Topology and its Applications 161 (2014) 364–376

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Topology and its Applications

www.elsevier.com/locate/topol

Axioms of separation in semitopological groups and relatedfunctors

M. Tkachenko 1

Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186,Col. Vicentina, C.P. 09340, Del. Iztapalapa, Mexico, D.F., Mexico

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 August 2013Accepted 25 October 2013

MSC:primary 54H99, 54D10secondary 54B30, 54C10

Keywords:Ti-reflectionCovariant functorSemitopological groupOpen homomorphismd-open mapping

We prove that for every semitopological group G and every i ∈ {0, 1, 2, 3, 3.5}, thereexists a continuous homomorphism ϕG,i : G → H onto a Ti (resp., Ti &T1 for i � 3)semitopological group H such that for every continuous mapping f : G → X to a Ti-(resp., Ti &T1- for i � 3) space X, one can find a continuous mapping h : H → Xsatisfying f = h ◦ ϕG,i. In other words, the semitopological group H = Ti(G)is a Ti-reflection of G. It turns out that all Ti-reflections of G are topologicallyisomorphic. These facts establish the existence of the covariant functors Ti for i =0, 1, 2, 3, 3.5, as well as the functors Reg and Tych in the category of semitopologicalgroups and their continuous homomorphisms.We also show that the canonical homomorphisms ϕG,i of G onto Ti(G) are openfor i = 0, 1, 2 and provide an internal description of the groups T0(G) and T1(G) byfinding the exact form of the kernels of ϕG,0 and ϕG,1. It is also established thatthe functors Reg and Ti ◦ T3, for i = 0, 1, 2 are naturally equivalent.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

It is difficult to exaggerate the role of universal constructions in Topology. The absolute of a space,Čech–Stone compactification of a Tychonoff space, or the Raıkov completion of a topological group, theHartman–Mycielski embedding of a space (topological group) into a connected and locally connected space(topological group), free topological groups, etc., are just a few examples of these constructions whichhave the common name of functor in the category theory. Categorical methods in Topology proved tobe a powerful tool of investigation [5,7]. The importance of functors resides, apart from the vast areas oftheir applicability, in the fact that functors act both on the objects of an appropriate category and on themorphisms of the same category. The results of this action can belong to another category, just like in thecase of the functor of taking the free topological group over a given Tychonoff space.

E-mail address: mich@xanum.uam.mx.1 The author was partially supported by the Mexican National Council for Sciences and Technology (CONACyT), Grant Number

CB-2012-01 178103.

0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.topol.2013.10.037

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 365

In this article we consider the category of semitopological groups, i.e., groups with a topology in which theleft and right translations are continuous; the morphisms of this category are continuous homomorphisms.No separation restrictions on the topology of the groups are imposed unless we mention them explicitly. Wealso work in the narrower category of paratopological groups in which multiplication is jointly continuous.Basic information on semitopological and paratopological groups can be found in [2], while the survey article[14] presents recent advances in this area.

Our aim is to define the so-called Ti-reflection Ti(G) of an arbitrary semitopological group G for i =0, 1, 2, 3, 3.5, as well as the regular reflection Reg(G) and Tychonoff reflection Tych(G) of G. All these objectsare again semitopological groups and their names indicate that Ti(G) satisfies the Ti separation axiom fori = 0, 1, 2, 3, 3.5, while Reg(G) and Tych(G) are, respectively, regular and Tychonoff. Each of these groupsis a continuous homomorphic image of G, and the respective homomorphisms ϕG,i : G → Ti(G) are openfor i = 0, 1, 2 (see Proposition 2.5). To describe the main property of the semitopological groups Ti(G),Reg(G), and Tych(G), we need two simple definitions.

Let us say that a class C of spaces is a PS-class if it contains arbitrary products of its elements, ishereditary with respect to taking subspaces, and contains a one-point space. It is clear that the class Ti

of Ti-spaces is a PS-class for each i ∈ {0, 1, 2, 3, 3.5} and so are the classes of regular (i.e., T1 &T3-) andTychonoff (i.e., T1 &T3.5-) spaces.

Definition 1.1. Let C be a PS-class of spaces and ϕCG : G → H a continuous surjective homomorphism of

semitopological groups. The pair (H,ϕCG) is called a C-reflection of G if H ∈ C and for every continuous

mapping f : G → X to a space X ∈ C, there exists a continuous mapping h : H → X such that f = h ◦ϕCG.

GϕC

G

f

H

h

X

Abusing terminology we will usually refer to H = ϕCG(G) as a C-reflection of G. It turns out that every

two C-reflections of a given semitopological group G are naturally equivalent (see Proposition 2.2). Our mainresult, Theorem 2.3, states that for every semitopological group G and every PS-class C of spaces, thereexists the C-reflection of G. Since the classes of spaces satisfying the usual axioms of separation (or theircombinations) are PS-classes, we deduce in Proposition 2.5, for a semitopological group G, the existenceof the Ti-reflection Ti(G) for i = 0, 1, 2, 3, 3.5, as well as the regular and Tychonoff reflections Reg(G) andTych(G), respectively.

In Corollary 2.8 we establish that ‘Ti’ for i = 0, 1, 2, 3, 3.5, ‘Reg’, and ‘Tych’ are covariant functors in thecategory of semitopological groups and their continuous homomorphisms, while Corollary 2.9 states thatthe functor ‘Ti’ preserves open and d-open homomorphisms for i = 0, 1, 2.

It turns out that for every semitopological group G, the reflections just mentioned are connected viacontinuous surjective homomorphisms

GϕG,0

ϕG,3

T0(G)ψ0

T1(G)ψ1

T2(G)ψ2 Reg(G)

ψ3 Tych(G)

T3(G)λG

of which ϕG,0, ψ0, and ψ1 are open, while ψ2, ψ3, and ϕG,3 are one-to-one (see Propositions 3.5, 3.7, andLemma 3.9).

366 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

Since Definition 1.1 introducing the notion of a C-reflection has a clear categorical nature, some of theresults in Section 2 can also be deduced using the methods of the category theory. We decided to presenta topological treatment of the subject here as all applications of our results given in [15,16] lie on thetopological side of topological algebra.

In the forthcoming article [15] we focus attention on Ti-reflections of paratopological groups. We give, inparticular, a ‘constructive’ description of the paratopological group T2(G), presenting the exact form of thekernel of the open homomorphism ϕG,2 : G → T2(G) and establishing several topological properties of thehomomorphisms ϕG,i for i = 0, 1, 2, 3 and ϕG,r : G → Reg(G).

Finally, we have to mention that the idea of the study of paratopological groups via topological groups wasconsidered by T. Banakh and O. Ravsky in [3,4]. They defined the topological group reflection and topologicalgroup coreflection of a given paratopological group and applied these concepts to establish several topologicalproperties of precompact and saturated paratopological groups. In [13], the author continued the study oftopological group reflection applying it to the categories of paratopological and semitopological groups.

2. Ti-reflections in the category of semitopological groups

We show first that all C-reflections of a semitopological group are equivalent, for each PS-class C. Thisrequires a lemma.

Lemma 2.1. Let G be a semitopological group, C a PS-class of spaces, and (H,ϕCG) a C-reflection of G.

If p : G → K is a continuous homomorphism of semitopological groups and K ∈ C, then there exists acontinuous homomorphism q : H → K satisfying p = q ◦ ϕC

G.

Proof. By Definition 1.1, there exists a continuous mapping q of H to K satisfying p = q ◦ ϕCG. We claim

that q is a homomorphism. Indeed, the latter equality implies that q(eH) = eK , where eH and eK are theneutral elements of H and K, respectively. Take arbitrary elements x, y ∈ H and pick x, y ∈ G satisfyingϕCG(x) = x and ϕC

G(y) = y. Since ϕCG and p are homomorphisms, we have:

q(x · y) = q(ϕCG(x) · ϕC

G(y))

= q(ϕCG(x · y)

)= p(x · y) = p(x) · p(y)

= q(ϕCG(x)

)· q(ϕCG(x)

)= q(x) · q(y).

This implies our claim and completes the proof. �Proposition 2.2. Let G be an arbitrary semitopological group and C a PS-class of spaces. If (H1, ϕ

C1 ) and

(H2, ϕC2 ) are C-reflections of G, then there exists a topological isomorphism q : H1 → H2 such that ϕC

2 =q ◦ ϕC

1 .

GϕC

1

ϕC2

H1q

H2

Proof. It follows from Lemma 2.1 that there exist continuous homomorphisms q1 : H1 → H2 and q2 : H2 →H1 such that ϕC

2 = q1 ◦ ϕC1 and ϕC

1 = q2 ◦ ϕC2 . These equalities imply that q1 and q2 are surjective and that

ϕC1 = q2 ◦ q1 ◦ϕC

1 . Hence q2 ◦ q1 is the identity mapping of H1 onto itself, and a similar argument shows thatq1 ◦ q2 is the identity mapping of H2. This yields that q1 and q2 are topological isomorphisms, so q = q1 isas required. �

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 367

The next existence theorem is the main result of this section.

Theorem 2.3. Let G be a semitopological group and C a PS-class of spaces. Then there exists the C-reflection(H,ϕC

G) of G.

Proof. Given a continuous mapping f of G to a space X ∈ C, we define a subgroup Kf of G by letting

Kf ={x ∈ G: f(axb) = f(ab) for all a, b ∈ G

}.

We leave to the reader a verification of the fact that Kf is invariant in G (the corresponding argument canbe found in [8, Example 37]). It also follows from the definition of Kf that f is constant on every coset ofKf in G.

For a, b ∈ G, denote by fa,b the mapping of G to X defined by fa,b(x) = f(axb), for each x ∈ G. Since G

is a semitopological group, the mappings fa,b are continuous. Let ϕf be the diagonal product of the family{fa,b: a, b ∈ G}. Then ϕf is a continuous mapping of G to the Cartesian product XG×G, and we denote byGf the image of G under the mapping ϕf considered with the topology inherited from XG×G. Then Gf ∈ C

since C is a PS-class.Let us define multiplication ∗ in Gf by letting ϕf (x) ∗ ϕf (y) = ϕf (xy), for all x, y ∈ G. To show that

this definition is correct, suppose that elements x, x′, y, y′ ∈ G satisfy ϕf (x′) = ϕf (x) and ϕf (y′) = ϕf (y).We have to verify that ϕf (x′y′) = ϕf (xy) or, equivalently, fa,b(x′y′) = fa,b(xy) for all a, b ∈ G. Takearbitrary elements a, b ∈ G. It follows from the definition of the mapping fa,b and our choice of x, x′, y, y′

that fa,b(xy) = f(axyb) = f(ax′yb) = f(ax′y′b) = fa,b(x′y′), as required.It is clear that the binary operation ∗ is associative and that ϕf (e) is the neutral element of Gf , where e

is the identity of G. Hence the inverse of ϕf (x) in Gf is ϕf (x−1), for each x ∈ Gf . We conclude that (Gf , ∗)is algebraically a group and ϕf is a homomorphism of G onto (Gf , ∗). An easy verification shows that thekernel of ϕf is the subgroup Kf of G, i.e., the group (Gf , ∗) is algebraically isomorphic to G/Kf .

We claim that multiplication in Gf is separately continuous with respect to the topology τf that Gf

inherits from the product space XG×G, i.e., (Gf , τf ) is a semitopological group. Indeed, take elementsg, x0 ∈ G and a neighborhood U of the element ϕf (g) ∗ ϕf (x0) = ϕf (gx0) in XG×G. We can assume thatU is a canonical open set in XG×G. Then there exist elements a1, b1 . . . , an, bn ∈ G and respective openneighborhoods V1, . . . , Vn of the points fa1,b1(gx0), . . . , fan,bn(gx0) in X such that U =

⋂ni=1 p

−1ai,bi

(Vi). Herepa,b is the projection of XG×G onto the factor X(a,b), where a, b ∈ G. Let O =

⋂ni=1 p

−1aig,bi

(Vi). ThenO′ = O ∩ Gf is an open neighborhood of ϕf (x0) in Gf . Suppose that ϕf (x) ∈ O′ for some x ∈ G. Thenfaig,bi(x) ∈ Vi for each i � n or, equivalently, fai,bi(gx) ∈ Vi for each i � n. This implies that ϕf (gx) ∈ U ,whence it follows that ϕf (g)∗O′ ⊆ U . Therefore, the left translation by ϕf (g) is continuous on Gf . A similarargument implies the continuity of the right translation by ϕf (g). This proves our claim.

It is clear that f = fe,e. Hence we have the equality f = pe,e ◦ ϕf or, equivalently, f = f ◦ ϕf , wheref = pe,e � Gf is a continuous mapping of Gf to X. This equality will be used later. Notice that if f(G) �= X,we can replace X with f(G), which results in the same semitopological group (Gf , τf ). Hence we will consideronly continuous mappings of G onto spaces in C.

Denote by F the class of all continuous mappings of G onto spaces in C. If f : G → X and h : G → Y arein F, we say that f and h are equivalent or, in symbols, f ∼ h, if there exists a homeomorphism ψ : X → Y

such that h = ψ ◦ f . Let κ = |G|. Since every continuous image of G can be identified as a set with a subsetof κ, the family of equivalence classes E = F/∼ is a set. We choose a representative from each equivalenceclass in E and denote the resulting set by E.

Let ϕCG be the diagonal product of the family {ϕf : f ∈ E}. Then ϕC

G is a continuous homomorphismof G to the product Π =

∏f∈E Gf of semitopological groups. Notice that Π ∈ C. Denote by H the image

ϕCG(G) considered as a subgroup and a subspace of Π. It is clear that the semitopological group H is in C,

and we claim that the pair (H,ϕCG) is the C-reflection of G.

368 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

Suppose that h : G → Y is a continuous mapping of G to a space Y ∈ C. By our definition of E, thereexists f ∈ E, say, f : G → X such that f ∼ h. Denote by ψ a homeomorphism of X onto Y such thath = ψ ◦f . Let πf be the projection of Π onto the factor Gf . Our definition of ϕC

G implies that ϕf = πf ◦ϕCG.

We have already mentioned above that f = f ◦ ϕf , so the following diagram commutes.

Ππf

Gf

f

Gf

h

ϕCG

ϕf

Xψ

Y

Denote by h the restriction to H of the continuous mapping ψ ◦ f ◦ πf . Then h = h ◦ ϕCG, which completes

the proof of the theorem. �In what follows we denote the continuous homomorphic image H of G defined in Theorem 2.3 by C(G).

Corollary 2.4. Let C and D be PS-classes of spaces, G be a semitopological group, and ϕCG : G → C(G),

ϕDG : G → D(G) the canonical continuous homomorphisms. If C ⊆ D, then there exists a continuous

surjective homomorphism p : D(G) → C(G) satisfying ϕCG = p ◦ ϕD

G . In particular, kerϕDG ⊆ kerϕC

G.

Proof. Since C(G) ∈ C ⊆ D, it follows from Lemma 2.1 that there exists a continuous homomorphismp : D(G) → C(G) satisfying the required equality. Since the homomorphisms ϕC

G and ϕDG are surjective, so

is p. The inclusion kerϕDG ⊆ kerϕC

G is evident. �In the following result we establish the existence of the Ti-reflection, for i ∈ {0, 1, 2, 3}, as well as the

regular and Tychonoff reflections in the class of semitopological groups.

Proposition 2.5. For every semitopological group G and every i ∈ {0, 1, 2, 3, 3.5}, there exists the Ti-reflection(Ti(G), ϕG,i) of G. Similarly, there exist the regular reflection (Reg(G), ϕG,r) and the Tychonoff reflection(Tych(G), ϕG,t) of G. The homomorphism ϕG,i is open for each i = 0, 1, 2.

Proof. Since the Ti-spaces form a PS-class, for each i ∈ {0, 1, 2, 3, 3.5}, and the same is valid for the classesof regular and Tychonoff spaces, all conclusions of the proposition, except for the last one, follow fromTheorem 2.3. Let us verify that the homomorphism ϕG,i : G → Ti(G) is open for i = 0, 1, 2.

First we note that if f : X → Y is a continuous one-to-one mapping and Y is a Ti-space for some i = 0, 1, 2,then so is X. Fix i ∈ {0, 1, 2} and consider a semitopological group G with the corresponding surjectivehomomorphism ϕG,i : G → Ti(G). Let N be the kernel of ϕG,i. Denote by π the quotient homomorphism ofG onto G/N . Since ϕG,i is continuous, there exists a continuous isomorphism j : G/N → Ti(G) satisfyingϕG,i = j ◦ π. Then G/N is a Ti-space. Therefore, by the main property of Ti(G), there exists a continuousmapping h : Ti(G) → G/N satisfying π = h ◦ ϕG,i. Hence ϕG,i = j ◦ h ◦ ϕG,i. It follows that h = j−1 and j

is a homeomorphism. Since the homomorphism π is open, so is ϕG,i = j ◦ π. �Given a semitopological group G and i ∈ {0, 1, 2, 3, 3.5}, we denote by Ti(G) the Ti-reflection of G defined

in Proposition 2.5, thus omitting the corresponding homomorphism ϕG,i. Similarly, Reg(G) and Tych(G)stand respectively for the regular and Tychonoff reflections of G.

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 369

The last assertion of Proposition 2.5 implies that for every semitopological group G, the groups T0(G),T1(G), and T2(G) are quotients of G. This is no more valid for T3(G) and Reg(G), even if G is a Hausdorffparatopological group (see [15, Corollary 2.9]).

It is worth noting that we are not aware of any ‘internal’ description of T2(G) and Reg(G), for anarbitrary semitopological group G, similar to those presented in [15, Theorems 2.1 and 2.6] in the case ofparatopological groups.

Our next step is to define the Ti-reflections of continuous homomorphisms of semitopological groups.Again, we do it in a more general setting involving PS-classes.

Proposition 2.6. Let f : G → H be a continuous homomorphism of semitopological groups and C be aPS-class of spaces. Then there exists a unique continuous homomorphism C(f) : C(G) → C(H) such thatthe diagram below commutes. If f is surjective, so is C(f).

Gf

ϕCG

H

ϕCH

C(G)C(f)

C(H)

Proof. Apply Lemma 2.1 to the continuous homomorphism p = ϕCH ◦ f of G to C(H) and find a continuous

homomorphism q : C(G) → C(H) satisfying q ◦ ϕCG = p. Clearly, C(f) = q is as required. Suppose that

ν : C(G) → C(H) is a homomorphism satisfying the commutativity condition ν ◦ ϕCG = ϕC

H ◦ f . Sinceq ◦ ϕC

G = ϕCH ◦ f and ϕC

G is surjective, we conclude that ν = q = C(f), i.e., the homomorphism C(f) isunique.

If the homomorphism f is surjective, then the surjectivity of ϕCG and ϕC

H implies the same property ofC(f). �Proposition 2.7. If f : G → H and g : H → K are continuous homomorphisms, then C(g ◦ f) = C(g) ◦C(f),i.e., the functor ‘C’ appeared in Proposition 2.6 is covariant.

Proof. According to Proposition 2.6, the following diagram commutes.

Gf

ϕCG

H

ϕCH

gK

ϕCK

C(G)C(f)

C(H)C(g)

C(K)

Hence, with h = g ◦ f , we have the equality ϕCK ◦ h = (C(g) ◦ C(f)) ◦ ϕC

G. Also the homomorphism C(h) ofC(G) to C(K) satisfies the equality ϕC

K ◦ h = C(h) ◦ ϕCG, so the uniqueness part of Proposition 2.6 implies

that C(h) = C(g) ◦ C(f). �Combining Propositions 2.6 and 2.7, we obtain the next fact:

Corollary 2.8. For every i = 0, 1, 2, 3, 3.5, there exists a covariant functor Ti in the category of semitopo-logical groups corresponding to the class C of Ti-spaces such that Ti(G) is the Ti-reflection of G, for eachsemitopological group G. Similarly, there exist the covariant functors Reg and Tych in the same categorycorresponding to the respective classes of regular and Tychonoff spaces.

370 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

In what follows we focus almost completely on the study of the functors Ti for i = 0, 1, 2, 3 and Reg, aswell as the relations between them.

Let us recall that, according to [12], a continuous mapping f : X → Y is called d-open if f(U) ⊆ Int f(U),for every open subset U of X or, equivalently, f−1(V ) = f−1(V ) for every open subset V of Y . It is easyto see that every open continuous mapping is d-open and that the composition of two d-open mappings isalso d-open.

It turns out that the functors Ti for i = 0, 1, 2 preserve open and d-open homomorphisms:

Corollary 2.9. Let f : G → H be a continuous homomorphism of semitopological groups. If f is open ord-open, then so is the homomorphism Ti(f) : Ti(G) → Ti(H) for i = 0, 1, 2.

Proof. Proposition 2.6 implies that the following diagram commutes, where g = Ti(f) and i ∈ {0, 1, 2}.

Gf

ϕG,i

H

ϕH,i

Ti(G)g

Ti(H)

The homomorphism ϕH,i is open by Proposition 2.5. Hence the composition ψ = ϕH,i ◦f is also open. Sinceψ = g ◦ ϕG,i and ϕG,i is continuous and surjective, we conclude that the homomorphism g is open as well.

Similarly, if f is d-open, then the composition ψ = ϕH,i ◦ f is d-open since ϕH,i is open. Hence the sameargument yields that g is d-open. �

The next fact complements Proposition 2.6; it will be frequently used in the rest of the article.

Corollary 2.10. Let C be a PS-class of spaces and suppose that p : G → H and q : H → C(G) are continuoussurjective homomorphisms of semitopological groups such that q ◦ p = ϕC

G. Then (C(G), q) is the C-reflectionof H and, in particular, C(H) ∼= C(G).

Proof. Let g : H → X be a continuous mapping of H to a space X ∈ C. Then f = g ◦ p is a continuousmapping of G to X, so there exists a continuous mapping h : C(G) → X satisfying f = h ◦ ϕC

G.

GϕC

G

p

C(G)

h

Hg

q

X

This implies the equality g = h ◦ q, and so (C(G), q) is the C-reflection of H. �3. Further properties of the functors Ti, Reg, and Tych

In the following theorem we obtain the T0-reflection of an arbitrary semitopological group G as thequotient group of G with respect to a well-defined subgroup. The family of open neighborhoods of theneutral element eG in G will be denoted by N(eG).

Theorem 3.1. Let G be a semitopological group, P =⋂N(eG), and K = P ∩ P−1. Then K is an invariant

subgroup of G and (G/K, π) is the T0-reflection of G, where π : G → G/K is the quotient homomorphism.Further, the homomorphism π satisfies U = π−1π(U), for each open set U ⊆ G.

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 371

Proof. It is clear from the definition of K that eG ∈ K and K−1 = K. Since internal automorphisms of Gare homeomorphisms of G onto itself, we see that K is invariant in G. Let us verify that K is a subgroupof G.

Take an arbitrary element x ∈ K. First we verify that xP ⊆ P and Px ⊆ P . Given an element U ∈ N(eG),we have that x ∈ K ⊆ U , so eG ∈ x−1U and x−1U ∈ N(eG). Hence our definition of P implies thatP ⊆

⋂U∈N(eG) x

−1U = x−1 ⋂N(eG) = x−1P or, equivalently, xP ⊆ P . A similar argument shows thatPx ⊆ P .

Again, let x be an element of K. Then x−1 ∈ K since K is symmetric. Hence Px−1 ⊆ P or, equivalently,xP−1 ⊆ P−1. We also know that xP ⊆ P , whence it follows that x(P ∩ P−1) ⊆ P ∩ P−1, i.e., xK ⊆ K.Hence K is a subgroup of G.

We claim that KU = U , for every open subset U of G. Indeed, suppose for a contradiction that thereare an element x ∈ K and a non-empty open set U in G such that xU \ U �= ∅. Then there exists a ∈ U

such that xa /∈ U . Hence V = Ua−1 is in N(eG) and x ∈ K \ V �= ∅, thus contradicting our definition of K.Let π : G → G/K be the quotient homomorphism. It follows from the above claim that U = π−1π(U),

for each open set U ⊆ G. It is easy to see that the quotient group G/K is a T0-space. Indeed, if an elementy ∈ G/K is distinct from the neutral element of the quotient group, take x ∈ G such that π(x) = y. Thenx /∈ K, so there exists U ∈ N(eG) with x /∈ U ∩U−1. If x /∈ U , then the equality U = π−1π(U) implies thaty /∈ π(U). If x /∈ U−1, then x−1 /∈ U and, as above, y−1 /∈ π(U). This implies that y /∈ π(U−1). In eithercase, y /∈ V ∩ V −1, where V = π(U) is an open neighborhood of the neutral element e in G/K.

Suppose that y1 and y2 are distinct elements of G/K. Then y1y−12 �= e, so there exists an open neighbor-

hood V of e in G/K such that y1y−12 /∈ V ∩ V −1. Then either y1 /∈ V y2 or y2 /∈ V y1, so G/K is a T0-space,

as claimed.Let ϕG,0 : G → T0(G) be the canonical quotient homomorphism. Since the T0-spaces constitute a

PS-class, Lemma 2.1 implies that there exists a continuous homomorphism q : T0(G) → G/K such thatq ◦ ϕG,0 = π. Hence the kernel N of the homomorphism ϕG,0 is contained in K. Let us show that K ⊆ N .Take an element x ∈ G\N . Then ϕG,0(x) is distinct from the identity e′ of T0(G). Since T0(G) is a T0-space,there exists an open neighborhood U of eG in G such that either ϕG,0(x) /∈ ϕG,0(U) or e′ /∈ ϕG,0(xU). Thenx /∈ U in the first case, and x−1 /∈ U in the second one. In either case, x /∈ U ∩ U−1, so x /∈ K. Therefore,K ⊆ N . We have thus proved that K = N .

Since the continuous open homomorphisms π and ϕG,0 have the same domain and the same kernels, theequality q ◦ϕG,0 = π implies that q is a topological isomorphisms between the groups T0(G) and G/K. Thiscompletes the proof. �

In what follows the canonical homomorphism of G onto T0(G) is always denoted by ϕG,0.In the next corollary we do not impose any separation restrictions on a semitopological group G, while

it is customary to require that the domain of a perfect mapping be a Hausdorff space (see e.g. [6]).

Corollary 3.2. The canonical homomorphism ϕG,0 : G → T0(G) is a perfect mapping, for every semitopo-logical group G. Hence the group G is either Lindelöf, or σ-compact, or countably compact iff so is T0(G).

Proof. Since, by Theorem 3.1, the equality U = ϕ−1G,0ϕG,0(U) holds for every open set U ⊆ G, the same

equality is valid for every closed subset F of G in place of U . Hence the quotient mapping ϕG,0 is closed.Notice that the kernel N of ϕG,0 is a compact subset of G since N ⊆ U or N ∩ U = ∅, for every open setU ⊆ G. Since ϕ−1

G,0(y) is homeomorphic to N , for every y ∈ T0(G), all fibers of ϕG,0 are compact. Hence themapping ϕG,0 is perfect. �Corollary 3.3. A paratopological group G is a topological group iff so is T0(G).

372 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

Proof. If G is topological group, then T0(G) is also a topological group as a quotient of G. Conversely,suppose that T0(G) is a topological group. Since the homomorphism ϕG,0 is perfect, G is a topologicalgroup by [1, Theorem 1.6]. Alternatively, the kernel N of the quotient homomorphism ϕG,0 carries theanti-discrete topology, so N is a topological group. Hence so is G by [9, Lemma 4]. �

In the next theorem we describe the T1-reflection of a semitopological group. A corresponding ‘internal’description of the Hausdorff reflection of a semitopological group is still unknown (see Problem 4.1).

Theorem 3.4. Let G be a semitopological group and K be the smallest closed subgroup of G. Then K isinvariant in G, the quotient mapping πG : G → G/K is a continuous open homomorphism, and G/K ∼=T1(G).

Proof. Since xKx−1 is a closed subgroup of G, for each x ∈ G, it follows from the minimality of K thatK ⊆ xKx−1. Hence K is invariant in G. So the quotient mapping πG : G → G/K is a continuous openhomomorphism.

Let ϕG,1 : G → T1(G) be the canonical quotient homomorphism. Since the T1-spaces constitute aPS-class, Lemma 2.1 implies that there exists a continuous homomorphism q : T1(G) → G/K such thatq ◦ϕG,1 = πG. Hence the kernel N of the homomorphism ϕG,1 is contained in K. Since T1(G) is a T1-space,N is closed in G. Therefore, the definition of K implies that K = N . The homomorphism πG is open,surjective, and continuous, and so is ϕG,1 by Proposition 2.5. Hence the equality q ◦ϕG,1 = πG implies thatq is a topological isomorphism. This completes the proof. �

It is not surprising, after all, that the functors Ti’s, Reg, and Tych are related to each other. This relationis described in the following proposition. As in Proposition 2.5, we denote the canonical homomorphism ofG onto Tych(G) by ϕG,t.

Proposition 3.5. Let f : G → H be a continuous homomorphism of semitopological groups. Then thereexist continuous surjective homomorphisms ψG

i and ψHi for i ∈ {0, 1, 2, 3} such that the following diagram

commutes.

GϕG,0

f

T0(G)ψG

0

T0(f)

T1(G)ψG

1

T1(f)

T2(G)ψG

2

T2(f)

Reg(G)

Reg(f)

ψG3 Tych(G)

Tych(f)

HϕH,0

T0(H)ψH

0T1(H)

ψH1

T2(H)ψH

2 Reg(H)ψH

3 Tych(H)

Furthermore, ψG0 ◦ϕG,0 = ϕG,1, ψG

1 ◦ϕG,1 = ϕG,2, ψG2 ◦ϕG,2 = ϕG,r, ψG

3 ◦ϕG,r = ϕG,t, and similar equalitiesare valid with H in place of G.

Proof. The existence of the continuous surjective homomorphisms ψGi and ψH

i for i = 0, 1, 2, 3 satisfying theequalities in the last part of the proposition follows directly from Corollary 2.4. Therefore, the commutativityof the above diagram is immediate from Proposition 2.6. �

The diagram in Proposition 3.5 does not contain information on the group T3(G). Proposition 3.7 belowfixes this detail. First we need a lemma.

Lemma 3.6. Let G be a semitopological group with identity e. If G satisfies the T3 separation axiom, thenthe closure of the singleton {e} in G, say, N is an invariant subgroup of G and (G/N, π) is the regularreflection of G, where π : G → G/N is the quotient homomorphism. Further, T0(G) ∼= G/N ∼= Reg(G).

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 373

Proof. It is well known that the closure of a subsemigroup of a semitopological group is again a subsemi-group. Since {e} is a subsemigroup of G, so is N . Let us show that N is symmetric.

Let N(e) be the family of open neighborhoods of e in G. Since G is a T3-space, x ∈ {y} implies thaty ∈ {x} for all x, y ∈ G. Hence, if x ∈ {e}, then e ∈ {x}, i.e., every U ∈ N(e) contains the element x.In turn, this implies that e ∈ x−1U , for each U ∈ N(e). The latter means that x−1 ∈ {e}. We have thusproved that N−1 ⊂ N . Taking the inverses in this inclusion, we obtain that N ⊂ N−1, whence the equalityN−1 = N follows.

As in Theorem 3.1, let P =⋂

N(e). We claim that N = P ∩ P−1. Indeed, x ∈ N iff x ∈ {e} iff x ∈ U−1

for each U ∈ N(e) iff x ∈ P−1. This implies the equality N = P−1. Since N is symmetric, we conclude thatN = P ∩ P−1. Hence Theorem 3.1 implies that (G/N, π) is the T0-reflection of G, where π : G → G/N isthe quotient homomorphism. In particular, T0(G) ∼= G/N .

Let us show that the space G/N is regular. The quotient group G/N is a T1-space since N is closed in G.It is easy to see that G/N is also a T3-space. Indeed, if O is a neighborhood of the neutral element in G/N ,take an open neighborhood U of e in G such that π(U) ⊆ O. Since G is a T3-space, there exists an openneighborhood V of e in G such that V ⊆ U . By Theorem 3.1, the sets U and V satisfy U = π−1π(U) andV = π−1π(V ). Since the mapping π is open, we also have the equality V = π−1π(V ). Therefore, the openneighborhood π(V ) of the identity in G/N satisfies π(V ) ⊆ π(U). This proves that G/N is a T3-space and,hence, is regular.

It remains to show that (G/N, π) is the regular reflection of G. Since the space Reg(G) is regular, thekernel of the continuous homomorphism ϕG,r is a closed subgroup of G. Hence the definition of N implies theinclusion N ⊆ kerϕG,r. Therefore, there exists a homomorphism ϕ : G/N → Reg(G) satisfying ϕG,r = ϕ◦π.Since ϕG,r is continuous and π is open, the homomorphism ϕ is continuous.

GϕG,r

π

Reg(G)

G/N

ϕ

Clearly, all homomorphisms in the above diagram are surjective. Since ϕG,r = ϕ ◦ π, Corollary 2.10 impliesthat (Reg(G), ϕ) is the regular reflection of G/N . But G/N is regular, so ϕ is a topological isomorphism.Hence (G/N, π) is the regular reflection of G, as claimed. �Proposition 3.7. Let f : G → H be a continuous homomorphism of semitopological groups. Then thereexist continuous open surjective homomorphisms λG and λH satisfying the equalities λG ◦ ϕG,3 = ϕG,r andλH ◦ ϕH,3 = ϕH,r and such that the following diagram commutes.

GϕG,3

f

T3(G)λG

T3(f)

Reg(G)

Reg(f)

HϕH,3

T3(H)λH Reg(H)

Furthermore, Reg(G) ∼= T0(T3(G)) and Reg(H) ∼= T0(T3(H)).

Proof. It is clear that regularity implies the T3 separation axiom. Hence we can apply Corollary 2.4 tofind continuous surjective homomorphisms λG : T3(G) → Reg(G) and λH : T3(H) → Reg(H) satisfyingλG ◦ ϕG,3 = ϕG,r and λH ◦ ϕH,3 = ϕH,r.

374 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

The equality T3(f) ◦ ϕG,3 = ϕH,3 ◦ f (i.e., the commutativity of the left part of the above diagram)was established in Proposition 2.6. The same result implies that Reg(f) ◦ ϕG,r = ϕH,r ◦ f . Hence thecommutativity of the right part of the diagram is immediate.

Since ϕG,r = λG ◦ ϕG,3, we apply Corollary 2.10 to conclude that (Reg(G), λG) is the regular reflectionof T3(G). The group T3(G) is a T3-space, so Lemma 3.6 implies that λG is the quotient homomorphism ofT3(G) onto T3(G)/N ∼= Reg(G) ∼= T0(T3(G)), where N is the closure of the neutral element in T3(G). Hencethe homomorphism λG is open and a similar argument shows that λH is open and Reg(H) ∼= T0(T3(H)).This completes the proof. �

Our next aim is to establish an equivalence of several functors.

Theorem 3.8. The functors Reg, T0 ◦ T3, T1 ◦ T3, and T2 ◦ T3 are naturally equivalent in the category ofsemitopological groups.

Proof. We will show that Reg(G) ∼= Ti(T3(G)) for each i ∈ {0, 1, 2}, leaving the corresponding verificationof equivalences for morphisms to the reader. For i = 0 this follows from Proposition 3.7, so it suffices toverify that Reg(G) ∼= Ti(T3(G)) for i = 1, 2.

Let ϕH,i be the canonical homomorphism of H onto Ti(H), where H = T3(G) and i ∈ {1, 2}. It fol-lows from Proposition 3.5 that there exists a continuous surjective homomorphism ψ0,i : T0(H) → Ti(H)satisfying ϕH,i = ψ0,i ◦ ϕH,0. Hence Corollary 2.10 implies that (Ti(H), ψ0,i) is the Ti-reflection of T0(H).Since the group T0(H) ∼= Reg(G) is regular, it satisfies the Ti separation axiom. Therefore, ψ0,i is a topo-logical isomorphism of T0(H) onto Ti(H). This proves that T0(H) ∼= Ti(H), i.e., Reg(G) ∼= Ti(T3(G)) fori ∈ {1, 2}. �

The conclusion of Theorem 3.8 is quite natural since regularity is the combination of the T1 and T3separation axioms. One can wonder, therefore, whether T3 ◦ T1 ∼= T1 ◦ T3. It turns out that the functorsT1 and T3 do not commute, even in the narrower class of paratopological groups. The explanation of thisphenomenon is given in Example 3.10 below. In Example 3.11 we show that the functors T0 ◦T3 and T3 ◦T0are not equivalent either. First we present a simple lemma.

Lemma 3.9. For every semitopological group G, the canonical homomorphism ϕG,3 : G → T3(G) is acontinuous bijection.

Proof. Denote by H the group G considered with the trivial anti-discrete topology. It is clear that H

is a topological group satisfying the T3 separation axiom and the identity mapping iG of G onto H iscontinuous. Hence there exists a continuous homomorphism h : T3(G) → H satisfying iG = h ◦ ϕG,3. SinceiG is a bijection, so is ϕG,3. �

Restricting the functors Ti for i ∈ {0, 1, 2} to paratopological groups, we obtain covariant functorsin the category of paratopological groups. In other words, Ti(G) is a paratopological group, for everyparatopological group G. Indeed, the homomorphism ϕG,i is open by Proposition 2.5, so it suffices to notethat every quotient of a paratopological group is again a paratopological group.

A similar fact for the functors T3 and Reg requires an additional argument. It turns out that T3(G) isthe semiregularization of G, for each paratopological group G (see [15, Theorem 2.5]). Hence T3(G) is aparatopological group, according to a result proved by Ravsky in [10]. Finally, since Reg(G) ∼= T0(T3(G))by Proposition 3.7, we see that the functor Reg preserves paratopological groups as well.

We do not know, however, whether Tych(G) has to be a paratopological group, for each paratopologicalgroup G (see Problem 4.2).

M. Tkachenko / Topology and its Applications 161 (2014) 364–376 375

We show in Examples 3.10 and 3.11 below that T1 ◦T3 � T3 ◦T1 and T0 ◦T3 � T3 ◦T0, even if we restrictthe functors T0, T1, and T3 to the narrower category of paratopological groups. Needless to say, we do notuse the fact that the functor T3 preserves paratopological groups.

Example 3.10. Let Z be the additive group of integers. For every integer n � 0, let Un = {0} ∪ {m ∈ Z:m > n}. It is easy to verify that the family {Un: n ∈ N} forms a base at zero for a paratopologicalgroup topology τ1 on Z and the paratopological group G = (Z, τ1) satisfies the T1 separation axiom. It isalso clear that every two non-empty open sets in G have a non-empty intersection, hence the Hausdorffreflection T2(G) and the regular reflection Reg(G) are trivial one-point groups. Thus, by Theorem 3.8,Reg(G) ∼= T1(T3(G)) is the trivial group. However, since G is a T1-space, we see that T1(G) = G and, hence,T3(T1(G)) = T3(G). By Lemma 3.9, the canonical homomorphism ϕG,3 of G onto T3(G) is a bijection,so T3(G) is an infinite group (carrying the anti-discrete topology). Therefore, the groups T1(T3(G)) andT3(T1(G)) are quite different. �Example 3.11. Let the additive group of integers, Z, carry the topology τ0 whose base consists of the setsVn = {k ∈ Z: k � n}, with n ∈ Z. Then H = (Z, τ0) is a T0 paratopological group, so T0(H) = H.Since Vn ⊆ Vm if m < n, we conclude that T3(H) is algebraically the group of integers which carries thetrivial anti-discrete topology. Hence, by Lemma 3.9, T3(T0(H)) = T3(H) is the group of integers with theanti-discrete topology, while T0(T3(H)) is a trivial one-point group. �Remark 3.12. It is easy to see that Ti ◦ Tj

∼= Tj ◦ Ti∼= Tj whenever 0 � i < j � 2. Indeed, let G be an

arbitrary semitopological group. Since Tj(G) satisfies the Tj separation axiom, we have that Ti(Tj(G)) =Tj(G). Further, it follows from Proposition 3.5 that there exists a continuous surjective homomorphismλi,j : Ti(G) → Tj(G) satisfying ϕG,j = λi,j ◦ ϕG,i, where ϕG,i and ϕG,j are canonical homomorphisms of Gonto Ti(G) and Tj(G), respectively. Hence Corollary 2.10 implies that Tj(Ti(G)) ∼= Tj(G). Similarly, we seethat Reg ◦ Ti

∼= Reg ∼= Ti ◦ Reg, for each i = 0, 1, 2, 3.

4. Open problems

In Theorems 3.1 and 3.4 we gave a description of the Ti-reflection, for i = 0, 1, of a semitopological groupG as the quotient of G with respect to a certain subgroup of G defined in ‘internal’ terms. It would beinteresting and important to find a description of the Hausdorff reflection of an arbitrary semitopologicalgroup G:

Problem 4.1. Describe in internal terms the kernel of the canonical homomorphism ϕG,2 of a semitopologicalgroup G onto T2(G).

In the special case when G is a paratopological group, Problem 4.1 is solved in [15, Theorem 2.1].

Problem 4.2. Let G be a paratopological group. Is Tych(G) a paratopological group? What if G is addition-ally first countable or separable?

According to Corollary 2.10, every semitopological group G satisfies Tych(G) ∼= Tych(Reg(G)). There-fore, it suffices to consider the case of a regular paratopological group G in Problem 4.2. It is worth noting,however, that all known examples of regular paratopological groups turn out to be Tychonoff. RecentlyI. Sánchez proved in [11, Corollary 2.7] that every regular totally ω-narrow paratopological group is Ty-chonoff. Hence Tych(G) is a paratopological group, for every totally ω-narrow paratopological group G.

376 M. Tkachenko / Topology and its Applications 161 (2014) 364–376

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