Net-convergence in fuzzy neighborhood spaces (Q1302231)

The authors introduce a concept of net convergence in Lowen-fuzzy neighborhood spaces. This notion seems to suit fuzzy neighborhood spaces well. For examples, it may be used to characterize Lowen's \(NT_2\) fuzzy neighborhood spaces and cluster points of a set, it is also a good extension of the corresponding notion in topological spaces. The authors prove that a net \((x_d)_{d\in D}\) converges to \(x\) in the product \(X=\prod _{j\in J}X_j\) of fuzzy neighborhood spaces \(\{X_j\}_{j\in J}\) if and only if the net \((p_j(x_d))_{d\in D}\) converges to \(p_j(x)\) in \(X_j\), where \(p_j\) is the projection from \(X\) to \(X_j\) (\(j\in J\)). The notion can also be applied to fuzzy neighborhood groups and fuzzy neighborhood linear spaces to prove certain characterization theorems.