To verify if the mapping f: R^3\{(x1,x2,x3):x1=0} → R^2 defined by (x1,x2,x3) |→ (x2/x1, x3/x1) is a quotient map, one must understand the definition of a quotient mapping and its required properties, including continuity and local injectivity. The discussion highlights the importance of specific topologies in both the domain and codomain for testing quotient maps. It suggests that demonstrating at least one nonzero partial derivative can establish local injectivity, which is a step toward proving the mapping is open and thus a quotient. Additionally, there is a query about the existence of continuous maps that are surjective but not open, indicating a need for further exploration of related theorems. The conversation also touches on whether a continuous map being open on any open subspace implies it is open everywhere, emphasizing the need for careful examination of subspaces.