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Let $(M^3,g)$ be a closed Riemannian manifold. Is there a “natural” topology on the space $\operatorname{Fol}(M)$ of smooth codimension $1$ foliations on $M$? Is there any other relevant structure on this set?

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A smooth codimension 1 foliation (in any dimension) is the same thing as an integrable codimension 1 distribution (tangent spaces to the leaves of the foliation). The space of smooth distributions is the space of smooth sections of the projectivization of the cotangent bundle of the manifold. If you have any smooth bundle $E\to M$ over any manifold, then the space of sections $\Gamma(M,E)$, has a family of topologies depending on how much smoothness you want to keep track of. In your case, for instance, if your foliation is of class $C^1$, the distribution will be of class $C^0$, hence, the natural topology to use on the space of continuous sections of $P(T^*M)$ is the compact-open topology (or topology of uniform convergence since $M$ is compact); then restrict this topology to the subspace topology on the subspace of integrable distributions. The Riemannian metric is mostly irrelevant here, but you can use it to identify $TM$ and $T^*M$ if you like.

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