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I find that in most definitions in differential geometry, such as those of defining a manifold, a smooth manifold,differentiable functions, diffeomorphisms on manifolds, an atlas,etc , open sets are used. From what I can guess it seems like these usage of open sets is being followed in line with notions in topology, ie a topological space and open sets covering it. Can anyone elaborate to a beginner in differential geometry why open sets are used?

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A topological manifold is by definition locally homeomorphic to $\mathbb{R}^n$. Locally meaning there is an open set containing each point that is homeomorphic. If you're talking about topology you would expect open sets to come up since the collection of open sets is by definition the topology on a set.

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Informally said, open subsets are those which represent “full pieces” of the space. They are “saturated” in the sense that they contain all the local information for each point they contain. Every local property for every point holds in open subset if and only if it holds in the whole space.

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Manifolds are spaces where we can make sense of differentiation.

Let $D\subset\mathbb{R}^n$. When does it make sense to say that a function $f:D\rightarrow \mathbb{R}$ is differentiable at a point $x\in D$? Precisely when $x$ lies in the interior of $D$. Manifolds are set up in such a way that each point in the manifold locally looks like an open subset of $\mathbb{R}^n$, so that we can make sense of differentiation (and integration and...).

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