A dichotomy result for countably based sober spaces

Cartesian closed categories have been playing fundamental roles in providing denotational semantic for higher-order programming languages. In this paper we try to identify Cartesian closed subcategories of the category $C{S}_{\perp }$ of pointed countably based sober spaces, and we present a conclusion known as the dichotomy result in the category $C{S}_{\perp }$. This result explains that any Cartesian closed full subcategory of $C{S}_{\perp }$ is contained within either the category of weakly compact open connected spaces or that of principally connected spaces.

To prove our dichotomy theorem, we first deduce that every pointed countably based sober space X is locally connected, if the space of all continuous functions from X to X is locally compact. Next, we demonstrate that a function space in $C{S}_{\perp }$ is locally connected only if its input space is either weakly compact open connected or its output space is principally connected.