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Def: An $n$ dimensional topological manifold is a paracompact Hausdorff topological space, say $M$, such that every point $p\in M$ is contained in some open set $U_p$ that is homeomorphic to an open subset of the euclidean space $\mathbb{R}^n$.

Using the above definition of a topological manifold, what conditions on a class $C^k$ atlas, $\mathcal{A}$ over $M$ do we need to guarantee that each chart $(V,y)\in \mathcal{A}$ is a homeomorphism? Is it enough to know that each chart transition map is $k$-times differentiable?

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If we know that all transition maps are differentiable (at least $C^1$) then we are fine. Since the transition map is the composition of a chart and the inverse of another chart. In particular having differentiable transition charts means that the charts are diffeomorphism and therefore homeomorphism.

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