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The main purpose of the paper is to establish a closedness theorem overHenselian valued fields $K$ of equicharacteristic zero (not necessarilyalgebraically closed) with separated analytic structure. It says that everyprojection with a projective fiber is a definably closed map. This remainsvalid also for valued fields with analytic structure induced by a strictlyconvergent Weierstrass systems, including the classical, complete rank onevalued fields with the Tate algebra of strictly convergent power series. Asapplication, we prove two theorems on existence of the limit and on piecewisecontinuity.

A closedness theorem over Henselian valued fields with analytic structure