COMPLETIONS OF NON-T 2 FILTER

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.


Introduction
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 T 2 FIL of T 2 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 T 2 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.

Preliminaries
For basic definitions and terminologies related to filters, the reader is referred to [11], though a few of the definitions will be mentioned here.Let X be a set and let F(X) be the set of all filters on X.If Ᏺ,Ᏻ ∈ F(X) and F ∩ G = φ for all F ∈ Ᏺ, G ∈ Ᏻ, then Ᏺ ∨ Ᏻ denotes the filter generated by {F ∩ G : F ∈ Ᏺ and G ∈ Ᏻ}.If there exist F ∈ Ᏺ and G ∈ Ᏻ such that F ∩ G = φ, we say that Ᏺ ∨ Ᏻ fails to exist.For each x ∈ X, we denote by ẋ the ultrafilter generated by {x}.If ζ ⊂ F(X) satisfies the conditions (c 1 ) ẋ ∈ ζ, for all x ∈ X, (c 2 ) Ᏺ ∈ ζ, Ᏻ ≥ Ᏺ imply that Ᏻ ∈ ζ, then the pair (X,ζ) is called a filter space.
The two filters Ᏺ,Ᏻ ∈ F(X) are said to be ζ-linked if there exist a finite number of filters Ᏼ 1 ,Ᏼ 2 ,...,Ᏼ n ∈ ζ such that Ᏺ ∨ Ᏼ 1 , Ᏼ 1 ∨ Ᏼ 2 ,...,Ᏼ n−1 ∨ Ᏼ n , Ᏼ n ∨ Ᏻ all exist.In − → x if and only if Ᏺ ∩ ẋ ∈ ζ.Note that for any filter space (X,ζ), p ζ ≤ q ζ (see [4]).A filter space (X,ζ) is a c-filter space if, in addition, (c 3 ) A convergence structure q on a set X is said to be compatible (resp., c-filter compatible, Cauchy compatible) if there exists a filter structure (resp., c-filter structure, Cauchy structure) ζ on X such that q = q ζ .As shown above, given a filter space (X,ζ), we can always associate a convergence q ζ .However, every convergence structure on X is not c-filter compatible.
Example 2.2.Let X = {a, b,c} and let q be the convergence structure on X defined by ȧ → q a, ḃ → q b, ṫ → q t, ȧ ∩ ḃ → q b, ḃ ∩ ṫ → q t, and all other filters fail to converge.If possible, let ζ be the filter structure on X such that q The following lemma states the necessary and sufficient conditions for such compatibilities of a convergence structure on X. Lemma 2.3.A convergence structure q on X is (a) compatible if and only if Ᏺ is q-convergent and ẋ ≥ Ᏺ ⇒ Ᏺ → q x; (b) c-filter compatible if and only if either q(x) = q(y) or q(x) ∩ q(y) = φ; (c) Cauchy compatible if and only if Ᏺ,Ᏻ ∈ q(x) ⇒ Ᏺ ∩ Ᏻ ∈ q(x) and for all x, y ∈ X, q(x) = q(y) or q(x) ∩ q(y) = φ.
Proof.The proof of (b) is similar to [4, Proposition 1.3] and the proof of (c) is well known.So we will prove only (a).Let q = q ζ , where ζ is a filter structure on X.Let Ᏺ be q-convergent, Ᏺ → q y (say), and ẋ ≥ Ᏺ.
Conversely, let q satisfy the given condition and let ζ q be the set of all qconvergent filters.We show that q = q ζq .If Compatibilities of a preconvergence structure on a filter space is defined in a similar way.The conditions for compatibility, c-filter compatibility, and Cauchy compatibility of a preconvergence structure p were established in [4].
Note that if q is a convergence structure on X and there is a filter structure ζ on X such that q = p ζ , then (X,ζ) is a c-filter space and q = q ζ .In particular, the following lemma holds when q is a pretopology.Lemma 2.4.A pretopology σ on X is compatible and σ = p ζ for some filter structure ζ on X if and only if ζ is a c-filter structure.
However, if the pretopology σ = q ζ for some filter structure ζ on X, then as illustrated in the following example, the above lemma may not hold in general.
Lemma 2.6.The following are true for any filter space (X,ζ): Lemma 2.7.Any regular filter space is a c-filter space.
Lemma 2.8.A filter space (X,ζ) is totally bounded and complete if and only if (X, We denote by FIL the category of all filter spaces and continuous maps as morphisms.Let CFIL and CHY be the full subcategories of FIL whose objects are cfilter spaces and Cauchy spaces, respectively.In [4], a completion of objects in T 2 FIL and a completion functor on T 2 CFIL and its bireflective subcategory C 3 FIL were constructed.The completion functor and the completion subcategory constructed in [4] deal with T 2 filter spaces.The underlying reason for this is the existence of unique limits for convergent filters which are also preserved by the continuous map.In this paper, we partially overcome that limitation by using a special type of continuous map called s-map which will be introduced later. The map f : (X,ζ) → (Y ,κ) between two filter spaces is a homeomorphism if f is bijective and both f and f −1 are continuous maps.In this case, (X,ζ) and (Y ,κ) are called homeomorphic filter spaces.Note that the underlying preconvergence spaces (X, p ζ ) and (Y , p κ ) are also homeomorphic.
In this paper, we construct completions for a filter space without the T 2 restriction on the space.Results obtained in this paper also generalise the completion theory developed for non-T 2 Cauchy spaces obtained in [8] where the author introduced the morphisms on the category CHY as s-map.The corresponding extension of the notion of s-maps to filter spaces can be used to form a completion subcategory of FIL.Definition 2.9.A continuous map between two filter spaces f : (X,ζ) → (Y ,κ) is said to be an s-map if it satisfies the following condition: There are several examples of s-maps.Any continuous map is an s-map if the codomain of the map is a T 2 filter space.The identity map on a filter space and the embedding map ϕ for a stable completion is also an s-map.Note that it follows from the definition of s-map that composition of two s-maps is an s-map.The class of all filter spaces with the s-maps as morphisms forms a category which we denote by FIL .Since every continuous map is not necessarily an s-map, FIL is not a full subcategory of FIL.

Completion and extension theorems for filter spaces
Throughout this section, (X,ζ) denotes a filter space (not necessarily T 2 ).A completion of a filter space (X,ζ) is a pair ((Y ,κ),φ) consisting of a complete filter space (Y ,κ) and an embedding φ : (X,ζ) → (Y ,κ) such that cl pκ (φ(X)) = Y .The completion ((Y ,κ),φ) is called a weak completion if (Y ,κ) is w-complete and cl qκ (φ(X)) = Y .A completion ((Y ,κ),ϕ) of a filter space (X,ζ) is said to be a ᏼ-completion if (Y ,κ) has the property ᏼ whenever (X,ζ) has the property ᏼ.For instance, a completion which is T 2 will be called a T 2 completion.The results related to completions in standard form and extension theorems are established here.
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.
We can define an equivalence relation for non-T 2 completions of filter spaces in the same way, but it is not a categorical equivalence in the sense of Preuss [6], since in this case the map h is not necessarily a unique homeomorphism.This motivates the introduction of a weakly stable completion.Definition 3.2.A completion ((Y ,κ),ϕ) of a filter space (X,ζ) is weakly stable if whenever z ∈ Y \φ(X) and φ(Ᏺ) pκ − → z, for some Ᏺ ∈ ζ, it follows that z is the unique limit of φ(Ᏺ) in Y.
Remark 3.3.(I) Stable completion of a T 2 filter space was defined in [4].Note that any stable completion of a T 2 filter space is always weakly stable.Also, if (X,ζ) is a T 2 c-filter space, then a weakly stable completion is stable.If (X,ζ) is a Cauchy space, then every weakly stable completion is stable.
In the following result, we show that the Wyler completion of a non-T 2 filter space has a property similar to the universal property of the T 2 completions.Theorem 3.4.(( X, ζ), j) is the finest weakly stable completion of the filter space (X,ζ) in standard form.
Proof.It is clear that ζ is a filter structure on X and the inclusion map j is an embedding.To show that ( X, ζ) is complete, let and this completion is in standard form.
Next we show that it is a weakly stable completion.
. This implies that j(Ᏺ) ∨ j(Ᏼ) exists from which it follows that Ᏺ ∨ Ᏼ exists.This leads to a contradiction since Ᏺ is non- ) exists from which it follows that either j(Ᏺ) ∨ j(ᏸ 1 ) or j(Ᏺ) ∨ [ᏸ 1 ] exists.Since the latter is an impossibility, j(Ᏺ) ∨ j(ᏸ 1 ) exists.This implies Following the same type of argument as above, we can show that there is no p ζ convergent filter ) exists.This implies at least one of j(ᏸ This proves that (( X, ζ), j) is a weakly stable completion.
Let (( X,β), j) be another weakly stable completion of (X,ζ) in standard form and let is the finest weakly stable completion in standard form.
We will refer to the completion (( X, ζ), j) as the Wyler completion of (X,ζ).Obviously, the mapping j in (( X, ζ), j) is an s-map.Note that if (X,ζ) is a c-filter space, then ( X, ζ) is a c-filter space.Also, if (X,ζ) is T 2 , then (( X, ζ), j) is a T 2 completion of (X,ζ).If we identify each x ∈ X with the equivalence class [ ẋ] of all filters which are p ζ convergent to x, then the Wyler completion coincides with ((X * ,ζ * ), j) in [4].We will refer to the latter completion as the T 2 Wyler completion of (X,ζ).Proposition 3.5.Any weakly stable completion ((Y ,κ),ϕ) of a filter space (X,ζ) is equivalent to one in standard form.
Note that h is well defined and bijective, because ((Y ,κ),ϕ) is a weakly stable completion.Let ζ κ be the quotient filter structure on X.Since j = hϕ, both j and j −1 are continuous maps and since (Y ,κ) is complete and ( X,ζ κ ) is the quotient space, the latter is also com- ) is a completion of (X,ζ) and by the same argument one can also show that it is in standard form.This proves that (( X,ζ κ ), j) is a completion of (X,ζ) in standard form.It remains to show that h is a homeomorphism.Since the category FIL is a topological category, h is an injective quotient map implies h is a monomorphism which is also an extremal epimorphism.Therefore, by [6, Proposition 0.2.7], h is an isomorphism.This completes the proof of Proposition 3.5 In view of Proposition 3.5, we may therefore assume without loss of generality that all weakly stable completions of a filter space (X,ζ) are in standard form.If (X,ζ) is T 2 , then any T 2 completion is always stable.Hence, we have the following corollary.
Corollary 3.6.Any T 2 completion of a filter space is equivalent to one in standard form.
, and let j : X → X * be defined by j(x) = [ ẋ] for all x ∈ X.
Following a similar argument as in [8,Proposition 3.13], we can show that the Wyler completion is the finest completion in CFIL .But it is not the finest completion in FIL.In fact, in [4] it was shown that there is no such finest completion whenever X \ j(X) is infinite.However, the following proposition states that we can uniquely extend any s-map on a non-T 2 c-filter space to its Wyler completion.and (Y ,β) is a c-filter space, then there is a unique extension f * : ( X, ζ) → ( Ỹ , β) which is also an s-map and f * • j X = j Y • f , where j X and j Y are the corresponding embedding maps.
The unique mapping f * in Proposition 3.8 is called the s-extension of f .Remark 3.9.(I) If f : (X,ζ) → (Y ,κ) is an s-map, where (Y ,κ) is a complete c-filter space, then there exists a unique s-extension f * : ( X, ζ) → (Y ,κ) such that f = f * • j X .If (Y ,κ) is a regular filter space, then (X * ,ζ * ) also has the same extension property (by Lemma 2.7).In either case, the s-extension f * is defined by f * (x) = f (x), for each x ∈ X and f * ([Ᏺ]) = y, where f (Ᏺ) pκ − → y. (II) If (X,ζ) is a T 2 filter space, then its T 2 Wyler completion has the extension property.Recall that if the codomain of an s-map is a T 2 space, then the s-map is simply a continuous map.If f : (X,ζ) → (Y ,κ) is a continuous map, where (Y ,κ) is a complete T 2 c-filter space [4] or a complete T 3 filter space, then there exists a unique extension f * : ( X, ζ) → (Y ,κ).
Since the composition of s-maps is an s-map and the identity map is an s-map, the class of all c-filter spaces with s-maps as morphisms forms a subcategory of FIL.We denote this category by CFIL .Let CFIL * be the subcategory of CFIL consisting of the complete objects of CFIL .Let W : CFIL → CFIL * be defined for objects by W(X,ζ) = ( X, ζ) and for morphisms by W( f ) = f * .Then W is a covariant functor on CFIL and is called the Wyler completion functor.
Note that a morphism f : (X,ζ) → (Y ,κ) in the category CFIL is an epimorphism if for each z ∈ Y \ f (X), there exists a non-q ζ convergent filter Ᏺ ∈ ζ such that f (Ᏺ) qκ − → y.For example, the embedding map j in the Wyler completion is an epimorphism.Since Wyler completion is the finest completion in the category CFIL , we have the following corollary.
Corollary 3.10.CFIL * is an epireflective subcategory of CFIL .However, CFIL is not a topological category, since it is not closed under initial structures.But it should be noted that T 2 CFIL for which we could construct a completion functor [4] also fails to be a topological category.The construction of a completion functor for a subcategory of FIL which is a topological category may need further investigation.Also, constructions of regular stable completions of filter spaces and the corresponding completion categories may lead to generalisation of the existing T 3 completions.