2
4. ZERO SELECTION ( ). Let f:
n+m
V. There exists
g:
n
m
V such that if f(x,y) = 0 then f(x,g(x)) = 0.
We use the adjective “special” if, in addition, we have
5. LIMIT (
). Every bounded f:
V has a limit at
infinity.
From model theory, an expansion of the field of real
numbers is a system ( ,<,0,1,+,•,V), where V obeys
MULTIVARIATE (
) above. (We identify relations with their
characteristic functions.)
An o-minimal expansion of the field of real numbers is an
expansion of the field of real numbers in which every first
order definable subset of is a finite union of open
intervals and points. Here and - are allowed as
endpoints. This is a special case of the more general
definition of o-minimal structure introduced in [PS86],
which we use in section 4.
We prove the following.
THEOREM A. Let M = ( ,<,0,1,+,•,V) be an expansion of the
field of real numbers. The following are equivalent.
i. M is o-minimal.
ii. V is a subset of some special rich class ( ).
ii. There is a least rich class ( ) containing V, and this
rich class (
) is special.
We conclude with a version for arbitrary ordered fields.
2. RICH CLASSES OF REAL FUNCTIONS.
The key to proving Theorem A is the characterization of
rich classes (
) in terms of first order definability. We
now fix V to be a rich class ( ). Let M = (
,<,0,1,+,•,V).
Let f:
n+m
. A zero selector of f is a g with the
property in the ZERO SELECTION ( ) clause.
LEMMA 2.1. There exist f
1
,...,f
8
:
V and f
9
,...,f
12
:
4
such that
i. f
1
(x) = 1/x if x ≠ 0.
ii. f
2
(x) = 0 if x ≠ 0; 1 otherwise.
iii. f
3
(0) = 0; f
3
(x) = 1 elsewhere.
iv. x
≥
0
´
(f
4
(x) = sqrt(x) f
4
(x) = -sqrt(x)).