Projection on Real Euclidean Plane is Open Mapping

Theorem
Let $\left({\R^2, d}\right)$ be the real Euclidean plane.

Let $\rho: \R^2 \to \R$ be the first projection on $\R^2$ defined as:
 * $\forall \left({x, y}\right) \in \R^2: \rho \left({x, y}\right) = x$

Then $\rho$ is an open mapping.

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

Proof
By definition, the real Euclidean plane on $\R^2$ is the topological product of $\R$ with $\R$.

The result follows from Projection from Product Topology is Open.