?-Complete filter spaces

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

“…Also, there is an equivalence relation defined between T 2 completions. For a T 2 Cauchy space 2 , and the two completions are equivalent, if each of these two completions is greater than or equal to the other. Note that in case of equivalence, h is a unique Cauchy homeomorphism.…”

“…Fric and Kent [3] have shown that any Cauchy map on a T 2 Cauchy space can be uniquely extended to its T 2 Wyler completion. The next proposition shows that the Wyler completion ((X,C), j) also enjoys this extension property with respect to the s-maps.…”

“…Now we can define a functor on the category CHY exactly the same as the T 2 Wyler completion functor defined in [8]. Let CHY be the subcategory of CHY consisting of all complete objects in CHY .…”

“…The completion of Cauchy spaces is already well known and familiar to most of us. In fact, since Keller [5] introduced the axiomatic definition of Cauchy spaces a very rich and extensive completion theory has been developed for Cauchy spaces during the last three decades [2,3,7,10,12]. It seems that Cauchy space rather than uniform convergence space is a natural generalization of completion of uniform space.…”

Completions of non‐T2 filter spaces

“…When (X,ζ) is a T 2 filter space, completion and extension theorems were established in [4] and later a few other classes of completions were constructed in [2,5]. In this section, we will construct non-T 2 completion and non-T 2 weak completion of a filter space and establish some extension theorems.…”