The density assumption on U implies U is Stein .Such pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse Theory.The title of the paper is "A Topological property of Runge pairs"
Mohan Ramachandran
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