Manifold

The Möbius strip is an example of a 2-manifold.

An n-dimensional manifold (or n-manifold) M is a topological space such that every point in M has a neighbourhood U that is homeomorphic to Rⁿ. The homeomorphisms $\phi _{U}:U\rightarrow \mathbf {R} ^{n}$ should be thought of as providing local coordinates in the neighborhood U. Whenever two such coordinate neighborhoods U and V intersect, we also require that the change of coordinate maps $\phi _{U}\circ \phi _{V}^{-1}:\phi _{V}(U\cap V)\rightarrow \phi _{U}(U\cap V)$ be homeomorphisms.

An n-dimensional differentiable manifold is an n-manifold with the property that the change of coordinate maps are differentiable functions.

An n-dimensional complex manifold N is a topological space such that every point in N has a neighbourhood that is homeomorphic to Cⁿ, and such that the change of coordinate functions are holomorphic.

Manifolds are Hausdorff and 2nd-countable.

Example: Consider the set of points $[x_{0}:x_{1}:x_{2}]$, where $x_{i}$ are not all zero, and where we declare $[x_{0}:x_{1}:x_{2}]$ to be equivalent to $[y_{0}:y_{1}:y_{2}]$ if $x_{i}=\lambda y_{i}$. This space describes the set $\mathbf {R} P^{2}$ of all one-dimensional subspaces of $R^{3}$. Let $U_{i}$ be the open set with $x_{i}\neq 0$. Define $\phi _{0}:U_{0}\rightarrow R^{2}$ to be the map given by $[x_{0}:x_{1}:x_{2}]\mapsto ({\frac {x_{1}}{x_{0}}},{\frac {x_{2}}{x_{0}}})$. Define $\phi _{1}$ and $\phi _{2}$ similarly. Then $\phi _{i}$ clearly define homeomorphisms from $U_{i}$ to $R^{2}$. It is simple to verify that the change of coordinate functions are also diffeomorphisms. Hence $\mathbf {R} P^{2}$ is a differentiable manifold. In fact, $\mathbf {R} P^{2}$ with a disk removed actually defines the mobius strip.