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In this paper we are concerned with definably, with or without
parameters, (Dedekind) complete expansions of ordered fields, i.e.
those with no definable gaps. We present several axiomatizations,
like being definably connected, in each of the two cases. As a
corollary, when parameters are allowed, expansions of ordered
fields are o-minimal if and only if all their definable subsets
are finite disjoint unions of definably connected (definable)
subset. We pay attention to how simply (in terms of the quantifier
complexity and/or usage of parameters) a definable gap in an
expansion is so. Next we prove that over parametrically definably
complete expansions of ordered fields, all one-to-one definable
(with parameters) continuous functions are monotone and open.
Moreover, in both parameter and parameter-free cases again,
definably complete expansions of ordered fields satisfy definable
versions of the Heine-Borel and Extreme Value theorems and also
Bounded Intersection Property for definable families of closed
bounded subsets.
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