Changes

Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].

Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]].

Example: Consider the set of points <math>[x_0:x_1:x_2]</math>, where <math>x_i</math> are not all zero, and where we declare <math>[x_0:x_1:x_2]</math> to be equivalent to <math>[y_0:y_1:y_2]</math> if <math>x_i = \lambda y_i</math>. This space describes the set <math>\mathbf{R}P^2</math> of all one-dimensional subspaces of <math>R^3</math>.

Example: Consider the set of points <math>[x_0:x_1:x_2]</math>, where <math>x_i</math> are not all zero, and where we declare <math>[x_0:x_1:x_2]</math> to be equivalent to <math>[y_0:y_1:y_2]</math> if <math>x_i = \lambda y_i</math>. This space describes the set <math>\mathbf{R}P^2</math> of all one-dimensional subspaces of <math>R^3</math>. Let <math>U_i</math> be the open set with <math>x_i \neq 0</math>. Define <math>\phi_0: U_0\rightarrow R^2</math> to be the map given by <math>[x_0:x_1:x_2]\mapsto (\frac{x_1}{x_0},\frac{x_2}{x_0})</math>. Define <math>\phi_1</math> and <math>\phi_2</math> similarly. Then <math>\phi_i</math> clearly define homeomorphisms from <math>U_i</math> to <math>R^2</math>. It is simple to verify that the change of coordinate functions are also diffeomorphisms. Hence <math>\mathbf{R}P^2</math> is a differentiable manifold. In fact, <math>\mathbf{R}P^2</math> with a disk removed actually defines the mobius strip.

[[Category:Mathematics]]

[[Category:Mathematics]]

[[category: Topology]]

[[category: Topology]]