.

Peterzil and Edmundo studied the subclass of linearly bounded o-minimal expansions of ordered groups [1, 5]. An o-minimal structure ${\mathcal {M}}$ is semi-bounded if any set definable in ${\mathcal {M}}$ is already definable in the o-minimal structure generated by the collection of all bounded sets definable in ${\mathcal {M}}$. Edmundo gave equivalent conditions for an o-minimal expansion of an ordered group to be semi-bounded in [1, Fact 1.6]. The condition (1) in our theorem is the negation of one of them. Theorem 1.2 gives a new equivalent condition. An o-minimal expansion of an ordered group is semi-bounded if and only if it does not have the definable Tietze extension property. In our proof, we use the following facts:

• In an o-minimal expansion ${\mathcal {M}}=(M,<,+,0,\ldots )$ of an ordered group which is not semi-bounded, we can define a real closed field whose universe is an unbounded subinterval of M and whose ordering agrees with < [1, Fact 1.6].

• An o-minimal expansion of an ordered field enjoys the definable Tietze extension property [7, Chapter 8, Corollary 3.10].

We introduce the terms and notations used in this paper. The term ‘definable’ means ‘definable in the given structure with parameters’ in this paper. For a linearly ordered structure $\mathcal M=(M,<,\ldots )$, an interval is a nonempty definable set of the form $\{x \in M\;|\; a \ * \ x \ *'\ b\}$, where $a,b \in M \cup \{\pm \infty \}$ and $*,*' \in \{<,\le \}$. The interval is denoted by ]a, b[ when both $*$ and $*'$ are the symbol <. It is denoted by [a, b] when both $*$ and $*'$ are $\le$. We define [a, b[ and ]a, b], similarly. An interval is called bounded if both a and b belong to M. It is called unbounded otherwise. We consider the order topology on M and its product topology on the Cartesian product $M^n$ in the paper. The notation $M_{>r}$ denotes the set $\{x \in M\;|\;x>r\}$ for any $r \in M$.

2 Proof

We now begin to prove Theorem 1.2. An o-minimal structure is always definably complete. We use this fact without notice. We first prove two lemmas.

Lemma 2.1

Consider an o-minimal structure. The structure has a definable bijection between a bounded interval and an unbounded interval if and only if it has a definable homeomorphism between a bounded interval and an unbounded interval.

Proof

It is immediate from the monotonicity theorem [7, Chapter 3, Theorem 1.2]. $\square$

Lemma 2.2

Consider a definably complete expansion of a densely linearly ordered abelian group ${\mathcal {M}}=(M,<,+,0,\ldots )$. If the structure ${\mathcal {M}}$ has a strictly monotone definable homeomorphism between a bounded open interval and an unbounded open interval, any two open intervals are definably homeomorphic and there exists a definable strictly increasing homeomorphism between them.

Proof

By the assumption, there exists a strictly monotone definable homeomorphism $\varphi :I \rightarrow J$, where I is a bounded open interval and J is an unbounded open interval. We may assume that $I=]0,u[$ for some $u>0$. In fact, an open interval $]u_1,u_2[$ is obviously definably homeomorphic to $]0,u_2-u_1[$. We may further assume that $\varphi$ is strictly increasing because the map $\tau :]0,u[ \rightarrow ]0,u[$ defined by $\tau (t)=u-t$ is a definable homeomorphism.

We next reduce to the case in which $J=]0,\infty [$. We have only three possibilities; that is $J=]v,+\infty [$, $J=]-\infty ,v[$ and $J=M$ for some $v \in M$. In the first and second cases, we may assume that $J=]0,\infty [$ because $J=]v,+\infty [$ and $J=]-\infty ,v[$ are obviously definably homeomorphic to $]0,\infty [$. In the last case, set $u'=\varphi ^{-1}(0)$. Then the restriction of $\varphi$ to the open interval $]0,u'[$ is a definable homeomorphism between $]0,u'[$ and $]-\infty ,0[$. Hence, we can reduce to the second case. We have constructed a strictly increasing definable homeomorphism $\varphi :]0,u[ \rightarrow ]0,\infty [$. We fix such a homeomorphism.

We next construct a definable strictly increasing homeomorphism between an arbitrary bounded open interval and $]0,\infty [$. We may assume that the bounded interval is of the form ]0, v[. We have nothing to do when $v=u$. When $v<u$, the map defined by $\varphi (t+u-v)-\varphi (u-v)$ for all $t \in ]0,v[$ is a definable homeomorphism between ]0, v[ and $]0,\infty [$. When $v>u$, consider the map $\psi :]0,v[ \rightarrow ]0,\infty [$ given by $\psi (t)=t$ for all $t \le v-u$ and $\psi (t)=\varphi (t+u-v)+v-u$ for the other case. The map $\psi$ is the desired definable homeomorphism. We have constructed a definable homeomorphism between ]0, u[ and all open intervals other than M.

The remaining task is to construct a definable homeomorphism between ]0, u[ and M. There exists a strictly increasing definable homeomorphisms $\psi _1:]0,u/2[ \rightarrow ]-\infty ,0[$ and $\psi _2:]u/2,u[ \rightarrow ]0,\infty [$. The definable map $\psi :]0,u[ \rightarrow M$ given by $\psi (t)=\psi _1(t)$ for $t<u/2$, $\psi (t)=0$ for $t=u/2$ and $\psi (t)=\psi _2(t)$ for $t>u/2$ is a definable homeomorphism. The function $\psi$ is well defined because $(M,+)$ is a divisible group by [3, Proposition 2.2]. $\square$

The following proposition is a part of [1, Fact 1.6].

Proposition 2.3

Consider an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots )$. The followings are equivalent:

• 1. There exists a definable bijection between a bounded interval and an unbounded interval.

• 2. In ${\mathcal {M}}$, we can define a real closed field whose universe is an unbounded subinterval of M and whose ordering agrees with <.

We now begin to prove Theorem 1.2.

Proof of Theorem 1.2

We first show that the condition (1) implies the condition (2).

There exist an unbounded subinterval I of M, two elements $0^*$ and $1^*$ in I, and definable functions $\oplus , \otimes : I \times I \rightarrow I$ such that $(I,0^*,1^*,\oplus ,\otimes )$ is a real closed field with the ordering < by Proposition 2.3. The subinterval I is obviously an open interval.

If $I=M$, the assertion (2) directly follows from the original definable Tietze extension theorem [7, Chapter 8, Corollary 3.10].

We next consider the other case. Consider a definable continuous function $f:A \rightarrow M$ defined on a definable closed subset A of $M^n$. We construct a definable continuous extension $F:M^n \rightarrow M$ of the function f. There exists a definable homeomorphism $\sigma :M \rightarrow I$ by Lemmas 2.1 and 2.2. The notation $\sigma _n$ denotes the homeomorphism from $M^n$ onto $I^n$ induced by $\sigma$. The definable set $\sigma _n(A)$ is contained $I^n$. Consider the definable continuous function $f_{\sigma }:\sigma _n(A) \rightarrow I$ defined by $f_{\sigma }(x)=(\sigma \circ f \circ \sigma _n^{-1})(x)$. Its graph is obviously contained in $I^{n+1}$.

We consider a new structure ${\mathcal {I}}$ whose universe is I. Let ${\mathfrak {S}}_n$ be the set of all subset of $I^n$ definable in ${\mathcal {M}}$. Set ${\mathfrak {S}} = \bigcup _{n \ge 0} {\mathfrak {S}}_n$. For any $S \in {\mathfrak {S}}$, we introduce new predicate symbol $R_S$ and we define ${\mathcal {I}} \models R_S(x)$ by $x \in S$. The structure ${\mathcal {I}}=(I, <, \{R_S\}_{S \in {\mathfrak {S}}})$ is obviously an o-minimal structure. Since the operators $\oplus$ and $\otimes$ are definable in ${\mathcal {I}}$, the structure ${\mathcal {I}}$ is an o-minimal expansion of an ordered field. The $\mathcal M$-definable set $\sigma _n(A)$ and the ${\mathcal {M}}$-definable function $f_{\sigma }$ are also definable in the structure $\mathcal I$. Note that the function $f_{\sigma }$ is also continuous under the topology induced by the ordering of the real closed field $(I,0^*,1^*,\oplus ,\otimes )$ because the two structures ${\mathcal {I}}$ and ${\mathcal {M}}$ share the same order <. There exists a continuous extension $F_\sigma :I^n \rightarrow I$ of $f_\sigma$ definable in ${\mathcal {I}}$ by the original definable Tietze extension theorem [7, Chapter 8, Corollary 3.10]. The function $F_\sigma$ is also definable in ${\mathcal {M}}$ by the definition of the structure ${\mathcal {I}}$. The function $F=\sigma ^{-1} \circ F_{\sigma } \circ \sigma _n$ is the desired definable continuous extension of f definable in ${\mathcal {M}}$.

We next show that the condition (2) implies the condition (1). We construct a definable bijection between a bounded interval and an unbounded interval. Take a positive element c in M. Consider the definable closed set $A=\{(x,y) \in M^2\;|\; x \le 0 \text { or } x \ge c\}$ and the definable continuous function $f:A \rightarrow M$ given by $f(x,y)=y$ if $x \ge c$ and $f(x,y)=0$ otherwise. By the condition (2), there exists a definable continuous extension $F:M^2 \rightarrow M$ of f. The notation g denotes the restriction of F to $[0,c] \times M$.

Consider the sets $S_{t,y}=\{x \in [0,c]\;|\;g(x,y)=t\}$ for all $t \ge 0$ and $y \ge 0$. The definable sets $S_{t,y}$ is not empty for $y>t$ by the intermediate value theorem [3, Corollary 1.5]. The definable function $\varphi _t:M_{>t} \rightarrow [0,c]$ is given by $\varphi _t(y)=\sup S_{t,y}$. For any $t>0$, there exists a nonnegative $u_t$ such that the restriction $\varphi _t|_{M_{>u_t}}$ of the function $\varphi _t(y)$ to $M_{>u_t}=\{y \in M\;|\; y>u_t\}$ is continuous and strictly monotone or constant for $y>u_t$ by the monotonicity theorem.

We consider the following two cases separately.

• (a) The restriction $\varphi _t|_{M_{>u_t}}$ is continuous and strictly monotone for some $t>0$.

• (b) The restriction $\varphi _t|_{M_{>u_t}}$ is constant for any $t>0$.

In the case (a), the restriction $\varphi _t|_{M_{>u_t}}$ gives a bijection between a bounded interval and an unbounded interval. We have finished the proof in this case. We next consider the case (b). By the definition of the function $\varphi _t$, the following assertion holds true:

For any $t>0$, there exist a point $x_t \in [0,c]$ and a nonnegative $u_t$ such that $g(x_t,y)=t$ for all $y>u_t$.

In fact, we have only to take $y'>u_t$ and set $x_t=\varphi _t(y')$. Consider the definable map $\psi :[0,\infty [ \rightarrow [0,c]$ given by $\psi (t)=x_t$, where $x_t$ is the point defined above. Since $\varphi _t|_{M_{>u_t}}$ is constant, the point $x_t$ is independent of the choice of $y'$. It means that $\psi$ is well defined. The map $\psi$ is injective. In fact, if $\psi (t)=\psi (t')$, we have $t=g(\psi (t),y')=g(\psi (t'),y')=t'$ for a sufficiently large $y'$. By the monotonicity theorem, there exists $c>0$ such that the restriction $\psi |_{]c,\infty [}$ of $\psi$ to $]c,\infty [$ is continuous and monotone. The restriction $\psi |_{]c,\infty [}$ is strictly monotone because $\psi$ is injective. Therefore, it gives a definable bijection between a bounded interval and an unbounded interval. $\square$

Corollary 2.4

An o-minimal expansion of an ordered group is semi-bounded if and only if it does not have the definable Tietze extension property.

Proof

The corollary follows from Theorem 1.2 and [1, Fact 1.6]. $\square$