User:Anghel/Sandbox

Theorem
Let $\C$ denote the complex plane.

Let $R^2$ denote the Euclidean plane.

Then the function $\phi : \R^2 \to \C$ defined by:


 * $\map \phi { x, y } = x + i y$

is a homeomorphism between $R^2$ and $\C$.

Proof
Define $\phi^{-1} : \C \to \R^2$ by:


 * $\map { \phi^{-1} }{ z } = \paren { \map \Re z, \map \Im z }$

Then $\phi^{-1}$ is the inverse of $\phi$, as:


 * $\map { \phi }{ \map { \phi^{-1} }{ z } } = \map \Re z + i \map \Im z = z $
 * $\map { \phi^{-1} }{ \map { \phi }{ x, y } } = \paren { \map \Re { x + i y } , \map \Im { x + i y } }$.

By definition of bijection, it follows that $\phi$ is bijective.