Grauert's theorem for subanalytic open sets in real analytic manifolds

By open neighbourhood of an open subset $Ω$ of $\mathbb{R}^n$ we mean an open subset $Ω'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\capΩ'=Ω.$ A well known result of H. Grauert implies that any open subset of $\mathbb{R}^n$ admits a fundamental system of Stein open neighbourhoods in $\mathbb{C}^n$. Another way to state this property is to say that each open subset of $\mathbb{R}^n$ is Stein. We shall prove a similar result in the subanalytic category, so, under the assumption that $Ω$ is a subanalytic relatively compact open subset in a real analytic manifold, we show that $Ω$ admits a fundamental system of subanalytic Stein open neighbourhoods in any of its complexifications.

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