Separation and compactness in topological categories

In previous papers, several extensions of the notions of closedness, separation properties, and compactness in a set-based topological category were introduced. In this paper, we develop further these extensions in a much larger non set-based topological categories. Moreover, we show that the categories KT2ConFCO (the category of KT2 constant filter convergence spaces and continuous functions) and Chy (the category of Cauchy spaces and Cauchy maps) are isomorphic. Also, we characterize strongly compact constant filter convergence spaces and investigate some invariance properties of them. Finally, we compare our results and give some applications.