Projection on Real Euclidean Plane is Open Mapping

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Theorem

Let $\struct{\R^2, d}$ be the real Euclidean plane.

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 Euclidean plane on $\R^2$ is the product space of $\R$ with $\R$.

The result follows from Projection from Product Topology is Open.

$\blacksquare$


Sources