Completions of non-T2 filter spaces

The categorical topologists Bentley et al. [1] have shown that the category FIL of filter spaces is isomorphic to the category of filter merotopic spaces which were introduced by Katětov [3]. The category CHY of Cauchy spaces is also known to be a bireflective, finally dense subcategory of FIL [7]. So the category FIL is an important category which deserves special discussion. A completion theory for filter spaces was introduced in [4], where a completion functor was defined on the subcategory T2 FIL of T2 filter spaces. This completion theory was later applied to completion of filter semigroups [9]. Several other types of completions and their properties were also studied by Minkler et al. [5] and Császár [2]. In this paper, a completion theory is developed for filter spaces without the T2 restriction on the spaces. Also, a completion functor is defined on a subcategory of FIL, which is constructed by taking all the filter spaces as objects and morphisms as certain special type of continuous maps which we call s-maps.