Topology Atlas
Document # ppae-20

## Using nets in Dedekind, monotone, or Scott incomplete ordered fields and definability issues

### Mojtaba Moniri and Jafar S. Eivazloo

Proceedings of the Ninth Prague Topological Symposium
(2001)
pp. 195-203
Given a Dedekind incomplete ordered field, a pair of convergent nets of
gaps which are respectively increasing or decreasing to the same point is
used to obtain a further equivalent criterion for Dedekind completeness of
ordered fields: Every continuous one-to-one function defined on a closed
bounded interval maps interior of that interval to the interior of the
image.
Next, it is shown that over all closed bounded intervals in any monotone
incomplete ordered field, there are continuous not uniformly continuous
unbounded functions whose ranges are not closed, and continuous 1-1
functions which map every interior point to an interior point (of the
image) but are not open.
These are achieved using appropriate nets cofinal in gaps or coinitial in
their complements.
In our third main theorem, an ordered field is constructed which has
parametrically definable regular gaps but no \emptyset-definable
divergent Cauchy functions (while we show that, in either of the two cases
where parameters are or are not allowed, any definable divergent Cauchy
function gives rise to a definable regular gap).
Our proof for the mentioned independence result uses existence of infinite
primes in the subring of the ordered field of generalized power series
with rational exponents and real coefficients consisting of series with no
infinitesimal terms, as recently established by D. Pitteloud.

Mathematics Subject Classification. 03C64, 12J15, 54F65.

Keywords. Ordered Fields, Gaps, Completeness Notions, Definable Regular
Gaps, Definable Cauchy Functions, Generalized Power Series.

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Copyright © 2002
Charles University and
Topology Atlas.
Published April 2002.