Abstract: The holomorphic closure dimension is one of the possible biholomorphic invariant that gives some information on how much of the complex ambient structure a subset of a complex manifold inherits. I will review a series of results by Rasul, Rasul and Janusz, and Janusz and myself that discuss the behaviour of the holomorphic dimension in a real analytic set on the one hand and in semialgebraic sets on the other hand.