= C1 openness of diffeomorphisms
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{title2=$\operatorname{Diff}(M)\text{ is }C^1\text{-open}$}
On a compact <smooth manifold> without boundary, every smooth self-map sufficiently $C^1$-close to a <diffeomorphism> is a <diffeomorphism>. Reducing to a map near the identity, invertibility of its differential follows uniformly by compactness. It is then a proper local <diffeomorphism>, hence a finite <covering map>. Sufficient $C^0$ closeness gives a <homotopy> to the identity on each component; its induced map on the <fundamental group> is surjective. The covering therefore has one sheet on each component, proving the assertion. This lets a nearby <Lagrangian submanifold> transverse to cotangent fibers be represented by a single global <differential one-form> graph.
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