  The ultrapower $T^{\ast}$ of an arbitrary ordered set $T$ is introduced as an
infinitesimal extension of $T$. It is obtained as the set of equivalence
classes of the sequences in $T$, where the corresponding relation is generated
by an ultrafilter on the set of natural numbers. It is established that
$T^{\ast}$ always satisfies Cantor's property, while one can give the necessary
and sufficient conditions for $T$ so that $T^{\ast}$ would be complete or it
would fulfill the open completeness property, respectively. Namely, the density
of the original set determines the open completeness of the extension, while
independently, the completeness of $T^{\ast}$ is determined by the cardinality
of $T$.
