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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.
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