Projection map for the product, uniform and box topologies

Question

That the canonical projection map (from say $\mathbb{R}^\mathbb{N}$ to $\mathbb{R}$) is continuous for all three topologies is straightforward. But is it also open for the uniform and box topologies? (That it is open for product I see).

Answer 1

It is well-known that $\mathscr{T}_{\text{prod}} \subseteq \mathscr{T}_{\text{unif}} \subseteq \mathscr{T}_{\text{box}}$

The projection is open in the box topology because a base for this topology is given by sets of the form $U=\prod_n U_n$ where all $U_n$ are open, and $\pi_n[U] = U_n$ which is open. So $\pi_n$ also open for all coarser topologies.