Manifolds

A manifold $M$ is a set of points that each has a continuous one-to-one map from an open neighborhood to an open subset of $\mathbb{R}^n$. $M$ is locally similar to standard $\mathbb{R}^n$. The map from an open neighborhood of $M$ to $\mathbb{R}^n$ associates each point $P$ in $M$ with an $n$-tuple $(x_1(P), \dots, x_n(P))$, where $x_1(P), \dots, x_n(P)$ are called the coordinates of point $P$. One way to think of an $n$-dimensional manifold is as a set in which every point has $n$ independent coordinates in some neighborhood.
For example, the surface of a sphere is a manifold. The surface of a 3D sphere $S^2$ is defined as a set of points in $\mathbb{R}^3$ such that $x^2 + y^2 + z^2 = r^2$ holds. One degree of freedom is eliminated, making it 2-dimensional. Its map onto $\mathbb{R}^2$ is given in usual spherical coordinates, and coordinates are identified as the combination $(\theta, \phi)$. At points like $0$ and $2\pi$ (problematic poles), a one-to-one correspondence is not possible. This shows that a globally valid coordinate system for all points on a manifold generally does not exist — only local coordinates in neighborhoods are valid. For instance, the spherical coordinate map is valid only in the open neighborhood $0 < \phi < \pi$, $0 < \theta < 2\pi$.

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