# Projection on Real Euclidean Plane is Open Mapping

## Theorem

Let $\struct {\R^2, d}$ be the real number plane with the usual (Euclidean) topology.

Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as:

$\forall \tuple{x, y} \in \R^2: \map \rho {x, y} = x$

Then $\rho$ is an open mapping.

The same applies with the second projection on $\R^2$.

## Proof

By definition, the real number plane with the usual (Euclidean) topology on $\R^2$ is the product space of $\struct {\R, d}$ with $\struct {\R, d}$, where $\struct {\R, d}$ is the real number line with the usual (Euclidean) topology

The result follows from Projection from Product Topology is Open.

$\blacksquare$