On the largest coreflective Cartesian closed subconstruct of Prtop (Q1364910)

Let \(\mathbf{Prtop}\) denote the construct of pretopological spaces and continuous mappings and let \(\mathbf{Fing}\) denote the full subconstruct of \(\mathbf{Prtop}\) consisting of all finitely generated pretopological spaces. \(\mathbf{Fing}\) is the largest Cartesian closed coreflective subconstruct of \(\mathbf{Prtop}\). Let \(\mathbf 3\) be the pretopological space on the set \(\{0,1,2\}\) such that the neighbourhood filters of \(0\) and \(2\) are the ultrafilters and the neighbourhood filter of \(1\) is \(\{\{1,2\},\{0,1,2\}\}\). If a subconstruct \(\mathcal A\) of \(\mathbf{Prtop}\) contains \(\mathbf 3\) then the following are equivalent: \(\mathcal A\) is initially structured; \(\mathcal A\) is coreflective in \(\mathbf{Prtop}\); \(\mathcal A\) is topological. Any exponential object of a coreflective subconstruct \(\mathcal C\) of \(\mathbf{Prtop}\) is finitely generated, if, moreover, \(\mathcal C\) is finitely productive in \(\mathbf{Prtop}\) then an object of \(\mathcal C\) is finitely generated if and only if it is exponential. It is shown that the assumption that \(\mathcal C\) is finitely productive cannot be omitted.