On Nash approximation of complex analytic sets in Runge domains

We prove that every complex analytic set X in a Runge domain Ω   can be approximated by Nash sets on any relatively compact subdomain Ω0Ω0 of Ω. Moreover, for every Nash subset Y of Ω   with Y⊂XY⊂X, the approximating sets can be chosen so that they contain Y∩Ω0Y∩Ω0. As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.