Exponential objects and Cartesian closedness in the construct \textit{Prtop}
An object \(X\) in a category \({\mathcal K}\) with finite products is called exponential if the functor \(X\times -: {\mathcal K}\to {\mathcal K}\) has a right adjoint. A pretopological space \(X\) is finitely generated if any point \(x\in X\) has a smallest neighborhood. A pretopological space \(X\) is exponential if and only if \(X\) is finitely generated. The full subconstruct of all finitely generated pretopological spaces is a bireflective Cartesian closed subconstruct of the construct Prtop of pretopological spaces. On the other hand, there exists no Cartesian closed coreflective full subconstruct of Prtop containing the space \(\{{1\over n}\); \(n>0\) is an integer\(\}\cup\{0\}\) with the natural topology. Thus the construct FrPrtop whose objects are pretopological spaces such that its pretopology is determined by convergent sequences is not Cartesian closed. A strong Fréchet axiom is given such that the full subconstruct of FrPrTop containing all objects in FrPrtop and satisfying one is a Cartesian closed bireflective subconstruct of FrPrtop.
scientific article; zbMATH DE number 554357