Completion of a Cauchy space without the T₂‐restriction on the space

Abstract

Abstract. A completion of a Cauchy space is obtained without the T 2 restriction on the space. This completion enjoys the universal property as well. The class of all Cauchy spaces with a special class of morphisms called s-maps form a subcategory CHY of CHY. A completion functor is defined for this subcategory. The completion subcategory of CHY turns out to be a bireflective subcategory of CHY . This theory is applied to obtain a characterization of Cauchy spaces which allow regular completion.

Quasi-Completion of Filter Spaces

“…One of the remarkable differences between these properties and the weaker quasi-properties is that the T 1 and T 2 properties are equivalent [9] for a filter space, whereas this is not true in general for q-T 1 and q-T 2 . The following example shows that there is a filter space which is q-T 1 , but not necessarily q-T 2 .…”

“…However, since this is not the case for all Cauchy spaces in general (see Example 3.2 [9]), the stable completions were introduced [9]. Since Cauchy spaces are special cases of filter spaces, Reed's result will also fail for non-T 2 filter spaces in general.…”

“…Extension theorems for filter spaces [3], regular filter spaces [10], filter semigroups [11] and Cauchy spaces ( not necessarily T 2 ) [9] have led to some interesting reflective subcategories of the categories F IL and CHY with some special type of morphisms called s-maps. In case of T 2 filter spaces, an unique extension of a Cauchy map f : (X, C) −→ (Y, D) to the corresponding completion space was possible only when the codomain was a c-filter space.…”

“…Also, the well-known completion theory for Cauchy spaces was extended to obtain a completion without the T 2 -restriction on the space by the author [9]. An s-map was introduced to form a special class of morphisms which led to a completion functor on a subcategory of CHY (the category of Cauchy spaces with Cauchy maps) with respect to a new class of morphisms.…”

On finest unitary extensions of topological monoids

“…Each Cauchy space has many completions which have been studied in papers of many authors (see [2][3][4][5][6][7][8][9][10][11]). The most important of them is perhaps the Wyler completion possessing the universal property over other completions.…”