I came across a proof where it's used one such map in the following way.
Let $V \subsetneq \mathbb{R}^{n}$ be an open, non empty, subset - we're using euclidean topology. We consider a projection map $\pi : V \rightarrow \mathbb{R}$ such that $\forall x \in V \ \pi(x) \geq 0$.

I just cannot convince myself that one such map always exists. Could anyone please explain why? I would also appreciate an example.

Thanks a lot.