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.

$\blacksquare$


Sources

  • 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $33$. Special Subsets of the Plane: $1$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.