  Let $X$ be a noncomplete metric space satisfying the usual (local)
assumptions of a doubling property and a Poincar\'e inequality. We study
extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of
$X$ and use them to obtain several results on $X$ itself, in particular
concerning minimal weak upper gradients, Lebesgue points, quasicontinuity,
regularity properties of the capacity and better Poincar\'e inequalities. We
also provide a discussion about possible applications of the completions and
extension results to $p$-harmonic functions on noncomplete spaces and show by
examples that this is a rather delicate issue opening for various
interpretations and new investigations.
