A characteristic of strong topology on function space and a sufficient condition of product of two covariant topology spaces to be covariant topology space (Q2752627)

Let \({\mathcal J}= \{f: X_f\to X\}\) be a family of functions and each \(X_f\) be a topological space. A topology \(\tau\) for \(X\) is called a covariant topology with respect to \({\mathcal J}\) if \(\tau= \{U\subset X: f^{-1}(U)\) is open in \(X_f\) for each \(f\in{\mathcal J}\}\). Let \({\mathcal G}= \{g: X\to X_g\}\) be a family of functions and each \(X_g\) be a topological space. A topology \(\lambda\) for \(X\) is called a contravariant topology with respect to \({\mathcal G}\) if \(\lambda\) has a subbase \(\{g^{-1}(U)\subset X: g\in{\mathcal G}\) and \(U\) is open in \(X_g\}\). In this paper, a strong topology on a function space can be better understood as analyzed from either covariant or contravariant point of view. Also, a sufficient condition is obtained that the product of two covariant topological spaces is a covariant topological space.