Is the projection map $\pi_1:\mathbb{R}^2 \to \mathbb{R}$ a quotient map?

Question

Is the projection map $\pi_1:\mathbb{R}^2 \to \mathbb{R}$ a quotient map? It is definitely a surjective and continuous map. I think it is also open and most likely not a closed one. Surjective, continuous and open should suffice it to be a quotient map, right? But is it necessary for a map to be open or closed to be a quotient one? What about saturated open sets should be mapped to open sets, is it necessary or sufficient?