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In this paper we restrict ourselves to the particular case where thepseudogroup is $\Gamma \ltimes G$ given by the action of a dense subgroup$\Gamma$ of a Lie group $G$ acting on $G$ by left translations. For aRiemannian foliation $F$ on a complete Riemannian manifold $M$ which istransversally parallelizable in the sense of Molino, let $X$ be the space ofleaves closures. The holonomy pseudogroup of $F$ is an example of a locallytrivial extension of $X$ by $\Gamma \ltimes G$. The study of a generalizationof this particular case shall be the main purpose of this paper.

On locally trivial extensions of topological spaces by a pseudogroup