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### Transcript of Free Topological Groups - UCM · topological groups): Theorem (1.2) Suppose that G 1 and G 2 are...

• Free Topological Groups

Mikhail Tkachenko

• Introduction

Let (G , ·) be an abstract group with multiplication

m : G × G → G , m(x , y) = x · y .

Suppose that G carries a Hausdorff topology τ such that m isjointly continuous as a mapping of G × G to G , and that theinversion

Inv : G → G , Inv(x) = x−1,

is also continuous. Than the triple (G , ·, τ) is called a topologicalgroup while τ is said to be a topological group topology.Usually, we will omit the symbols designated to the multiplicationand topology on G and will say that G is a topological group, if itis not ambiguous. By Pontryagin’s theorem, every topologicalgroup G is a Tychonoff space, i.e., the topology of G is generatedby the family of all continuous real-valued functions on X .Therefore, every subspace of a topological group is Tychonoff.

• Definitions. Graev’s extension of pseudometrics

By several reasons, the free topological groups constitute a veryimportant and interesting subclass of topological groups.

Definition (1.1)

Suppose that X is a subspace of a topological group G . We saythat G is a free topological group over X if the following hold:

1. X generates algebraically a dense subgroup of G ;

2. for every continuous mapping f : X → H to an arbitrarytopological group H, there exists an extension of f to acontinuous homomorphism f̃ : G → H.

Xid //

f��

G

f̃~~~~

~~~

H

• Unicity Theorem

It turns out that a free topological group over a space X is, in asense, unique (we postpone the discussion of the existence of freetopological groups):

Theorem (1.2)

Suppose that G1 and G2 are free topological groups over aTychonoff space X . Then there exists a topological isomorphismϕ : G1 → G2 such that ϕ(x) = x, for each x ∈ X. Furthermore,both groups G1 and G2 are algebraically generated by X , i.e., X isa set of generators for G1 and G2.

• Proof.By Definition 1.1, there exist continuous homomorphismsϕ1 : G1 → G2 and ϕ2 : G2 → G1 which extend the identity mappingof X onto itself. Let ψ1 = ϕ1 ◦ ϕ2 and ψ2 = ϕ2 ◦ ϕ1. Then ψ1 is acontinuous homomorphism of G1 to itself whose restriction to X isthe identity mapping of X . Since X generates a dense subgroup ofG1 (and G1 is Hausdorff), it follows that ψ1 is the identityautomorphism of G1. Similarly, ψ2 is the identity automorphism ofG2. Hence, ϕ1 and ϕ2 are topological isomorphisms that do notmove the points of X .Finally, it is clear that the dense subgroup 〈X 〉 of G1 generated byX satisfies all conditions of Definition 1.1. Hence, it follows fromthe first claim of the theorem that 〈X 〉 = G1.The uniqueness of a free topological group over a Tychonoff spaceX enables us to introduce a name for this object, say F (X ).If all groups in Definition 1.1 are assumed Abelian, we obtain thedefinition of the free Abelian group over X which is denoted byA(X ).

• Let us now explain the meaning of the word “free” in the name ofthe group F (X ).Since, by Theorem 1.2, X algebraically generates the group F (X ),every element of g ∈ F (X ) can be written in the form

g = x�11 · . . . · x�nn ,

where x1, . . . , xn ∈ X and �1, . . . , �n ∈ {−1, 1}. The expressionx�11 · . . . · x�nn is called a word. A word x

�11 · . . . · x�nn is said to be

reduced if it contains no pair of consecutive symbols of the formx · x−1 or x−1 · x . It turns out that if a word is reduced andnon-empty, then it is different from the identity of F (X ). Usingthe language of the theory of groups, we say that X is a set of freegenerators for F (X ). Equivalently, from the algebraic point ofview, F (X ) is a free group over the set X .

• In the Abelian case, the meaning of the word “free” is, of course,slightly different. Again, one can show that if x1, . . . , xn arepairwise distinct elements of X and k1, . . . , kn are arbitraryintegers, then the equality

k1x1 + k2x2 + · · ·+ knxn = 0A(X )

implies that k1 = k2 = · · · = kn = 0. Therefore, the group A(X ) istorsion-free and, again, X is a set of free generators for A(X ).Equivalently, A(X ) is a free Abelian group over X .

This is a good point to turn back to the problem of the existenceof free topological groups. Let us consider the non-Abelian casefirst. It follows from Definition 1.1 that the topology of the groupF (X ) (when the latter exists) is maximal in some sense. Here isthe exact mathematical formulation of this fact.

• Theorem (1.3)

The topology of F (X ) is maximal among all topological grouptopologies on F (X ) that induce on X its original topology.

We now explain Graev’s approach to the existence proof. Supposethat G is a family of topological group topologies (not necessarilyHausdorff) on an abstract group G . Then the upper bound

∨G of

the topologies in G is again a topological group topology on G .This simple fact tells us that it suffices to find any Hausdorfftopological group topology on F (X ) that agrees with theoriginal topology of X . Then, by Theorem 1.3, the upper boundof all these topologies will be the topology of the free topologicalgroup F (X ).To find such a topology on F (X ), Graev proved the followingremarkable theorem. We need just one definition prior to Graev’stheorem.

Definition (1.4)

A pseudometric % on a group G is called invariant if%(xa, xb) = %(a, b) = %(ax , bx), for all a, b, x ∈ G .

• Theorem (1.5)

Every pseudometric % on a Tychonoff space X can be extended toan invariant pseudometric %̂ on F (X ).

Here are some lines explaining Graev’s extension procedure.Suppose that % is a pseudometric on a non-empty set X . First, weextend % to a pseudometric %∗ onto X ∗ = X ∪ {e} ∪ X−1 ⊆ F (X ),where e is the identity of F (X ). Clearly, we write F (X ) in place ofthe free Abelian group on X .Choose a point x0 ∈ X and for every x ∈ X , put

%∗(e, x) = %∗(e, x−1) = 1 + %(x0, x).

Then for x , y ∈ X , define the distances %∗(x−1, y−1), %∗(x−1, y)and %∗(x , y−1) by

%∗(x−1, y−1) = %∗(x , y) = %(x , y),

%∗(x−1, y) = %∗(x , y−1) = %∗(x , e) + %∗(e, y).

From our definition it follows immediately that %∗ extends % and%∗(z , t) = %∗(t, z) ≥ 0, for all z , t ∈ X̃ . It is easy to verify that %∗satisfies the triangle inequality, i.e., it is a pseudometric on X ∗.

• Let g , h be arbitrary elements of F (X ). We write these elementsas words, not necessarily reduced, of one the same length, in allpossible forms:

g = x�11 · . . . · x�nn

andh = y δ11 · . . . · y

δnn ,

where xi , yi ∈ X ∪ {e} and �i , δi ∈ {−1, 1}. Given a couple ofwords representing g and h, as above, we calculate the number

D =n∑

i=1

%∗(x�ii , yδii ).

The upper lower bound of the numbers D is denoted by %̂(g , h).Clearly, %̂(g , h) ≥ 0 and %̂(h, g) = %̂(g , h), for all g , h ∈ F (X ). Themost difficult part of the job is to verify that %̂ satisfies the triangleinequality, which we must skip by obvious reasons. Thus, %̂ is apseudometric. Our definition of %̂ implies almost immediately that%̂ is invariant.

• The final part of the existence proof goes as follows. For everycontinuous pseudometric % on X , consider the topology τ% on theabstract free group F (X ) generated by %̂. Since the pseudometric%̂ is invariant, one can easily verify that τ% is a topological grouptopology on F (X ), not necessarily Hausdorff. Since % is continuousand %̂ extends %, the restriction of the topology τ% to X is coarserthan the original topology of X . Therefore, the upper boundτ =

∨% τ% of these topologies is again a topological group topology

on F (X ). The advantage of τ , compared to the topologies τ%, isthat this topology is Hausdorff and generates on X its originaltopology. Taking the upper bound of the topological grouptopologies on F (X ) with the same properties, we get the required“free” topological group topology on F (X ). This finishes theexistence proof in the non-Abelian case.A similar argument, with obvious modifications, implies theexistence of the free Abelian topological group A(X ) over aTychonoff space X .

• Once the existence and the uniqueness theorems are established,one can wonder what the free topological groups serve for andwhat topological properties they possess. At this point we come tothe origins of this topic. In the early thirties of the twentiethcentury L. S. Pontryagin showed that every Hausdorff topologicalgroup is completely regular, i.e., Tychonoff. This gave rise to theproblem whether every Hausdorff topological group is a normalspace. To solve the problem, A. A. Markov invented the theory offree topological groups. His idea was to embed every Tychonoffspace X as a closed subspace of an appropriate topological groupGX , which he constructed as the free topological group over X .So, he proved the following:

Theorem (1.6)

Every Tychonoff space X is closed in the free topological groupF (X ) and in the free Abelian topological group A(X ).

This fact enabled Markov to give an elegant solution to thenormality problem.

• Indeed, take any Tychonoff space X which is not normal, forexample, the Tychonoff plank (ω1 + 1)× (ω+ 1) \ {(ω1, ω)}. Thenthe free topological group F (X ) contains X as a closed subset and,therefore, F (X ) cannot be a normal space — otherwise X wouldbe normal.

Theorem 1.6 can be given a sharper form in the sense that thegroup “envelope” of a space can be chosen σ-compact. Thisrequires a small modification in the original Markov’s argument.

Theorem (1.7)

Every Tychonoff space X can be embedded as a closed subspaceinto a topological group which is a dense subgroup of a σ-compacttopological (Abelian) group.

• Proof.Let bX be any compactification of the space X . Then A(bX ) is atopological Abelian group which, by Theorem 1.3, contains bX as aclosed subspace. Let G = 〈X 〉 be a subgroup of A(bX ) generatedby the set X . Clearly, G is dense in A(bX ) since X is dense in bX ,and the group A(bX ) is σ-compact. In addition, we have that

X = G ∩ bX .

It follows that X is closed in G .

A non-compact Tychonoff space X cannot be closed in a compacttopological group. However, we can refine Theorem 1.7 as well.We recall that a topological group is precompact if it is a (dense)subgroup of a compact topological group.

Theorem (1.8)

Every Tychonoff space X can be embedded as a closed subspaceinto a precompact topological (Abelian) group.

• Proof.Let Y = C (X ) be the family of all continuous real-valued functionson X , and ϕ the diagonal product of the functions in C (X ),ϕ : X → TY . Then ϕ is a topological embedding of X into thecompact topological Abelian group TY , and one can verify thatthe subspace ϕ(X ) of the precompact topological groupG = 〈ϕ(X )〉 generated by ϕ(X ) is closed in G .Surprisingly, Theorem 1.8 is not the strongest result about closedembeddings:

Theorem (1.9)

Every Tychonoff space X can be embedded as a closed subspaceinto a pseudocompact Abelian topological group.

• Proof.Let X be a closed subspace of a precompact topological Abeliangroup H (Theorem 1.8). By a theorem of M. Ursul, H is a closedtopological subgroup of a pseudocompact topological Abeliangroup G . Hence, X is a closed subspace of G .

• Lecture 2.

1. The direct limit property in free topological groups.

2. Completeness of free topological groups.

3. Bounded subsets of free topological groups.

• The main difficulty in working with free topological groups is anon-constructive way we introduced them. It is almost impossibleto describe constructively open subsets of a free topological groupF (X ) for a non-discrete space X .

Theorem (2.1)

The following conditions are equivalent for a Tychonoff space X :

(a) F (X ) is locally compact;

(b) F (X ) is first countable;

(c) F (X ) is discrete;

(d) X is discrete.

In particular, F (X ) is metrizable if and only if X is discrete.In the case of a compact space X , however, there is a good chanceto understand the “shape” of open subsets of F (X ). This requiresone useful concept.

• Definition (2.2)

Suppose that a space X is the union of a family ξ = {Xn : n ∈ ω}of its subspaces, where Xn ⊆ Xn+1 for each n ∈ ω. We say that Xis the inductive limit of the sequence ξ provided that a set F ⊆ Xis closed in X if and only if F ∩ Xn is closed in Xn, for each n ∈ ω.If a space X is the inductive limit of a sequence ξ of compactsubsets, then X is called a kω-space. The sequence ξ is called akω-decomposition of X .

Every kω-space is σ-compact. Every locally compact σ-compactspace is a kω-space. However, the space Q of rationals with thetopology inherited from R is evidently σ-compact but is not akω-space.

• The following important result has to be attributed to Graev andto Mack–Morris–Ordman.For a given integer n ≥ 0, we denote by Fn(X ) the subspace of thefree topological group F (X ) consisting of all elements g of theform

g = x�11 · . . . · x�nn ,

where x1, . . . , xn ∈ X and �1, . . . , �n ∈ {−1, 1}.

Theorem (2.3)

If X is a kω-space with a kω-decomposition {Xn : n ∈ ω}, then thefree topological group F (X ) is the inductive limit of the sequence{Fn(Xn) : n ∈ ω}, i.e., the latter sequence is a kω-decompositionfor F (X ).

• Note that each set Fn(Xn) in the above theorem is a compactsubset of F (X ), since the sets Xn are compact.Sketch of the proof. Denote by τ the family of all sets O ⊆ F (X )such that O ∩ Fn(Xn) is open in Fn(Xn), for each n ∈ ω. It is clearthat τ is a topology for F (X ) and that τ induces the originaltopology on X . It is also clear that O−1 ∈ τ , for each O ∈ τ .One can also verify that if g · h ∈ U ∈ τ , then there are V ∈ τ andW ∈ τ such that g ∈ V , h ∈ W , and VW ⊆ U. In other words,the multiplication in (F (X ), τ) is continuous and, therefore, F (X )endowed with the topology τ is a (Hausdorff) topological group.Obviously, τ is finer than the topology T of the free topologicalgroup F (X ). However, since τ is a topological group topology onF (X ) and it induces the original topology on X , we conclude, byTheorem 1.3, that τ = T.

• Corollary (2.4)

If X is a compact space, then the free topological group F (X ) isthe inductive limit of the sequence {Fn(X ) : n ∈ ω} of its compactsubspaces.

Corollary (2.5)

If X is a Tychonoff space and a set K ⊆ F (X ) satisfies|K ∩ Fn(X )| < ω for each n ∈ ω, then K is closed and discrete inF (X ).

Proof.Let bX be a compactification of X . The identity embeddingX ↪→ bX extends to a continuous monomorphismf : F (X ) → F (bX ). Clearly, |f (K ) ∩ Fn(bX )| < ω for each n ∈ ω.Since, by Corollary 2.4, F (bX ) is the inductive limit of thesubspaces Fn(bX ), it follows that f (K ) is closed in F (bX ), and thesame argument implies that every subset of f (K ) is closed inF (bX ). Hence f (K ) is discrete. Since f is a continuousmonomorphism, K is closed and discrete in F (X ).

• Corollary (2.6)

Let X be an arbitrary Tychonoff space. If C is a compact subset ofF (X ), then C ⊆ Fn(X ), for some n ∈ ω.Here are more applications of Theorem 2.3.

Theorem (2.7)

Let Y be a closed subset of a Tychonoff space X . Then thesubgroup F (Y ,X ) = 〈Y 〉 of F (X ) generated by Y is closed inF (X ).

Sketch of the proof. Modify the argument given in the proof ofCorollary 2.5.

After Theorem 2.7 we come to the following:

ProblemIf Y is a closed subspace of a Tychonoff space X , is thenF (Y ) ∼= F (Y ,X )?

• Corollary (2.8)

If K is a compact subset of a Tychonoff space X , thenF (K ) ∼= F (K ,X ).

Proof.Denote by bX an arbitrary compactification of X and letf : K ↪→ X and g : X ↪→ bX be the natural embeddings. Extend fand g to continuous monomorphisms f̂ : F (K ) → F (X ) andĝ : F (X ) → F (bX ), respectively. By Corollary 2.4, F (K ) andF (bX ) are kω-spaces with kω-decompositions F (K ) =

⋃∞n=0 Fn(K )

and F (bX ) =⋃∞

n=0 Fn(bX ).

Clearly, ϕ = ĝ ◦ f̂ is a continuous monomorphism of F (K ) toF (bX ). Let C be an arbitrary closed set in F (K ). ThenCn = C ∩ Fn(K ) is compact and, hence, ϕ(C ) ∩ Fn(bX ) = ϕ(Cn)is closed in Fn(bX ), for each n ∈ ω. We conclude, therefore, thatϕ(C ) is closed in F (bX ). This proves that ϕ is a closed mapping,whence it follows that ϕ is a topological isomorphism betweenF (K ) and F (K , bX ). Finally, the equality ϕ = ĝ ◦ f̂ implies that f̂is a topological isomorphism between F (K ) and F (K ,X ).

• Definition (2.9)

We say that a topological group G is complete (meaning Răıkovcomplete) if G is closed in any topological group that contains Gas a topological subgroup. Equivalently, G is complete if everyCauchy filter in G converges.

Here is yet another application of Theorem 2.3 given by Graev andHunt–Morris.

Theorem (2.10)

If X is a kω-space, then the group F (X ) is complete.

Corollary (2.11)

The free topological group F (X ) is complete, for every compactspace X .

• ProblemCharacterize the spaces X such that the free Abelian topologicalgroup A(X ) or the free topological group F (X ) is complete.

The solution to this problem is very far from obvious.

Definition (2.12)

A space X is called Dieudonné complete if X is homeomorphicto a closed subspace of a product of metrizable spaces.

All metrizable spaces and arbitrary products of metrizable spacesare Dieudonné complete. Less trivially: If a Tychonoff space Xadmits a continuous one-to-one mapping onto a metrizable space,then X is Dieudonné complete (such a space X is calledsubmetrizable).

• Theorem (2.13)

The free Abelian topological group A(X ) is complete if and only ifthe space X is Dieudonné complete. Furthermore, the equality%A(X ) ∼= A(µX ) holds for every Tychonoff space X .Here % stands for the Răıkov completion of a topological group,while µX is the Dieudonné completion of the space X .Therefore, % ◦ A ∼= A ◦ µ, in the language of functors.

Corollary (2.14)

The free Abelian topological group A(X ) is complete for anysubmetrizable space X .

Every paracompact space is Dieudonné complete, so we have:

Corollary (2.15)

The free Abelian topological group A(X ) is complete for anyparacompact space X .

• The non-Abelian case requires a very different technique, but theresults are the same in nature:

Theorem (2.16)

The free topological group F (X ) is complete if and only if thespace X is Dieudonné complete. In addition, %F (X ) ∼= F (µX ), foran arbitrary Tychonoff space X .

We now deduce the following result (reformulation ofTheorem 2.16):

Corollary (2.17)

If X is a closed subspace of a product of metrizable spaces, thenthe free topological group F (X ) is complete.

• Bounded subsets of free topological groups

Definition (2.18)

A subset B of a Tychonoff space X ia called bounded in Xprovided every continuous real-valued function defined on X isbounded on B.

Every compact subset of a space X is bounded in X , but not viceversa. In fact, there exist bounded, closed, discrete, infinite subsets(a closed copy of N) of Tychonoff spaces.

Lemma (2.19)

If B is a bounded subset of a Dieudonné complete space X , thenthe closure of B in X is compact.

Proposition (2.20)

Every bounded subset B of a topological group G is precompact inG, i.e., for every neighbourhood U of the identity in G, there existsa finite set F ⊆ G such that B ⊆ FU and B ⊆ UF.

• Here is another useful fact about bounded subsets of topologicalgroups:

Theorem (2.21)

If A and B are bounded subsets of a topological group G, thenA · B is a bounded subset of G.

ProblemDoes Theorem 2.21 remain valid for paratopological groups?

• Precompact subsets of free topological groups are, in a sense, veryspecial.For an element g ∈ F (X ), let

g = xε11 · · · xεnn

be the irreducible representation of g , where x1, . . . , xn ∈ X andε1, . . . , εn ∈ {−1, 1}. Then we put

supp (g) = {x1, . . . , xn} and supp (B) =⋃g∈B

supp (g).

Lemma (2.22)

If B is a precompact subset of a free topological group F (X ), thenY = supp (B) is bounded in X , and B ⊆ Fn(Y ,X ), for somen ∈ N. In other words, the lengthes of the elements in B withrespect to the basis X are uniformly bounded.

• Finally, we can characterize bounded subsets of free topologicalgroups:

Theorem (2.23)

The following condition are equivalent for a subset B of a freetopological group F (X ):

(a) B is bounded in F (X );

(b) B is precompact in F (X );

(c) there exist a bounded subset Y of X and an integer n ∈ Nsuch that B ⊆ Fn(Y ,X ).

Proof.The implication (a) ⇒ (b) follows from Proposition 2.20, while(b) ⇒ (c) is Lemma 2.22. Finally, (c) ⇒ (a) follows fromTheorem 2.21.

• Lecture 3.

1. Free groups over metrizable spaces.

2. Quotients of zero-dimensional groups.

3. Commercial propaganda.

• We know that for every Tychonoff space X and any integer n ∈ N,the subset Fn(X ) of F (X ) consisting of all words of reduced length≤ n is closed. Here is a stronger result in the case when the spaceX is metric (essentially due to V. K. Bel’nov):

Proposition (3.1)

Let (X , d) be a metric space, and Td be the topology on F (X )generated by the Graev extension d̂ of d over F (X ). Then Fn(X )is closed in (F (X ),Td) for each integer n ∈ N.

Proof.Apply the definition of Graev’s extension d̂ of the metric d .

We have to complement Proposition 3.1, which requires somesimple definitions.

• Given an integer n ∈ N and a space X , we put

Cn(X ) = Fn(X ) \ Fn−1(X ).

Let also X̃ = X ∪ {e} ∪ X−1 and consider the multiplicationmapping

in : X̃n → F (X ), in(y1, y2, . . . , yn) = y1 · y2 · · · yn.

Theorem (3.2)

Let (X , d) be a metric space and Td the topology on F (X )generated by the Graev extension d̂ of d. Then the mapping in is ahomeomorphism of C ∗n (X ) = i

−1n (Cn) ⊆ X̃ n onto the subspace

Cn(X ) of (F (X ),Td). In particular, Cn(X ) is metrizable as asubspace of (F (X ),Td).

Corollary (3.3)

Let (X , d) be a metric space. Then the topology on Cn(X )inherited from F (X ) coincides with the one induced by the Graevextension d̂ of the metric d.

• Our aim now is to establish that a free topological group over ametrizable space is paracompact (Arhangel’skii’s theorem). To thisend, we need the definition of an s-approximation.

Definition (3.4)

Let T1 and T2 be two topologies on a set X such that T1 is finerthan T2. If the space (X ,T2) has a σ-discrete family of subsetswhich is a network for (X ,T1), then the topology T2 is called ans-approximation for T1.

Lemma (3.5)

Suppose that T1 and T2 are topologies on a set X such that T2 isan s-approximation for the topology T1. Then the following hold:

(a) for every subset Y of X , T2 �Y is an s-approximation forT1 �Y ;

(b) for every integer n > 0, the topology of the product (X ,T2)n

is an s-approximation for the topology of (X ,T1)n.

• Sometimes the existence of an s-approximation for a giventopology T on a set X implies that the space (X ,T) isparacompact.

Lemma (3.6)

Let T1 and T2 be regular topologies on a set X such that T2 is ans-approximation for T1 and the space (X ,T2) is collectionwisenormal. Then the space (X ,T1) is paracompact.

A paracompact space with a σ-discrete network is said to be aparacompact σ-space. All metric spaces, as well as their arbitraryimages under closed continuous mappings, are paracompactσ-spaces. Paracompact σ-spaces can be characterized in terms ofs-approximations.

Theorem (3.7)

A regular T1-space (X ,T) is a paracompact σ-space iff T admits ametrizable s-approximation.

• Here is the promised Arhangel’skii’s theorem, given in a slightlymore general form:

Theorem (3.8)

The free topological group F (X ) is a paracompact σ-space iff X isa paracompact σ-space.

• Proof.If F (X ) is a paracompact σ-space, then X is closed in F (X ) and,hence, X is also a paracompact σ-space.Conversely, if X is a paracompact σ-space, then Theorem 3.7implies that the topology τ of X admits a metrizables-approximation τ1. Choose a metric % on X which generates thetopology τ1 and consider the topology T% on F (X ) generated bythe Graev extension %̂ of % to F (X ). Let F%(X ) = (F (X ),T%). ByProposition 3.1, the sets Fn(X ) are closed in F%(X ), so each setCn(X ) = Fn(X ) \ Fn−1(X ) is open in the subspace Fn(X ) ofF%(X ). Therefore, Cn(X ) =

⋃k∈ω Cn,k , where each Cn,k is closed

in F%(X ). For all n, k ∈ ω, choose a σ-discrete network γn,k for(Cn,k , %̂). By Corollary 3.3, the topology on Cn(X ) inherited fromF (X ) coincides with the one inherited from F%(X ), so the familyγ =

⋃n,k∈ω γn,k is a network for F (X ). Since γ is σ-discrete in

F%(X ), we conclude that T% is a metrizable s-approximation for theoriginal topology T of F (X ). Therefore, Theorem 3.7 implies thatF (X ) is a paracompact σ-space.

• We say that a Tychonoff space X is σ-closed-metrizable if it canbe represented as the union of countably many closed metrizablesubspaces. The next result complements Theorem 3.7.

Theorem (3.8)

The free topological group F (X ) is σ-closed-metrizable andparacompact if and only if the space X is σ-closed-metrizable andparacompact.

Corollary (3.9)

The free topological group F (X ) on a metrizable space X isσ-closed-metrizable and paracompact.

• ProblemWhen is the group F (X ) zero-dimensional?

Definition (3.10)

A continuous mapping f : X → Y will be called gentle if there is anetwork S in the space X such that its image {f (P) : P ∈ S} is aσ-discrete family of sets in Y .

We also recall that dim X denotes the covering dimension of aTychonoff space X defined in terms of finite cozero coverings of X ,while indX stands for the small inductive dimension of X .

Theorem (3.11)

Let X be a paracompact σ-space such that indX = 0. Thendim X = 0 if and only if there is a one-to-one gentle mapping of Xonto a metrizable space Y such that dim Y = 0.

• The next fact is slightly more general than the corresponding partof Theorem 3.11:

Theorem (3.12)

Suppose that g is a one-to-one gentle mapping of a space X withindX = 0 onto a metrizable space Y satisfying dim Y = 0. Thendim X = 0 and X is paracompact.

Note that the paracompactness of X in Theorem 3.12 follows evenwithout assumption that indX = 0 or dim X = 0.The following lemma is evident:

Lemma (3.13)

The product of any countable family of (one-to-one) gentlemappings is a (one-to-one) gentle mapping.

• Proposition (3.14)

Let Xn be a paracompact σ-space such that dim Xn = 0, for eachn ∈ ω. Then the product space X =

∏n∈ω Xn satisfies dim X = 0,

and X is paracompact.

Proof.By Theorem 3.11, we can fix, for each n ∈ ω, a one-to-one gentlemapping gn of X onto a metrizable space Yn such thatdim Yn = 0. Then IndYn = 0, by Theorem 7.1.10 in [Engelking].For the product space Y =

∏n∈ω Yn, we have dim Y = IndY = 0,

since each Yn is metrizable. Obviously, indX = 0. It remains torefer to Lemma 3.13 and Theorem 3.12.

• Theorem (3.15)

Let X be a non-empty paracompact σ-space. Then dim F (X ) = 0if and only if dim X = 0.

Proof.Since every continuous function f : X → R extends to a continuoushomomorphism f̃ : F (X ) → R, the set X is C -embedded in F (X ).Therefore, dim X ≤ dim F (X ) by Theorem 7.1.8 of [Engelking].Now, suppose that dim X = 0. Since X̃ = X ⊕ {e} ⊕ X−1 is aparacompact σ-space, Theorem 3.2 and Lemma 3.5 imply thatCn(X ) = Fn(X ) \ Fn−1(X ) is a paracompact σ-space. As in theproof of Theorem 3.8, one can represent every Cn(X ) as the unionCn(X ) =

⋃k∈ω Cn,k , where each Cn,k is closed in Fn(X ) and in

F (X ). Then Cn,k is homeomorphic to a closed subspace of X̃n.

Hence, by Proposition 3.14,

dim Cn,k ≤ dim X̃ n = dim X̃ = 0.

Since F (X ) =⋃

n,k∈ω Cn,k , the countable sum theorem for thedimension dim implies that dim F (X ) = 0.

• Here is the main result of the lecture.

Theorem (3.16)

Every topological group G is a quotient of a topological group Hsatisfying the following conditions:

(a) dim H = 0 and dim Y = 0 for every Y ⊆ H;(b) H is a paracompact σ-space hereditarily;

(c) H admits a continuous isomorphism onto a metrizabletopological group.

All we need now is the following lemma:

Lemma (3.17)

Every T1-space X is an image under a quotient mapping of atopological sum Z = ⊕α∈AZα, where each Zα is a normal space ofcountable pseudocharacter with at most one non-isolated point.

• TO APPEAR SOON:

TOPOLOGICAL GROUPS AND RELATED STRUCTURES

by

A. V. ARHANGEL’SKII and M. G. TKACHENKO

[ATLANTIS STUDIES in MATHEMATICS, Vol. I, ATLANTISPRESS/WORLD SCIENTIFIC (2008); ISBN: 978-90-78677-06-2;ISSN: 1875-7634]

• CONTENTS

Chapter 1. INTRODUCTION TO TOPOLOGICAL GROUPSAND SEMIGROUPSChapter 2. RIGHT (SEMI)TOPOLOGICAL GROUPSChapter 3. TOPOLOGICAL GROUPS: BASICCONSTRUCTIONSChapter 4. SPECIAL CLASSES OF TOPOLOGICAL GROUPSChapter 5. CARDINAL INVARIANTS OF TOPOLOGICALGROUPSChapter 6. MOSCOW TOPOLOGICAL GROUPS,COMPLETIONSChapter 7. FREE TOPOLOGICAL GROUPSChapter 8. R-FACTORIZABLE TOPOLOGICAL GROUPSChapter 9. COMPACTNESS IN TOPOLOGICAL GROUPSChapter 10. ACTIONS OF TOPOLOGICAL GROUPS ONTOPOLOGICAL SPACES

• Chapter 1. INTRODUCTION TO TOPOLOGICAL GROUPSAND SEMIGROUPS1.1 Some algebraic concepts1.2 Groups and semigroups with topologies1.3 Neighbourhoods of the identity in groups1.4 Open sets, closures, connected sets1.5 Quotients of topological groups1.6 Products, σ-products, and Σ-products1.7 Factorization theorems1.8 Uniformities on topological groups1.9 Markov’s Theorem1.10 Historical comments to Chapter 1

• Chapter 2. RIGHT (SEMI)TOPOLOGICAL GROUPS2.1 From discrete semigroups to compact2.2 Idempotents in compact semigroups2.3 Joint continuity, continuity of the inverse2.4 Pseudocompact semitopological groups2.5 Cancellative topological semigroups2.6 Historical comments to Chapter 2

• 3. TOPOLOGICAL GROUPS: BASIC CONSTRUCTIONS3.1 Locally compact topological groups3.2 Quotients w.r.t. locally compact subgroups3.3 Prenorms, metrization3.4 ω-narrow and ω-balanced groups3.5 Groups of isometries and groups of homeomorphisms3.6 Răıkov completion of a topological group3.7 Precompact topological groups3.8 Embeddings into locally connected groups3.9 Historical comments to Chapter 3

• 4. SPECIAL CLASSES OF TOPOLOGICAL GROUPS4.1 Ivanovskij–Kuz’minov Theorem4.2 Embedding into a compact group4.3 Čech-complete and feathered groups4.4 P-groups4.5 Extremally disconnected groups4.6 Perfect mappings and topological groups4.7 Convergence phenomena in topological groups4.8 Historical comments to Chapter 4

• 5. CARDINAL INVARIANTS OF TOPOLOGICAL GROUPS5.1 Embeddings into products of groups5.2 Basic invariants of topological groups5.3 Lindelöf Σ-groups and the Nagami number5.4 Cellularity and weak precalibres5.5 o-tightness in topological groups5.6 Steady and stable topological groups5.7 Cardinal invariants in paratopological groups5.8 Historical comments to Chapter 5

• 6. MOSCOW TOPOLOGICAL GROUPS, COMPLETIONS6.1 Moscow spaces and C -embeddings6.2 Moscow spaces, P-spaces, and extremal disconnectedness6.3 Products and mappings of Moscow spaces6.4 The breadth of the class of Moscow groups6.5 When the Dieudonné completion of a topological group is agroup?6.6 Pseudocompact groups and their completions6.7 Moscow groups and the formula υX × υY = υ(X × Y )6.8 Subgroups of Moscow groups6.9 Pointwise pseudocompact and feathered groups6.10 Bounded and C -compact sets6.11 Historical comments to Chapter 6

• 7. FREE TOPOLOGICAL GROUPS7.1 Definition and basic properties7.2 Extending pseudometrics from X to F (X )7.3 Extension of metrizable groups7.4 Direct limit property and completeness7.5 Precompact and bounded sets in free topological groups7.6 Free groups on metrizable spaces7.7 Nummela–Pestov theorem7.8 Direct limit property and countable compactness7.9 Completeness of free Abelian groups7.10 M-equivalent spaces7.11 Historical comments to Chapter 7

• 8. R-FACTORIZABLE TOPOLOGICAL GROUPS8.1 Basic properties8.2 Subgroups of R-factorizable groups8.3 Dieudonné completion of R-factorizable groups8.4 Homomorphic images of R-factorizable groups8.5 Products with a compact factor and m-factorizability8.6 R-factorizability of P-groups8.7 Factorizable groups and projectively Moscow groups8.8 Zero-dimensionality of R-factorizable groups8.9 Historical comments to Chapter 8

• 9. COMPACTNESS IN TOPOLOGICAL GROUPS9.1 A proof of the Gel’fand–Masur Theorem9.2 Invariant integral on a compact group9.3 Non-trivial characters on compact Abelian groups9.4 Pontryagin duality theory for discrete and for compact groups9.5 Some applications of the Pontryagin duality9.6 Non-trivial characters on locally compact Abelian groups9.7 Varopoulos’ theorem: the Abelian case9.8 Bohr topologies on discrete groups9.9 Pseudocompact group topologies on Abelian groups9.10 Countably compact topologies on Abelian groups9.11 Historical comments to Chapter 9

• 10. ACTIONS OF TOPOLOGICAL GROUPS ONTOPOLOGICAL SPACES10.1 Dugundji spaces and 0-soft mappings10.2 Continuous actions of topological groups on spaces10.3 Uspenskij’s theorem on continuous transitive actions ofω-narrow groups on compacta10.4 Continuous actions of compact groups and some classes ofspaces10.5 Historical comments to Chapter 10