Monotone determined spaces via $\mathbb{C}$-generated spaces

Abstract

The category of monotone determined spaces is an extended topological framework for dcpos in domain theory. We first show that monotone determined spaces are exactly the spaces generated by one-point convergence spaces, and then naturally form a convenient Cartesian closed category of $\mathbb{C}$-generated spaces. We then show that monotone determined spaces are not always compact Hausdorff generated, answering the question raised by Ingo Battenfeld in 2013. Moreover, we generalize the notion of monotone determined spaces by introducing $\mathcal{C}$-determined spaces and showing that categories of $\mathcal{C}$-determined spaces correspond to coreflective subcategories of topological spaces. This yields a uniform construction of several convenient categories determined by directed, chain and monotone sequential convergence classes. We finally discuss the relationships among them, including the categories generated by continuous spaces, quasicontinuous spaces and Scott spaces of dcpos.