Symmetric \(g\)-functions (Q1414251)

Several classes of generalized metric spaces \((X,\tau)\) were characterized by \(g\)-functions \(g:\mathbb N \times X \rightarrow \tau\) such that \(x\in g(n,x)\) for each \(x\in X\) and each \(n\in \mathbb N\). The authors of this paper study the role of symmetric \(g\)-functions, i.e., such \(g\)-functions which satisfy \(x\in g(n,y)\) if and only if \(y\in g(n,x)\) for each \(n\in \mathbb N\) and all \(x,y \in X\). They identify and consider six classes of such functions, give relationships between them and explain their role in the theory of generalized metric spaces and especially in metrization theory.