  This paper is concerned with algebraic geometry over complete discretely
valued fields $K$ of equicharacteristic zero. Several results are given
including: the canonical projection $K^{n} \times K\mathbb{P}^{m}
\longrightarrow K^{n}$ and blow-ups of the $K$-points of smooth $K$-varieties
are definably closed maps, a descent principle, a version of the Lojasiewicz
inequality for continuous rational functions, curve selection for semialgebraic
sets and the theorem on extending continuous hereditarily rational functions,
established for the real field in our joint paper with J. Kollar. Our approach
applies the quantifier elimination due to Pas. By the transfer principle of
Ax-Kochen-Ershov, all these results carry over to the case of Henselian
discretely valued fields. Using different arguments in our subsequent paper, we
establish them over Henselian real valued fields of equicharacteristic zero.
