T(α,β)-SPACES AND THE WALLMAN COMPACTIFICATION

Some new separation axioms are introduced and studied. We also deal 
with maps having an extension to a homeomorphism between the 
Wallman compactifications of their domains and ranges.

1. Introduction.Among the oldest separation axioms in topology there are three famous ones, T 0 , T 1 , and T 2 .
The T 0 -axiom is usually credited to Kolmogoroff and the T 1 -axiom to Fréchet or Riesz (and spaces satisfying the axioms are sometimes called Kolmogoroff spaces, Fréchet spaces, or Riesz spaces, accordingly).The T 2 -axiom is included in the original list of axioms for a topology given by Hausdorff [10].
We denote by Top the category of topological spaces with continuous maps as morphisms, and by Top i the full subcategory of Top whose object is T i -spaces.It is a part of the folklore of topology that Top i+1 is a reflective subcategory of Top i , for i = −1, 0, 1, with Top −1 = Top.Thus Top i is reflective in Top, for each i = 0, 1, 2 (see MacLane [17]).In other words, there is a universal T i -space for every topological space X; we denote it by T i (X).The assignment X T i (X) defines a functor T i from Top onto Top i , which is a left adjoint functor of the inclusion functor Top i Top.
The first section of this paper is devoted to the characterization of morphisms in Top rendered invertible by the functor T 0 .
Let X be a topological space.Then T i (X) is a T i -space; moreover, T i (X) may be a T i+1 -space.The second section deals with space X such that T i (X) is a T i+1 -space.Definition 1.1.Let i, j be two integers such that 0 ≤ i < j ≤ 2. A topological space X is said to be a T (i,j) -space if T i (X) is a T j -space (thus there are three new types of separation axioms; namely, T (0,1) , T (0,2) , and T (1,2) ).
More generally, one may introduce the following categorical concept.Definition 1.2.Let C be a category and F, G two (covariant) functors from C to itself.
(1) An object X of C is said to be a T (F,G) -object if G(F(X)) is isomorphic with F(X).
(2) Let P be a topological property.An object X of C is said to be a T (F,P) -object if F(X) satisfies the property P.
Recall that a topological space X is said to be a T D -space [1] if every one-point set of X is locally closed.For the separation axioms T 0 , T 1 , T 2 , T D , we classically have the following implications: (1.1) Following Definition 1.2, one may define another new separation axiom; namely, T (0,D) .Unfortunately, we have no intrinsic topological characterization of T (1,2) -spaces.However, T (0,D) -, T (0,1) -, and T (0,2) -spaces are completely characterized in Section 3.
Section 4 deals with the separation axioms T (0,S) , T (S,D) , T (S,1) , and T (S, 2) , where S is the functor of soberification from Top to itself (following Definition 1.2, a space X is said to be T (S,D) if S(X) is a T D -space).
One of the two anonymous referees of this paper has notified that the T D property is not reflective in Top; the second author has asked Professor H. P. Kunzi (University of Cape Town) for an explanation of this fact.We give this explanation as communicated by Kunzi.
In [5, Remark 4.2, page 408], Brümmer has proved that the countable product of the Sierpinski space is not a T D -space.On the other hand, according to Herrlich and Strecker [12], if a subcategory A is reflective in a category B, then for each category I, A is closed under the formation of I-limits in B (see [12,Theorem 36.13]).(Taking I a discrete category, you see that in particular A is closed under products in B.) Therefore the full subcategory Top D of Top whose objects are T D -spaces is not reflective in Top.
The importance and usefulness of compactness properties in topology and functional analysis is universally recognized.Compactifications of topological spaces have been studied extensively, as well as the associated Stone-Čech compactification.
In [11], Herrlich has stated that it is of interest to determine if the Wallman compactification may be regarded as a functor, especially as an epireflection functor, on a suitable category of spaces.This problem was solved affirmatively by Harris in [9].
Let X, Y be two T 1 -topological spaces and f : where wX is the Wallman compactification of X and ω X : X → wX is the canonical embedding of X into its Wallman compactification wX.
In Section 5, we attempt to characterize when Wallman extensions of maps are homeomorphisms.[8], where ᏻ(Y ) is the set of all open subsets of the space Y .A subset S of a topological space X is said to be strongly dense in X if S meets every nonempty locally closed subset of X [8].Thus a subset S of X is strongly dense if and only if the canonical injection S X is a quasihomeomorphism.It is well known that a continuous map q : X → Y is a quasihomeomorphism if and only if the topology of X is the inverse image by q of that of Y and the subset q(X) is strongly dense in Y [8].The notion of quasihomeomorphism is used in algebraic geometry and it has recently been shown that this notion arises naturally in the theory of some foliations associated to closed connected manifolds (see [3,4]).Now, we give some straightforward remarks about quasihomeomorphisms.

Topologically onto quasihomeomorphisms. Recall that a continuous map
Remark 2.1.(1) If f : X → Y , g : Y → Z are continuous maps and two of the three maps f , g, g • f are quasihomeomorphisms, then so is the third one.
(2) Let q : X → Y be a continuous onto map.Then the following statements are equivalent (see [7,Lemma 1.1]): (i) q is a quasihomeomorphism; (ii) q is open and for each open subset U of X, we have q −1 (q(U)) = U; (iii) q is closed and for each closed subset C of X, we have q −1 (q(C)) = C.
We introduce the concept of "topologically onto (resp., one-to-one) maps" as follows.
Definition 2.2.Let q : X → Y be a continuous map.
(1) It is said that q is topologically onto if, for each y ∈ Y , there exists x ∈ X such that {y} = {q(x)}.
(2) q is said to be topologically one-to-one if, for each y,x ∈ X such that q(x) = q(y), {y} = {x}.
(3) q is said to be topologically bijective if it is topologically onto and topologically one-to-one.
We recall the T 0 -identification of a topological space which is done by Stone [18].Let X be a topological space and define ∼ on X by x ∼ y if and only if {x} = {y}.Then ∼ is an equivalence relation on X and the resulting quotient space X/ ∼ is a T 0space.This procedure and the space it produces are referred to as the T 0 -identification of X.Clearly, T 0 (X) = X/ ∼.The canonical onto map from X onto its T 0 -identification T 0 (X) will be denoted by µ X .Of course, µ X is an onto quasihomeomorphism.
As recalled in the introduction, T 0 defines a (covariant) functor from Top to itself.If q : X → Y is a continuous map, then the diagram (1) Every one-to-one continuous map is topologically one-to-one.
(2) Every onto continuous map is topologically onto.
(3) A topologically bijective map need not be one-to-one.Let X be a topological space which is not T 0 .Of course, µ X is topologically bijective and µ X is not one-to-one.
For any functor F : C → D between two given categories, the set of all arrows in C rendered invertible by F has, sometimes, important applications.The following result characterizes morphisms in Top rendered invertible by the functor T 0 .Theorem 2.4.Let q : X → Y be a continuous map.Then the following statements are equivalent: (i) q is a topologically onto quasihomeomorphism; (ii) T 0 (q) is a homeomorphism.
Therefore, T 0 (q) is a bijective quasihomeomorphism.But one may check easily that bijective quasihomeomorphisms are homeomorphisms.
(ii)⇒(i).The equality T 0 (q) • µ X = µ Y • q forces q to be a quasihomeomorphism, by Remark 2.1 (1).It remains to prove that q is topologically onto.To do so, let y ∈ Y .Then there exists x ∈ X such that As a direct consequence of [6, Lemma 1.2, Theorem 1.3] and Theorem 2.4, one may give an external characterization of T 0 -spaces.Theorem 2.5.(1) For any topological space X, the following statements are equivalent: (i) X is a T 0 -space; (ii) for each topologically onto quasihomeomorphism q : Y → Z and each continuous map f : Y → X, there is a unique continuous map f : Z → X such that f • q = f .(2) Let q : Y → Z be a continuous map.Then the following statements are equivalent: (i) q is a topologically onto quasihomeomorphism; (ii) for each T 0 -space X and each continuous map f : Y → X, there is a unique continuous map f : Z → X such that f • q = f .Question 2.6.Give an intrinsic topological characterization of morphisms in Top rendered invertible by the functor F , where F ∈ {T 1 , T 2 }.

T (0,D
) -, T (0,1) -, and T (0,2) -spaces.We begin by recalling the T 1 -reflection.Let X be a topological space and R the intersection of all closed equivalence relations on X (an equivalence relation on X is said to be closed if its equivalence classes are closed in X).The quotient space X/R is homeomorphic to the T 1 -reflection of X.
We begin with some straightforward examples and remarks.
(4) There is a T (0,1) -space which is not T (1,2) : it suffices to consider a T 1 -space which is not T 2 .
(5) There is a T (0,1) -space which is not T (0,2) : take a T 1 -space which is not T 2 .( 6) There is a T (1,2) -space which is not T (0,2) : the Sierpinski space does the job.(7) There is a T (0,D) -space which is not T (0,1) : it suffices to consider a T D -space which is not T 1 .
(8) There is a T (0,D) -space which is not T (1,2) .The example in (4) does the job.( 9) There is a T (1,2) -space which is not T (0,D) : let Y be an infinite set.Let w ∉ Y and X = Y ∪ {w}.Equip X with the topology whose closed sets are X and all finite subsets of Y .Clearly, For each point x of a space X, we denote by γ(x) the set {x}\ y ∈ X : {y} = {x} . (3.3) With this notation, we have the following.
Theorem 3.3.Let X be a topological space.Then the following statements are equivalent: The proof needs the following lemma.
Lemma 3.4.Let q : X → Y be an onto quasihomeomorphism and C a subset of Y .Then the following statements are equivalent: Proof.Let ᏸᏯ(X), ᏸᏯ(Y ) be the sets of all locally closed subsets of X and Y , respectively.
It is well known [8] that the map ᏸᏯ(Y ) → ᏸᏯ(X) defined by F q −1 (F ) is bijective.
It is sufficient to show (ii)⇒(i).Indeed, if q −1 (C) is locally closed in X, then there is an L ∈ ᏸᏯ(Y ) such that q −1 (C) = q −1 (L); and since q is onto, we get C = L, proving that C is locally closed.
Proof of Theorem 3.3.Let µ X : X → T 0 (X) be the canonical map from X to its T 0 -reflection T 0 (X).
According to Lemma 3.4, X is a T (0,D) -space if and only if µ −1 X (µ X ({x})) is locally closed in X, for each x ∈ X.
One may check easily that µ −1 But it is well known that a subset S of a space X is locally closed if and only if S \ S is closed, completing the proof.
The following result gives a characterization of T (0,1) -spaces.Theorem 3.5.Let X be a topological space.Then the following statements are equivalent: (i) X is a T (0,1) -space; (ii) for each x, y ∈ X such that {x} ≠ {y}, there is a neighborhood of x not containing y; of all open subsets of X containing x and ᐂ(x) is the set of all neighborhoods of X.
(ii)⇒(i).Let µ X (x), µ X (y) be two distinct points in T 0 (X).Then {x} ≠ {y}.Hence there is a neighborhood U of x not containing y. Since µ X is a quasihomeomorphism, there exists a neighborhood U in T 0 (X) such that U = µ −1 X (U ) and thus U is a neighborhood of µ X (x) not containing µ X (y).Therefore T 0 (X) is a T 1 -space.
(i)⇒(iii).Let C be a closed subset of X and x ∈ X such that {x}∩C ≠ ∅.Since µ X is a quasihomeomorphism, there exists a closed subset (iii)⇒(iv).Let U be an open subset of X and, x ∈ U .Then {x} ∩ X\U = ∅, and consequently, {x} ⊆ U .
Let C be a closed subset of a space X.We say that C has a generic point if there exists x ∈ C such that C = {x}.
Recall that a topological space X is said to be quasisober [14] (resp., sober [8]) if any nonempty irreducible closed subset of X has a generic point (resp., a unique generic point).
Then there exists an open subset U of X such that, for example, x 1 ∈ U and x 2 ∉ U .Since there exists an open subset V of Y satisfying q −1 (V ) = U, we get q(x 1 ) ∈ V and q(x 2 ) ∉ V , which is impossible.It follows that q is one-to-one.
(2) Let y ∈ Y .Then {y} is a locally closed subset of Y .Hence {y} ∩ q(X) ≠ ∅, since q(X) is strongly dense in Y .Thus y ∈ q(X), proving that q is onto.
(3) One may check easily that bijective quasihomeomorphisms are homeomorphisms.( 4) By (1), q is one-to-one.Now, observe that if S is a closed subset of Y , then S is irreducible if and only if so is q −1 (S).
We prove that q is onto.To this end, let y ∈ Y .According to the above observation, q −1 ({y}) is a nonempty irreducible closed subset of X. Hence q −1 ({y}) has a generic point x.Thus we have the containments {x} ⊆ q −1 {q(x)} ⊆ q −1 {y} = {x}. (3.5) So that q −1 ({q(x)}) = q −1 ({y}).It follows from the fact that q is a quasihomeomorphism that {q(x)} = {y}.Since Y is a T 0 -space, we get q(x) = y.This proves that q is onto, and thus q is bijective.But a bijective quasihomeomorphism is a homeomorphism.
Proposition 3.8.Let q : X → Y be a quasihomeomorphism.If Y is a T (0,1) -space, then so is X.
Proof.Clearly, T 0 (q) : T 0 (X) → T 0 (Y ) is a quasihomeomorphism.Hence, since T 0 (X) is a T 0 -space and T 0 (Y ) is a T 1 -space, T 0 (q) is a homeomorphism, by Lemma 3.7.Thus T 0 (X) is a T 1 -space, proving that X is a T (0,1) -space.Example 3.9.A quasihomeomorphism q : Y → X such that Y is a T (0,1) -space but X is not.Take Y and X as in Remark 3.1 (9).Then each nonempty locally closed subset of X meets Y .Hence, the canonical embedding q : Y → X is a quasihomeomorphism.
The following proposition follows immediately from Theorem 2.4.
It is well known that a space X is a T 2 -space if and only if, for each x ∈ X, {U : U ∈ ᐂ(x)} = {x}, where ᐂ(x) is the set of all neighborhoods of x.
Before giving a characterization of T (0,2) -spaces, we need a technical lemma.
Lemma 3.11.Let q : X → Y be an onto quasihomeomorphism.Then the following properties hold. ( Proof.(1) We observe that a continuous map q : X → Y is open if and only if, for each subset B of Y , we have q −1 (B) = q −1 (B) (see [8, Chapter 0, (2.10.1)]).Now by Remark 2.1(2) an onto quasihomeomorphism is open, so that (1) follows immediately.
Theorem 3.12.Let X be a topological space.Then the following statements are equivalent: (i) X is a T (0,2) -space; (ii) for each x, y ∈ X such that {x} ≠ {y}, there are two disjoint open sets U and V in X with x ∈ U and y ∈ V ; (ii)⇒(i).Let µ X (x), µ X (y) be two distinct points in T 0 (X).Then {x} ≠ {y} and, by (ii), there are disjoint open sets U and V of X with x ∈ U and y ∈ V .Since µ X is a quasihomeomorphism, there exist two disjoint open sets U , V of T 0 (X) with µ X (x) ∈ U and µ X (y) ∈ V and such that U = µ −1 X (U ) and V = µ −1 X (V ), so that T 0 (X) is a T 2 -space.
Corollary 3.13.It is clear that the T (0,2) -property is a productive and hereditary property.
Proposition 3.14.Let q : X → Y be a quasihomeomorphism.Then the following statements are equivalent: Proof.(i)⇒(ii).Clearly, T 0 (q) : T 0 (X) → T 0 (Y ) is a quasihomeomorphism.On the other hand, since T 0 (X) is a T 2 -space, it is a sober space; and since in addition T 0 (Y ) is a T 0 -space, then T 0 (q) is a homeomorphism, by Lemma 3.7.Therefore, T 0 (Y ) is a T 2 -space.This means that Y is a T (0,2) -space.
Example 3.15.A quasihomeomorphism q : X → Y such that X is a T 2 -space and Y is not a T 2 -space.
Let Y = {0, 1, 2} equipped with the topology {∅,Y ,{1, 2}, {0}} and let X = {0, 1} be provided with a discrete topology.Then X is a T 2 -space and Y is not a T 2 -space.The canonical embedding of X into Y does the job.

T (0,S) -, T (S,D) -, T (S,1) -, and T (S,2) -spaces.
Let X be a topological space and S(X) the set of all nonempty irreducible closed subset of X [8].Let U be an open subset of X; set U = {C ∈ S(X) : U ∩ C ≠ ∅}; then the collection { U: U is an open subset of X} provides a topology on S(X) and the following properties hold [8].
(i) The map η X : X → S(X) which carries x ∈ X to η X (x) = {x} is a quasihomeomorphism.
(ii) S(X) is a sober space.
(iii) The topological space S(X) is called the soberification of X, and the assignment S(X) defines a functor from the category Top to itself.
(iv) Let q : X → Y be a continuous map, then the diagram Hong has proved that a space is quasisober if and only if its T 0 -reflection is sober.The following result makes [14, Proposition 2.2] more precise.Theorem 4.1.Let q : X → Y be a quasihomeomorphism.Then the following properties hold.
(1) If X is a T (0,S) -space, then so is Y .
(2) Suppose that Y is a T (0,S) -space.Then the following statements are equivalent: (i) X is a T (0,S) -space; (ii) q is topologically onto.(3) If q is topologically onto, then the following statements are equivalent: (i) X is a T (0,S) -space; (ii) Y is a T (0,S) -space.
Proposition 4.2 [14,Proposition 2.2].A topological space is quasisober if and only if its T 0 -reflection is sober.
Proof.The canonical map µ X : X → T 0 (X) is an onto quasihomeomorphism.Then, applying Theorem 4.1(3), the proof is complete.
(ii)⇒(i).By Proposition 4.2, T 0 (X) is a sober space.On the other hand, X is quasihomeomorphic to T 0 (X); then S(X) is homeomorphic to S(T 0 (X)), by [2, Theorem 2.2].Thus S(X) is homeomorphic to T 0 (X) so that S(X) is a T D -space, proving that X is a T (S,D) -space.Now, we give a characterization of T (S,1) -spaces.Theorem 4.6.Let X be a topological space.Then the following statements are equivalent: (i) X is a T (S,1) -space; (ii) whenever F and G are distinct nonempty irreducible closed subsets of Proof.(i)⇒(ii).Let F and G be two distinct nonempty irreducible closed subsets of X.Since S(X) is a T 1 -space and F ≠ G, there exists an open set U of S(X) such that We finish this section by characterizing T (S,2) -spaces.Theorem 4.7.Let X be a topological space.Then the following statements are equivalent: (i) X is a T (S,2) -space; (ii) whenever F and G are distinct nonempty irreducible closed subsets of X, there are disjoint open subsets U and V of X such that F ∩ U ≠ ∅ and G ∩ V ≠ ∅.
Proof.(i)⇒(ii).Let F and G be two distinct nonempty irreducible closed subsets of X.Since S(X) is a T 2 -space and F ≠ G, then there exist two disjoint open subsets U and V in S(X) such that F ∈ U and G ∈ V , so that the two open sets U and V satisfy (ii).
Proof.This follows easily from the fact that S(q) : S(X) → S(Y ) is a homeomorphism, by [2, Theorem 2.2].

The Wallman compactification.
The Wallman compactification of a T 1 -space is introduced and studied by Wallman [20] as follows.
Let X be a T 1 -space, and let wX be the collection of all closed ultrafilters on X.For each closed set D ⊆ X, define D * to be the set D * = {ᐁ ∈ wX : D ∈ ᐁ} if D ≠ ∅ and ∅ * = ∅.Then {D * : D is a closed subset of X} is a base for the closed sets of a topology on wX.Let U be an open subset of X; we define U * ⊆ wX to be the set U * = {ᐁ ∈ wX : A ⊆ U for some A in ᐁ}.The class {U * : U is an open subset of X} is a base for the open sets of the topology of wX.
The following properties are well known and may be found in any standard textbook on general topology (see, e.g., Kelley [16]).
Properties 5.1.Let X be a T 1 -space.Consider the map ω X : X → wX which takes x ∈ X to ω X (x) = {A : A is a closed subset of X and x ∈ A}.Then the following properties hold.
(1) If D is closed in X, then ω X (D) = D * .In particular, ω X (X) is dense in wX.
(2) ω X is continuous and it is an embedding of X in wX if and only if X is a T 1 -space.
(3) If A and B are closed subsets of X, then ω X (A ∩ B) = ω X (A) ∩ ω X (B).
(5) Every continuous map on X to a compact Hausdorff space K can be extended to wX.
For a T (0,1) -space X, we define W X = w(T 0 (X)) and we call it the Wallman compactification of X.The notation wX is reserved only for T 1 -spaces so that it is better to use some other notation for T (0,1) -spaces; the same for ω X : ω X is reserved for T 1 -spaces; for T (0,1) -spaces, we define w X = ω T 0 (X) • µ X .
Since µ X is an onto quasihomeomorphism, one obtains immediately that W X can be described exactly as wX is for T 1 -spaces.Properties 5.1 are also true for T (0,1)spaces.
Recall from [9] that a continuous map q : X → Y between T 1 -spaces is said to be a wextension if there is a continuous map w(q) : wX → wY such that ω Y • q = w(q) • ω X .In an analogous manner, one may define W -extensions for T (0,1) -spaces.
The following gives a class of morphisms q : X → Y which yield a W -extension W (q) : W X → W Y that is a homeomorphism.Proposition 5.14.Let X, Y be two T (0,1) -spaces and q : X → Y a W -morphism.Then q has a W -extension which is a homeomorphism.
If we denote W (q) = w(T 0 (q)), then the above diagrams indicate clearly that W (q) is a W -extension of q which is a homeomorphism.
It is well known that the Wallman compactification of a T 1 -space X is Hausdorff if and only if X is normal and in this case wX = β(X) (the Stone-Čech compactification of X) (see, e.g., Wallman [20]).
Corollary 5.15.W X is Hausdorff if and only if T 0 (X) is a normal space.In this case W X = β(T 0 (X)).Remark 5.16.If a continuous map q : X → Y has a W -extension which is a homeomorphism, then q need not be a homeomorphism.To see this it suffices to take a noncompact T 1 -space X.Of course, 1 wX is a w-extension of ω X ; however, ω X is not a homeomorphism.Definition 5.17.Let X be a T (0,1) -space and Y a subspace of X.
(1) It is said that Y is a Wallman generator (W -generator, for short) of X, if W Y is homeomorphic to W X. Theorem 5.20.Let X, Y be two T (0,1) -spaces and q : X → Y a continuous map.Then the following statements are equivalent: (i) q has a W -extension which is a homeomorphism; (ii) q(X) is an sW -generator of Y and the topology of X is the inverse image of that of Y by q.
Proof.(i)⇒(ii).(a) The topology of X is the inverse image of that of Y by q.Let C be a closed subset of X.Since W (q) is a homeomorphism, W (q)(C * ) = K is a closed subset of W Y .
(b) q(X) is an sW -generator of Y .According to (a), the induced map q 1 : X → q(X) by q is a W -morphism.Hence q 1 has a W -extension W (q 1 ) which is a homeomorphism, by Proposition 5.14.Thus the diagrams X w X q 1 q(X) w q(X) W X W (q 1 ) W q(X) Let j : q(X) Y be the canonical embedding.Clearly, the diagram q(X) w q(X) j Y w Y wq(X) W (q)•(W (q 1 )) −1 W Y (5.4) commutes.Therefore, j has W (q)• (W (q 1 )) −1 as a W -extension which is a homeomorphism.This means that q(X) is an sW -generator of Y .
(ii)⇒(i).Under the assumptions of (ii), the induced map q 1 : X → q(X) by q is a Wmorphism.Thus, according to Proposition 5.14, q 1 has a W -extension W (q 1 ) which is a homeomorphism.On the other hand, the canonical embedding j : q(X) Y has a W -extension which is a homeomorphism, by Proposition 5.14.It follows that the two diagrams W q(X) W (j) W Y (5.5) commute.Therefore, W (j)•W (q 1 ) is a W -extension of q : X → Y which is a homeomorphism.
Question 5.21.Is it possible to replace the word "sW -generator" in Theorem 5.20 by "W -generator"?

( 1 )Proposition 5 . 3 .
For each open subset U of X, we have w X (U) ⊆ U * .(2) For each closed subset D of X, we have w X (D) ⊆ D * .(3) Let U be open and D closed in a T (0,1) -space.Then U ∩ D ≠ ∅ if and only if U * ∩ D * ≠ ∅.Now we give some new observations about Wallman compactifications.Let X be a T (0,1) -space and U an open or closed subset