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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} are not all zero, and where we declare Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_0:x_1:x_2]} to be equivalent to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{R}P^2} with a disk removed actually defines the mobius strip.