Differentiable Manifolds/What is a manifold?

In this section, the important concepts of manifolds shall be introduced.

Charts, compatibility of charts, atlases and manifolds
In this subsection, we define a manifold and all the things which are necessary to define it. It's a bit lengthy for a definition, but manifolds are such an important concept in mathematics that it's far more than worth it.

Differentiable functions on manifolds
In this subsection, we shall define what differentiable maps, which map from a manifold or to a manifold or both, are.

Instead of writing $$\varphi \cdot \vartheta$$, we will in the following write $$\varphi \vartheta$$; just omitting the dot. This is often also done for the multiplication of variables (for instance $$xy$$ stands for $$x \cdot y$$ if $$x, y \in \mathbb R$$).

Tangent vectors, tangent spaces and tangent bundles
Tangents, in the classical sense, are lines which touch a geometrical object at exactly one point. The following definition of a tangent of a manifold, in this context called tangent vector to a manifold, is somewhat strange.