In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms and . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.