The character of free topological groups I

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 6, No. 1, 2005 pp. 15-41 The character of free topological groups I Peter Nickolas and Mikhail Tkachenko 1 Abstract. A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if Xis a compact Hausdorff space, then the character of the free abelian topological group on Xlies between w(X) and w(X)ℵ0, where w(X) denotes the weight of X; (ii) if the Tychonoff space Xis not a P-space, then the character of the free abelian topological group is bounded below by the “small cardinal” d; and (iii) if Xis an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces. 2000 AMS Classification: Primary 22A05, 54H11, 54A25; Secondary 54D30, 54D45. Keywords: Free (abelian) topological group, uniform space, entourage of the diagonal, character, cardinal invariant, compact, locally compact, pseudocompact, metrizable, kω-space, dominating family 1The first author wishes to thank the second author, and his department, for hospitality extended during the course of this work. The second author was supported by Mexican National Council of Sciences and Technology (CONACyT), Grant No. 400200-5-28411-E. 16 P. Nickolas and M. Tkachenko 1. Introduction In 1948, Graev [5] proved that the free topological group F(X) and the free abelian topological group A(X) on a Tychonoff space Xare metrizable only if Xis discrete, in which case the groups are themselves discrete. For our present purposes, we may rephrase Graev’s result as saying that when Xis a non-discrete Tychonoff space, the groups F(X) and A(X) have uncountable character (= minimal size of a base at the identity of the group). It appears that no other estimates of the characters of these groups (except for those valid in the context of general topological groups) have been found to date. In this paper and its sequel [14], we investigate these characters systematically and in some detail. Most of our results are in fact for free and free abelian topological groups on uniform spaces, since this gives maximum generality and allows the derivation at will of bounds for the characters of free and free abelian groups over topological spaces. In the abelian case, the free topology has a rather straightforward description in terms of the family of uniformly continuous pseudometrics on the given uniform space, and another in terms of the given uniformity itself, and our initial results express the character of the corresponding group in terms of simple cardinal invariants of the family of pseudometrics (Theorem 2.3) or of the uniformity (Theorem 2.9). An immediate corollary of these results is that if Xis a compact Hausdorff space, then the character of A(X) lies between w(X) and w(X)ℵ0, where w(X) denotes the weight of X. The well known “small cardinal” d(see [3] and our discussion below) plays a role in several other results. For example, we show that if the Tychonoff space Xis not a P-space, then dis a lower bound for the character of A(X) (Corollary 2.16); this applies in particular if Xcontains an infinite compact subset or a proper dense Lindel¨of subspace. Further, if Xis an infinite compact metrizable space, then the character of A(X) is precisely d(Corollary 2.22). In the non-abelian case, the situation is intrinsically more complex and description of the free topology in terms of pseudometrics or entourages, for example, is now far from straightforward (see [21] and [16]). Our main results on the characters in the non-abelian case make use of a new description of a neighborhood base at the identity in the free topological group on an arbitrary uniform space (Theorem 3.6). While it is easy to see that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group (Lemma 3.1), the inequality is in general strict, and the two characters may indeed differ arbitrarily largely (see [14]). Using our new description of the topology of the free topological group, however, we show that the characters of the free abelian and the free (non-abelian) topological groups are equal whenever the underlying uniform space is ω-narrow (Theorem 3.15). The sequel [14] to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces. The character of free topological groups I 17 1.1. Notation and terminology. We denote by Nthe set of positive integers and by Rthe set of real numbers. All topological spaces are hypothesised implicitly to be completely regular, but are not taken to be Hausdorff (and therefore Tychonoff) without explicit indication. Similarly, our uniform spaces are not taken to be Hausdorff (or separated) unless this is explicitly signalled. We next establish our conventions for certain cardinal invariants of topological spaces and uniform spaces. In some formulations, such as that of [8], for example, cardinal invariants are taken always to be infinite, that is, to have a minimum value of ℵ0, because this tends to simplify the statements of theorems. For us, however, it is convenient not to follow this convention. Thus, if x is a point in a space X, then χ(x, X) denotes the minimum cardinal of a local base at x, and then χ(X), the character of X, is the supremum of the values χ(x, X) for x∈X. More generally, if Yis a subspace of X, then we write χ(Y, X) for the minimum cardinal of a base at Yin X. We introduce the ad hoc notation χ∆(X) to denote the least cardinal of a basis at the diagonal ∆ in X×X; that is, χ∆(X) = χ(∆, X ×X). For a space X, the weight w(X) of X is defined to be the smallest cardinal of an open base for X, and for a uniform space (X, U), we denote by w(X, U) the least cardinal of a base of (X, U). A space Xis a kω-space if there exists a sequence of compact subsets Xn⊆X for n∈Nsuch that X=S∞ n=1 Xnand such that Xhas the weak topology with respect to the family {Xn:n∈N}(that is, such that U⊆Xis open in Xif and only if U∩Xnis open in Xnfor each n∈N). In this situation, we call the collection {Xn:n∈N}, or the corresponding expression X=S∞ n=1 Xnfor X, akω-decomposition of X. We may take the sets Xnwithout loss of generality to be non-decreasing. If Uis a uniformity on a set X, and if U∈ U and n≥1, we use nU to denote the n-fold relational composite U◦U◦ · · · ◦ U. If −Udenotes the relational inverse of U, then we call Usymmetric if U=−U. For x∈Xand U∈ U, we denote by B(x, U) the set {y∈X: (x, y)∈U}. If (X, U) is a uniform space and τ≥ ℵ0is a cardinal, then we say that (X, U) is τ-narrow if for each U∈ U, there is a set {xα:α < τ} ⊆ Xsuch that X=Sα<τ B(xα, U). Similarly, we say that a topological group Gis τ-narrow if Gcan be covered by τmany translates of every neighborhood of the identity (the groups with this property was called τ-bounded in [6]). For a non-empty set X, we use Fa(X) and Aa(X) to denote the abstract free group and abstract free abelian group on X. If n∈ω, then Fn(X) and An(X) are the subsets of Fa(X) and Aa(X), respectively, which consist of all elements whose length with respect to the basis Xdoes not exceed n. If Xis a space, then F(X) and A(X) stand respectively for the free topological group and free abelian topological group on X. In this case, Fn(X) and An(X) refer to the corresponding subspaces of F(X) and A(X), respectively. Given a subset Yof X,F(Y, X) denotes the subgroup of F(X) generated by Y, and A(Y, X) has a similar meaning. 18 P. Nickolas and M. Tkachenko If (X, U) is a uniform space, then F(X, U) and A(X, U) stand for the free topological group and free abelian topological group on (X, U), and other related notations follow those already introduced for the free and free abelian topological groups on a topological space. If Xand Yare sets, we write XYfor the set of functions from Xto Y. If αand βare cardinals, we write cardinal exponentiation in the form βα; thus,  XY =|Y||X|. Also, c= 2ℵ0is the power of the continuum. In any topological group G, we denote by N(e) the family of all open neighborhoods of the neutral element ein G. 1.2. Quasi-ordered sets. Many of the arguments below make use of the notion of a quasi-ordered set, and related ideas. We establish here the relevant terminology and notation. We say that a pair (P, ≤) is a quasi-ordered set if ≤is a reflexive transitive relation on the set P. If (P, ≤) has the additional property of antisymmetry, then it is a partially ordered set. A set D⊆Pis called dominating or cofinal in the quasi-ordered set (P, ≤) if for every p∈Pthere exists q∈Dsuch that p≤q. Similarly, a subset Eof Pis said to be dense in (P, ≤) if for every p∈Pthere exists q∈Ewith q≤p. The minimal cardinality of a dominating family in (P, ≤) is denoted by D(P, ≤), while we use d(P, ≤) for the minimal cardinality of a dense set in (P, ≤). The notions of dominating and dense sets are dual: if a set Sis dense in (P, ≤), then it is dominating in (P, ≥) and vice versa. Therefore, d(P, ≤) = D(P, ≥) and D(P, ≤) = d(P, ≥). Note in any topological group Gwe have d(N(e),⊆) = χ(G). If (P, ≤) and (Q, ≪) are quasi-ordered sets, then a mapping f:P→Qis called order-preserving if x≤yimplies f(x)≪f(y), where x, y ∈P. Similarly, fis order-reversing if x≤yimplies f(x)≫f(y). If the mapping fis a bijection between Pand Qand if fand f−1both are order-preserving, then fis called an (order-)isomorphism of (P, ≤) onto (Q, ≪). Lemma 1.1. Let (P, ≤)and (Q, ≪)be quasi-ordered sets, and let f:P→ Qbe an order-preserving mapping. If f(P)is a dominating set in Q, then D(Q)≤D(P). Proof. Let Dbe an arbitrary dominating set in P. For each q∈Q, there exists p∈Psuch that q≤f(p). Also, there exists d∈Dsuch that p≤d, from which we have q≤f(p)≤f(d). Hence f(D) is a dominating set in Q, and so D(Q)≤ |f(D)| ≤ |D|. Finally, taking Dto be a dominating set in Pof minimal cardinality gives D(Q)≤D(P), as required.  Observe that because of the duality noted above between dominating sets and dense sets, there are dualised versions of the lemma: a version for dense sets rather than dominating sets, and others for order-reversing mappings rather than order-preserving ones. We will make frequent use of the lemma and its unstated variants, usually without explicit reference. We will deal later with many different quasi-ordered sets, most, though not all, of which will in fact be partially ordered sets. Generally, we will The character of free topological groups I 19 give each ordering distinctive notation by using an appropriate subscript, since some arguments make use of several quasi-orderings simultaneously. If ≪is an ordering on a set X, then we will usually denote the order defined coordinatewise on a set of functions YXby attaching an asterisk as a superscript, as in ≪∗. An exception is when X=ω, when the ordering defined coordinatewise using ≤on sets such as ωωand Nωwill again be denoted just by ≤. Consider ωω, the collection of all functions from ωto ω. Then following [3], we define two quasi-orders ≤∗and ≤on ωω, by specifying that if f, g ∈ωω, then f≤∗gif f(n)≤g(n) for all except finitely many n∈ω, and f≤gif f(n)≤g(n) for all n∈ω; the quasi-order ≤is of course in fact a partial order. (The use of the asterisk in ≤∗is inconsistent with the notational convention just established, but we will not in fact use this ordering again after this paragraph.) The least cardinal of a dominating set in the quasi-ordered set (ωω, ≤∗) is denoted by d, and in the partially ordered set (ωω, ≤) by d1. It is shown in Theorem 3.6 of [3] that d=d1. It is also known that ℵ1≤d≤c, but that the value of ddepends on extra axioms of set theory [3]. Below, we will find it most useful to use the characterization of das the least cardinal of a dominating set with respect to the relation ≤: thus, d=D(ωω, ≤). 2. The character of free abelian topological groups Graev’s proof in [5] of the fact that the free topological group on a Tychonoff space Xis Hausdorff proceeds by a construction which extends each continuous pseudometric on Xto an invariant pseudometric on the underlying abstract free group Fa(X), and an argument which shows that the group topology induced by all the extensions (referred to as Graev’s topology) is Hausdorff and weaker than the free topology. It is well known that Graev’s topology is only equal to the free topology in somewhat pathological cases [9, 19, 22]. In the abelian case, a parallel construction was also outlined by Graev, and in this case Graev’s topology is always the free topology (see [13, 11, 20]; cf. also 4.4 and 4.8 of [10]). (It seems unlikely that this fact was unknown to Graev, but it is not mentioned in his paper.) In what follows, we deal mostly with free and free abelian topological groups on uniform spaces, for convenience and generality, deducing applications to free and free abelian topological groups on topological spaces when appropriate. We therefore need the uniform space analog of the result just noted: that Graev’s pseudometrics generate a neighborhood base at the neutral element of the free abelian topological group on a uniform space. For a given pseudometric don a set X, we denote by b dand b dAGraev’s extension [5] of dto the abstract groups Fa(X) and Aa(X), respectively. Note that algebraically the free uniform groups F(X, U) and A(X, U) are Fa(X) and Aa(X), respectively [12, 15]. If (X, U) is a uniform space and dis a pseudometric on X, then we write Vd={g∈A(X, U) : b dA(g, 0) <1}, 20 P. Nickolas and M. Tkachenko where 0 is the neutral element of A(X, U). The following theorem is the uniform space analog of the result of [13, 20]; the arguments in the topological case apply with minimal adjustment in the uniform space case, and we therefore omit the proof. Theorem 2.1. Let (X, U)be a uniform space. Then for every uniformly continuous pseudometric don (X, U)the set Vdis open in A(X, U), and for every neighborhood Uof the neutral element in A(X, U)there exists a uniformly continuous pseudometric don (X, U)such that Vd⊆U. Instead of using uniformly continuous pseudometrics, one can directly use the entourages U∈ U belonging to a uniform space (X, U) to construct a neighborhood base at the identity of A(X, U). The theorem below gives a simple, explicit description in these terms of the topology of the free abelian topological group on a uniform space, equivalent in a sense to Theorem 2.1. This result has certainly been known since the early 1980s, when the first description of the topology of the (non-abelian) free topological group on a uniform space was published [16]. Though the paper [20] contains a result equivalent to ours stated in the language of pseudometrics, the first occurrence of essentially our result in the literature appears to have been in [26], though it is formulated there in a context less general than ours. First, we introduce the following notation. If (X, U) is a uniform space and if hU0, U1,...iis a sequence of elements of U, then we denote by N(hU0, U1,...i) the set of all elements of A(X, U) of the form x0−y0+x1−y1+···+xn−yn, where n∈ωand where (xi, yi)∈Uifor i= 0,1,...,n. Theorem 2.2. Let (X, U)be a uniform space. Then the collection of all sets of the form N(hU0, U1,...i), as hU0, U1,...iruns over all sequences of elements of U, is an open basis at the neutral element in A(X, U). Denote by P(X, U) the family of all uniformly continuous pseudometrics on (X, U) bounded by 1. For d1, d2∈ P(X, U), we write d1≤d2if d1(x, y)≤ d2(x, y) for all x, y ∈X. We express our first result on the character of A(X, U)) in terms of the partially ordered set (P(X, U),≤). Theorem 2.3. If (X, U)is a uniform space, then χ(A(X, U)) = D(P(X, U),≤). Proof. By Theorem 2.1, there exists a natural correspondence between the family P(X, U) and a base at the neutral element 0 of A(X, U). In fact, the mapping d7→ Vdfrom (P(X, U),≤) to the partially ordered set (N(0),⊇) of open neighborhoods of 0 in A(X, U) ordered by reverse inclusion is orderpreserving and maps P(X, U) to a base at 0 in A(X, U), that is, a dominating set in (N(0),⊇). This immediately implies (by Lemma 1.1) that χ(A(X, U)) ≤ D(P(X, U),≤). Suppose that a subset Q ⊆ P(X, U) is such that {Vd:d∈ Q} is a base at the neutral element in A(X, U). We claim that for every ∈ P(X, U) there exists The character of free topological groups I 21 d∈Qsuch that ≤2d. Indeed, given ∈ P(X, U), we can find d∈ Q such that Vd⊆V. Let x, y ∈X. If d(x, y)<1, then x−y∈Vd⊆V, whence (x, y)<1. Similarly, if n∈Nand d(x, y)<2−n, then b dA(0,2n(x−y)) = 2nd(x, y)<1 by the linearity of the pseudometric b dA(see [23]). Therefore, 2n(x−y)∈Vd⊆ V, whence (x, y)<2−n. We have thus proved that d(x, y)<2−nimplies (x, y)<2−n, for n= 0,1,.... In particular, d(x, y) = 0 implies (x, y) = 0. Therefore, the inequality (x, y)≤2d(x, y) holds if d(x, y) = 0. It is also obvious that the same inequality holds if d(x, y) = 1. If 0 < d(x, y)<1, choose n∈ωsuch that 2−n−1≤d(x, y)<2−n. Then (x, y)<2−n, so that (x, y)≤2d(x, y), and the inequality again holds. Thus, ≤2d, proving our claim. For d∈ Q, define d∗∈ P(X, U) by setting d∗= min{2d, 1}, and consider the family Q∗={d∗:d∈ Q} ⊆ P(X, U). By the claim just verified, for all ∈ P(X, U) there exists d∗∈ Q∗such that ≤d∗, and so Q∗is a dominating family in P(X, U). Therefore, D(P(X, U),≤)≤ |Q∗| ≤ |Q|. Let Bbe any base at 0 in A(X, U). Then for every N∈ B, Theorem 2.1 shows that there exists dN∈ P(X, U) such that VdN⊆N. Set Q={dN: N∈ B}. Then {Vd:d∈ Q} is a base at 0 and satisfies |Q| ≤ |B|, so that D(P(X, U),≤)≤ |Q| ≤ |B|. Hence D(P(X, U),≤)≤χ(A(X, U)). Combining this with the reverse inequality obtained earlier, we conclude that χ(A(X, U)) = D(P(X, U),≤).  For a topological space X, denote by PXthe family of all continuous pseudometrics on Xbounded by 1. It is clear that PX=P(X, UX), where UXis the fine uniformity of X. Since A(X) = A(X, UX), from Theorem 2.3 we have: Corollary 2.4. The equality χ(A(X)) = D(PX,≤)holds for all Tychonoff spaces X. We record in passing a couple of consequences of these last results, not otherwise directly related to our main concerns in this paper. First, let (X, U) be an arbitrary uniform space, and let FG(X, U) denote the abstract group Fa(X) equipped with the Graev topology, the topology generated by Graev’s extensions of all the uniformly continuous pseudometrics on (X, U). Then we have: Theorem 2.5. If (X, U)is a uniform space, then χ(FG(X, U)) = χ(A(X, U)). Proof. By Theorem 2.3, χ(A(X, U)) = D(P(X, U),≤). Now if {dα:α∈A}is a dominating family of pseudometrics in (P(X, U),≤), then it is clear that the corresponding family of Graev extensions {c dα:α∈A}on the group Fa(X) induces an open base at the identity, giving χ(FG(X, U)) ≤χ(A(X, U)). Conversely, the natural continuous homomorphism from FG(X, U)) onto A(X, U) is a quotient, from which the inequality χ(A(X, U)) ≤χ(FG(X, U)) follows, giving the result.  In particular, if Xis an arbitrary topological space and FG(X) denotes the group Fa(X) equipped with the Graev topology, then we have: 22 P. Nickolas and M. Tkachenko Corollary 2.6. χ(FG(X)) = χ(A(X)). Second, as we have noted, it is well known [13, 20] that Graev’s topology is the free topology on a free abelian topological group A(X), and Theorem 2.1 extended this to the case of a free abelian topological group A(X, U). On the other hand, it has been noted more than once [13, 17] that the fact that a certain family of (uniformly) continuous pseudometrics are sufficient together to define the topology (or uniformity) of Xdoes not imply in general that the corresponding family of Graev extensions defines the free topology. This is clear, for example, from the observation of Graev [5] noted at the start of the first section: since the free abelian topological group A(X) on a Tychonoff space Xis metrizable only when Xis discrete, the metrizability of Ximplies the metrizability of A(X) only when Xis discrete. We can adapt the proof of Theorem 2.3 to obtain a necessary and sufficient condition on a family of uniformly continuous pseudometrics under which their Graev extensions do indeed define the free topology. The proof is essentially a recapitulation of the second paragraph of the proof of Theorem 2.3, and we omit the details. Theorem 2.7. Let (X, U)be a uniform space and let Q ⊆ P(X, U). Then the collection of open sets {g∈A(X, U) : b dA(g, 0) < ǫ},for d∈ Q and ǫ > 0, is a base at 0for the topology of the free abelian topological group A(X, U)if and only if for every ∈ P(X, U)there exists d∈Qsuch that ≤2d. Let (X, U) be a uniform space. Given two sequences s=hUn:n∈ωiand t=hVn:n∈ωiin ωU, we write s≤tprovided that Un⊆Vnfor each n∈ω. It is easy to see that the correspondence s7→ N(s), where N(s) is as defined immediately before Theorem 2.2, is an order-reversing mapping of (ωU,≤) to (N(0),⊇). Since by Theorem 2.2 the family {N(s) : s∈ωU} is a base at 0 in A(X, U), we conclude that χ(A(X, U)) ≤d(ωU,≤). In fact, we will show shortly that the latter inequality is equality. Lemma 2.8. d(ωU,≤)≤D(P(X, U),≤)for every uniform space (X, U). Proof. For d∈ P(X, U) and n∈ω, put On(d) = {(x, y)∈X×X:d(x, y)≤2−n}. It is clear that the correspondence d7→ hOn(d) : n∈ωiis an order-reversing mapping of (P(X, U),≤) to (ωU,≤). Consider an arbitrary sequence hUn:n∈ωi ∈ ωU. Clearly, there exists a sequence hVn:n∈ωi ∈ ωUsuch that Vnis symmetric and 3Vn+1 ⊆Vn⊆Un for each n∈ω. By Theorem 8.1.10 of [4], we can find d∈ P(X, U) such that On(d)⊆Vnfor each n∈ω, so that the function d7→ hOn(d) : n∈ωi maps P(X, U) to a dense set in ωU. It follows, as required, that d(ωU,≤)≤ D(P(X, U),≤). (We note that [4] uses the standing assumption that all uniform spaces are Hausdorff (or separated), but it is well known and easy to see that this assumption is unnecessary for the validity of Theorem 8.1.10.)  The character of free topological groups I 23 Using the observation made before the lemma together with the lemma itself, we have χ(A(X, U)) ≤d(ωU,≤)≤D(P(X, U),≤). But by Theorem 2.3, we also have χ(A(X, U)) = D(P(X, U),≤), so we have proved both of the following results. Theorem 2.9. If (X, U)is a uniform space, then χ(A(X, U)) = d(ωU,≤). Theorem 2.10. If (X, U)is a uniform space, then d(ωU,≤) = D(P(X, U),≤). It is clear that every uniform space (X, U) satisfies w(X, U) = d(U,⊆)≤d(ωU,≤)≤d(U,⊆)ℵ0=w(X, U)ℵ0. Hence Theorem 2.9 implies the following bounds for the character of the free abelian topological group on (X, U): Corollary 2.11. Let (X, U)be a uniform space. Then w(X, U)≤χ(A(X, U)) ≤w(X, U)ℵ0. In the case of a compact Hausdorff space X, we have w(X, U) = w(X), so the bounds can be simplified as follows. Corollary 2.12. If Xis a compact Hausdorff space, then w(X)≤χ(A(X)) ≤w(X)ℵ0. In Theorem 2.15 and Corollary 2.16 below we present a different lower bound for the character of most free abelian topological groups. We shall see later that this bound, the cardinal d, is the exact value of the character of A(X) (and indeed of F(X)) when Xis an infinite compact metrizable space. First, we recall two useful notions. If every Gδ-set in Xis open, then X is said to be a P-space. Similarly, given a uniform space (X, U), we say that (X, U) is a uniform P-space if the intersection of countably many elements of Uis again an element of U. Note that if (X, U) is a uniform P-space, then the underlying topological space Xis a P-space. In some of the arguments which follow, it is natural to consider separately the cases when (X, U) is a uniform P-space and when (X, U) is not a uniform P- space. In fact, the character of A(X, U) in the “exceptional” case of a uniform P-space (X, U) can be dealt with simply and conclusively in the following form. Theorem 2.13. If (X, U)is a uniform P-space, then χ(A(X, U)) = w(X, U). Proof. For any uniform space (X, U), the mapping U7→ hU, U, . . .ifrom (U,⊆) to (ωU,≤) is an order-preserving embedding, and if (X, U) is also a uniform P-space, then Uis mapped to a dense set in ωU, since for an arbitrary hU0, U1,...i ∈ ωUwe have hU, U, . . .i ≤ hU0, U1,...i, where U=Tn∈ωUn∈ U. It follows that d(U,⊆) = d(ωU,≤), and then the conclusion that χ(A(X, U)) = w(X, U) follows from Theorem 2.9.  We now turn to the “usual” case when (X, U) is not a uniform P-space. 30 P. Nickolas and M. Tkachenko We only check (2), (3) and (4), since (1) is immediate from Lemma 3.5. Note that by definition, the set WPis symmetric for each P∈T(X, U), and so is Osfor each s∈NT(X, U). Let U∈Σ be arbitrary. Then U=Osfor some s∈NT(X, U), say s=hPn:n∈Ni. Let us check (2). By Lemma 3.5, we can find a sequence t=hQn:n∈Ni ∈ NT(X, U) such that WQn⊆WP2n−1∩WP2nfor each n∈N. We claim that O−1 t·Ot⊆Os. Thus, we take m, n ∈Nand π∈Smand ∈Sn, and show that WQπ(1) ···WQπ(m)WQ(1) ···WQ(n)⊆Os. In fact, however, it suffices to assume that m=nhere, since each WQpcontains e. Thus, let n∈Nand π,  ∈Sn. Define σ∈S2nby σ(i) = 2π(i) if 1≤i≤nand σ(i) = 2(i−n)−1 if n < i ≤2n. Then from our definition of σand tit follows that WQπ(1) ···WQπ(n)WQ(1) ···WQ(n) ⊆WPσ(1) ···WPσ(n)WPσ(n+1) ···WPσ(2n)⊆Os. This along with (3.3) implies that O−1 t·Ot=Ot·Ot⊆Os, as claimed, and the set V=Ot∈Σ is as required. To verify (3), take an arbitrary g∈U=Os. Then g∈WPπ(1) ···WPπ(k) for some k∈Nand π∈Sk. Put Qn=Pn+kfor each n∈Nand consider t=hQn:n∈Ni. Then t∈NT(X, U) and the set V=Otsatisfies V·g⊆U. Indeed, for n∈Nand σ∈Sn, define ∈Sn+kby (i) = σ(i) + kif i≤nand (i) = π(i−n) if n < i ≤n+k. Then WQσ(1) ···WQσ(n)·g⊆WQσ(1) ···WQσ(n)WPπ(1) ···WPπ(k) =WP(1) ···WP(n+k) ⊆Os, so that Ot·g⊆Osor, equivalently, V·g⊆U. The verification of (4) is similar. Let gbe an arbitrary element of Fa(X). By Lemma 3.5, for every n∈Nthere exists Qn∈T(X, U) such that g−1·WQn·g⊆ WPn. Put t=hQn:n∈Ni. Then t∈NT(X, U) and g−1·Ot·g⊆Os. Indeed, if n∈Nand π∈Sn, then we have g−1·WQπ(1) · · · WQπ(n)·g= (g−1·WQπ(1) ·g)···(g−1·WQπ(n)·g) ⊆WPπ(1) ···WPπ(n) ⊆Os. This implies that g−1·Ot·g⊆Os, and so the set V=Ot∈Σ is as required. We conclude, therefore, that Σ is a base for a group topology T∗on Fa(X). This proves (a). (b) Let Obe an arbitrary neighborhood of ein F(X, U). Choose a sequence hVn:n∈ωiof open symmetric neighborhoods of ein F(X, U) such that V0⊆Oand V3 n+1 ⊆Vnfor each n∈ω. By Lemma 3.3, for every n∈Nthere The character of free topological groups I 31 exists Pn∈T(X, U) such that WPn⊆Vn. Note that if n∈Nand π∈Sn, then Vπ(1) ···Vπ(n)⊆V0by Lemma 1.3 of [21]. This immediately implies that WPπ(1) ···WPπ(n)⊆Vπ(1) ···Vπ(n)⊆V0, whence it follows that Os⊆V0⊆O, where s=hPn:n∈Ni. This proves that the topology T∗generated by the family Σ is finer than T. (c) Let V=Os, where s=hPn:n∈Ni ∈ NT(X, U) is arbitrary. Put W=[ (U)∈P1(0) U×U. Then W∈ U, and from the definition of WP1it follows that W⊆ {(x, y)∈X×X:x−1y∈WP1, xy−1∈WP1}. Since WP1⊆Os, we conclude that W⊆ {(x, y)∈X×X:x−1y∈Os, xy−1∈Os}. Therefore, the restriction of the two-sided uniformity of the group (Fa(X),T∗) to the set X⊆Fa(X) is coarser than U. The proof is complete.  Suppose that (X, U) is an ℵ0-narrow uniform space. Roughly speaking, Theorem 3.6 and Lemma 3.3 show that one has to use only countably many elements of Uto produce a basic neighborhood of the identity in F(X, U). We use this fact as well as the next two lemmas in the proof of Theorem 3.10. For any uniform space (X, U), we denote by C(U) the family of all U-uniform covers of X, and for γ, λ ∈ C(U), we write γ≺λprovided that γrefines λ. Lemma 3.7. If (X, U)is a uniform space, then each of the partially ordered sets (U,⊆)and (C(U),≺)admits an order-preserving mapping onto a dense subset of the other, and hence d(U,⊆) = d(C(U),≺). Proof. For each U∈ U, we set γU={B(x, U) : x∈X}, where we recall that B(x, U) denotes the set {y∈X: (x, y)∈U}for each x∈X. Clearly, γUis a U-uniform cover of X. It is easy to see that the mapping U7→ γUof (U,⊆) to (C(U),≺) is order-preserving, and the set {γU:U∈ U} is obviously dense in (C(U),≺). Conversely, for every γ∈ C(U), put Wγ=S{V×V:V∈γ}. It is clear that the mapping γ7→ Wγof (C(U),≺) to (U,⊆) is order-preserving. To show that {Wγ:γ∈ C(U)}is dense in (U,⊆), let U∈ U and take V∈ U which is symmetric and satisfies 2V⊆U. Then WγV=[ x∈X B(x, V )×B(x, V ), and it is easy to check that WγV⊆U, as required. The equality d(U,⊆) = d(C(U),≺) now follows from (a version of) Lemma 1.1.  32 P. Nickolas and M. Tkachenko Suppose that s=hγn:n∈ωiand t=hλn:n∈ωiare two sequences of U-uniform covers of X, that is, that s, t ∈ωC(U). We write s≺tif γn≺λn for each n∈ω. This defines the partially ordered set (ωC(U),≺), and the next result follows directly from Lemma 3.7 and Theorem 2.10. Corollary 3.8. d(ωC(U),≺) = d(ωU,≤) = D(P(X, U),≤)for every uniform space (X, U). Our next step is to show that the difference between the characters of the groups A(X, U) and F(X, U) cannot be too big for any ℵ0-narrow uniform space (X, U) (see Theorem 3.10). First, we deal with the special case when the uniform space has a countable base. Technically, this is the most difficult part of the work. Lemma 3.9. Let an ℵ0-narrow uniform space (X, U)satisfy w(X, U)≤ ℵ0. Then χ(F(X, U)) ≤d. Proof. (I) Let {Un:n∈ω}be a countable base for the uniformity U. Since (X, U) is ℵ0-narrow, we can choose a sequence Γ = hγn:n∈ωiof countable U-uniform covers of Xsuch that γn+1 ≺γnand S{V×V:V∈γn} ⊆ Un for each n∈ω. Note that the space (X, U) is pseudometrizable (and hence paracompact, using the term without the assumption of separation which most authors include in the definition of paracompactness), and so each cover γn can be additionally chosen to be locally finite. We can further assume that if U∈γmand V∈γnwhere m > n, then Udoes not properly contain V, that is, that U⊇Vimplies U=V. (II) We define an order ≤ton the collection T(X, U) of U-covering trees by specifying that for P, Q ∈T(X, U), we have P≤tQif WP⊇WQ(see equation (3.1)). Clearly, ≤tis a quasi-order on T(X, U), though not a partial order. Denote by T(Γ) ⊆T(X, U) the set of all U-covering trees Pwith the property that γP∈Γ and γP(x)∈Γ for each x∈P(see (iii) and (iv) of Definition 3.2). Also, set γ∗=Sn∈ωγn. Then our definition of the sequence Γ = hγn:n∈ωiimplies that every V∈γ∗is contained only in finitely many distinct elements of γ∗. Claim 1. T(Γ) is a dominating subset of (T(X, U),≤t). For subsets U1,...,Un−1, Unof X, we write bπ(U1,...,Un−1, Un) = (U1,...,Un−1) if n≥2, and we write π(U1,...,Un−1, Un) = Un if n≥1. To prove the claim, let P∈T(X, U). We construct a tree Qas follows. Since γPis a U-uniform cover of X, there is δ∈Γ such that δ≺γP. We set Q(0) = {x∈(γ∗)1:π(x)∈δ}. Since δ≺γP, we can choose a function p0:Q(0) →P(0) such that π(x)⊆π(p0(x)) for all x∈Q(0). Now for each x∈Q(0), we pick δx∈Γ such δx≺γP(p0(x)). Then we put Q(1) = {y∈ The character of free topological groups I 33 (γ∗)2:x=bπ(y)∈Q(0), π(y)∈δx}. Since δx≺γP(p0(x)) for each x∈Q(0), we can choose a function p1:Q(1) →P(1) such that bπ(p1(y)) = p0(bπ(y)) and π(y)⊆π(p1(y)) for all y∈Q(1). Now for each y∈Q(1), we pick δy∈Γ such δy≺γP(p1(y)), and then put Q(2) = {z∈(γ∗)3:y=bπ(z)∈Q(1), π(z)∈δy}. Continuing in this way, we finally obtain the tree Q=Sn∈ωQ(n), ordered according to (ii) of Definition 3.2. We clearly have Q∈T(Γ). Also, if u= (U0,...,Un)∈Q(n), then, by construction, pn(u)∈P(n), and if we write pn(u) = (V0,...,Vn), say, then we have U0⊆V0,...,Un⊆Vn, so that each product in the union of the form (3.1) defining WQis contained in some product in the corresponding union defining WP, giving WQ⊆WP, and hence P≤tQ, as required to prove the claim. (III) If we define E={(V0,...,Vn) : V0,...,Vn∈γ∗, n ∈ω}, then it is clear that |E| ≤ ℵ0. Let F=Eωbe the family of all mappings from Eto ω. Define a partial order ≤eon Eas follows: for p= (U0,...,Uk), q = (V0,...,Vl)∈ E, we define p≤eqif and only if k≤land Ui⊇Vifor i= 0,...,k. Now we define a “strong” partial order ≤son F=Eω, as follows (the name and notation distinguish the relation from another that we will define later on the same set). For f, g ∈ F, we write f≤sgif f=gor if p≤eqin E always implies f(p)≤g(q) in ω. It is straightforward to verify that ≤sis indeed a partial order. We also define a partial order ≪son ω× F coordinate-wise, using the usual order ≤on ωand the order ≤son F. We define a function Π: ω× F → T(X, U). For a pair (m, f)∈ω× F, the U-covering tree P= Π(m, f) is defined as follows. Define the initial level of P by P(0) = {(U) : U∈γm}. Suppose that we have defined P(n)⊆(γ∗)n+1 for some n∈ω. Given an element p= (U0,...,Un)∈P(n), put p+={(U0,...,Un, V ) : V∈γf(p)} and then set P(n+ 1) = [{p+:p∈P(n)}. To finish the construction, we put P=Sn∈ωP(n) and define the partial order ≤on Pas in Definition 3.2. Note that the tree P= Π(m, f) is in fact an element of T(Γ) ⊆T(X, U). Claim 2. The mapping Πis order-preserving as a mapping from (ω×F,≪s) to (T(X, U),≤t). Thus, we suppose that m, n ∈ωand m≤nand that f, g ∈ F and f≤sg, and we show that WΠ(n,g)⊆WΠ(m,f). Indeed, by (3.1), WΠ(n,g)is the union over all kof all the sets of the form V−ε0 0·V−ε1 1···V−εk k·Vεk k···Vε1 1·Vε0 0, where V0∈γn, V1∈γg(q1), V2∈γg(q2), . . . , Vk∈γg(qk), 34 P. Nickolas and M. Tkachenko where q1= (V0), q2= (V0, V1), . . . , qk= (V0, V1,...,Vk−1), and where ε0, ε1, . . . , εk=±1. Fix one such set. Since γnrefines γm, there exists U0∈γmsuch that V0⊆U0. Put p1= (U0). Then from f≤sgit follows that f(p1)≤g(q1). Hence, γg(q1)refines γf(p1), so there exists U1∈ γf(p1)such that V1⊆U1. From f≤sgit follows that f(p2)≤g(q2), where p2= (U0, U1). Again, γg(q2)refines γf(p2), so there exists U2∈γf(p2)such that V2⊆U2. Continuing this way, we finally obtain Uk∈γf(pk)such that Vk⊆Uk, where pk= (U0, U1,...,Uk−1). Clearly, pk∈Π(m, f), so that the set U−ε0 0·U−ε1 1···U−εk k·Uεk k···Uε1 1·Uε0 0is a summand in the union of the form (3.1) corresponding to WΠ(m,f). By construction, we have Vi⊆Uifor each i= 0,...,k, whence it follows that V−ε0 0· · · V−εk k·Vεk k···Vε0 0⊆U−ε0 0···U−εk k·Uεk k···Uε0 0. We conclude, therefore, that WΠ(n,g)⊆WΠ(m,f), proving our claim. We claim next that Π(ω× F) = T(Γ). That Π(ω× F)⊆T(Γ) is clear. For the reverse inclusion, let P∈T(Γ). Now γP∈Γ, so that γP=γn∈Γ for some n∈N. Also, for all x∈P, we have γP(x)∈Γ, so that γP(x) = γnx∈Γ for some nx∈N. Define f:E → ω, that is, f∈ F, by setting f(x) = nxfor all x∈Pand f(x) = 0 for all x /∈P. Then it is easy to see that Π(n, f) = P, proving that Π(ω× F)⊇T(Γ), and hence our claim. We now consider the quasi-ordered sets (N(ω× F),≪∗ s) and (NT(X, U),≤∗ t), where the orders ≪∗ sand ≤∗ tare defined coordinate-wise in terms of ≪sand ≤t, respectively. We likewise consider the mapping Π∗: (N(ω× F),≪∗ s)→(NT(X, U),≤∗ t) defined coordinate-wise in terms of Π. It is immediate from what we have shown above that the mapping Π∗is order-preserving, and maps N(ω× F) to a dominating subset of (NT(X, U),≤∗ t). Thus, we conclude that (3.4) D(NT(X, U),≤∗ t)≤D(N(ω× F),≪∗ s). (IV) We wish to obtain a different expression for the right-hand side of (3.4). To this end, we define a “weak” partial order ≤won F=Eω, as follows. For f, g ∈ F, we write f≤wgif f(p)≤g(p) for each p∈ E. It is clear that ≤wis a partial order. It is also clear that f≤sgimplies f≤wg. Now we define ≪w on ω× F and then ≪∗ won N(ω× F) in exact analogy to the earlier definitions of ≪sand ≪∗ s. Claim 3. D(N(ω× F),≪∗ s) = D(N(ω× F),≪∗ w). Indeed, it is clear that D(N(ω× F),≪∗ w)≤D(N(ω× F),≪∗ s), since every dominating set in (N(ω×F),≪∗ s) remains dominating in (N(ω×F),≪∗ w). Next, we have to verify that D(N(ω× F),≪∗ s)≤D(N(ω× F),≪∗ w). For q∈ E, put E(q) = {p∈ E :p≤eq}. The character of free topological groups I 35 From our assumptions about the covers γn, it follows that E(q) is finite for every q∈ E. In fact, suppose that there exists q= (V0,...,Vl)∈ E such that E(q) is infinite. Then for some k≤l, the set Ek(q) = {p∈ E :p= (U0,...,Uk), p ≤eq} is infinite. Then for some i≤k, we can find a sequence of elements pn= (U(n) 0,...,U(n) i,...,U(n) k)∈ Ek(q) for n∈Nsuch that the sets U(n) iare distinct for all nand such that U(n) i⊇Vi for all n. If n0∈Nis such that Vi∈γn0, then U(n) imust properly contain Vi for infinitely many n, and it follows that there is n1≤n0such that infinitely many of the U(n) iare in γn1, contradicting the local finiteness of γn1. This contradiction shows that E(q) is finite, as claimed. It is clear furthermore that each E(q) is also non-empty. Now for every f∈ F and q∈ E, put e f(q) = max{f(p) : p∈ E(q)}, noting that by the argument above the value e f(q) is defined validly. Therefore, we obtain a function e f:E → ω, that is, e f∈ F. It is easy to see that the mapping f7→ e ffrom (F,≤w) to (F,≤s) is order-preserving, and from our definition of the order ≤sand the function e f, it is also easily checked that f≤se ffor each f∈ F, so that the image of the mapping is a dominating subset of (F,≤s). For ϕ∈N(ω× F), we have ϕ(n)∈ω× F for each n∈N, and we have ϕ(n) = (π1(ϕ(n)), π2(ϕ(n))), where π1and π2are the projections from ω× F into ωand F, respectively. We then define eϕ∈N(ω× F) by setting eϕ(n) = (π1(ϕ(n)),^ π2(ϕ(n))) for each n∈N. Then by the argument above, the mapping ϕ7→ eϕfrom (N(ω× F),≪∗ w) to (N(ω× F),≪∗ s) is order-preserving and has as its image a dominating set, proving the inequality D(N(ω×F),≪∗ s)≤D(N(ω× F),≪∗ w), and hence our claim. (V) We can now complete the proof of the lemma. Equation (3.3) (immediately preceding Theorem 3.6) defines for us a mapping s7→ Osfrom NT(X, U) to N(e), the family of open neighborhoods of the identity ein F(X, U). Further, it is immediate from the relevant definitions that this mapping is order-reversing from (NT(X, U),≤∗ t) to (N(e),⊆), where ≤∗ tis defined coordinate-wise in terms of ≤t. Moreover, rephrased in this terminology, Theorem 3.6 states that the mapping has a dense subset of (N(e),⊆) as its image. Hence we have (3.5) d(N(e),⊆)≤D(NT(X, U),≤∗ t). 36 P. Nickolas and M. Tkachenko In addition, we have obvious order-isomorphisms as follows: (N(ω× F),≪∗ w)∼ =(Nω, ≤)×(NF,≤∗ w) ∼ =(Nω, ≤)×(N×E ω, ≤) ∼ =(ωω, ≤)×(ωω, ≤) ∼ =(ωω, ≤)(3.6) (where ≤∗ wis the coordinate-wise extension of ≤wfrom Fto NF). Therefore, from (3.5), (3.4), Claim 3 and (3.6), we have χ(F(X, U)) = d(N(e),⊆) ≤D(NT(X, U),≤∗ t) ≤D(N(ω× F),≪∗ s) =D(N(ω× F),≪∗ w) =D(ωω, ≤) =d. This finishes the proof.  The conclusion of the next theorem will be strengthened in Theorem 3.15. Theorem 3.10. If (X, U)is an ℵ0-narrow uniform space, then χ(F(X, U)) ≤ d·χ(A(X, U)). Proof. According to Theorem 2.3 and Theorem 2.10, we have (3.7) χ(A(X, U)) = D(P(X, U),≤) = d(ωU,≤), where P(X, U) is the family of uniformly continuous pseudometrics on (X, U) bounded by 1. Therefore, all we have to verify is that (3.8) χ(F(X, U)) ≤d·d(ωU,≤). Denote by Sthe family of all sequences V=hUn:n∈ωi ∈ ωUsuch that 3Un+1 ⊆Unfor each n∈ω. It is clear that Sis dense in (ωU,≤), whence d(S,≤) = d(ωU,≤). Choose a dense subset Dof (S,≤) of the minimal cardinality. Then Dis also dense in (ωU,≤). Note that the set of terms of the sequence Vis a base for a (non-Hausdorff) uniformity e Von Xfor each V ∈ S, and hence Lemma 3.9 implies that χ(F(X, e V)) ≤d. So, for every V ∈ D, we can find a base B(V) at the identity in F(X, e V) such that |B(V)| ≤ d. It is clear that the family B=SV∈D B(V) satisfies |B| ≤ d·d(ωU,≤), and we claim that Bis a base at the identity in the group F(X, U). By assumption, the space (X, U) is ℵ0-narrow, so the group F(X, U) is ℵ0- narrow according to [2, Lemma 3.2] or [6]. Hence the topology of F(X, U) is generated by continuous homomorphisms to second countable topological groups (see [6] or [25, Lemma 3.7]). In other words, given a neighborhood Uof the identity in F(X, U), one can find a continuous homomorphism f:F(X, U)→ Gto a second countable topological group Gand an open neighborhood Vof the identity in Gsuch that f−1(V)⊆U. Choose a countable base {Vn:n∈ω} The character of free topological groups I 37 at the identity of Gsuch that V0=Vand V3 n+1 ⊆Vnfor each n∈ω. For n= 0,1,..., put (3.9) Un={(x, y)∈X×X:f(x)−1·f(y)∈Vn, f(x)·f(y)−1∈Vn}. Evidently 3Un+1 ⊆Unfor each n∈ω, so that V=hUn:n∈ωi ∈ S. Since Dis dense in (D,≤), we can assume that V ∈ D. Our choice of V(see (3.9)) guarantees that the restriction of fto Xis a uniformly continuous mapping of (X, e V) to (G, ∗V∗), where ∗V∗is the two-sided uniformity of the group G. Hence the homomorphism f:F(X, e V)→Gremains continuous. Take an element W∈ B(V) such that f(W)⊆V. Then W⊆f−1(V)⊆U, and hence Bis a base at the identity in F(X, U). This proves (3.8) and the theorem.  Combining Corollary 2.17 and Theorem 3.10, we obtain the following result, which will be given its final form in Theorem 3.15. Corollary 3.11. If an ℵ0-narrow Hausdorff uniform space (X, U)contains an infinite precompact set, then χ(A(X, U)) = χ(F(X, U)). Now we proceed to show, in Theorem 3.15, that the existence of an infinite precompact set in (X, U) can be omitted in the assumptions of Corollary 3.11. The main additional information we need is given in the following result. Lemma 3.12. If (X, U)is an ℵ0-narrow uniform P-space, then the group F(X, U)has a base at the identity consisting of open normal subgroups. In particular, the topology of F(X, U)is generated by Graev’s extensions of the uniformly continuous pseudometrics on (X, U). Proof. The first assertion follows from the uniform analog of [22, Th. 4]. For the second assertion, suppose that (X, U) is a uniform P-space, and let U be an open neighborhood of the identity ein F(X, U). Then there exists an open normal subgroup Vof F(X, U) with V⊆U. Consider the open cover γ={X∩xV :x∈X}of the space X. It is clear that γis a U-uniform cover. Since Vis a subgroup of F(X, U), the family γis a partition of X, i.e., every two elements of γare disjoint or coincide. Define a pseudometric on Xby setting (x, y) = 0 if x, y ∈Ofor some O∈γ, and (x, y) = 1 otherwise. Clearly, is uniformly continuous. Let bbe Graev’s extension of to the maximal invariant pseudometric on Fa(X) (see [5, Section 3]). Then the set W={g∈Fa(X) : b(g, e)<1} is an open neighborhood of ein F(X, U) by the continuity of bon F(X, U). It remains to show that W⊆V. From the fact that the pseudometric is {0,1}-valued, it is immediate from Graev’s construction that the extension bis integer-valued. We therefore have W={g∈Fa(X) : b(g, e) = 0}. 38 P. Nickolas and M. Tkachenko (Indeed, it follows that Wis an open normal subgroup of F(X, U).) Further, Graev’s construction shows straightforwardly that for g∈W, there exist (nonreduced) representations g=xε1 1···xε2n 2nand e=yε1 1···yε2n 2n of gand e, for some n∈N, some x1,...x2n, y1,...y2n∈X, and ε1,...ε2n= ±1, such that (xi, yi) = 0 for i= 1,...,2n. Now we clearly have yεi iyεi+1 i+1 =e for some i, from which it follows that xεi ixεi+1 i+1 ∈V. Set g1=xε1 1···xεi−1 i−1 and g2=xεi+2 i+2 ···xε2n 2n, and put bg=g1g2. Note that ˆg∈W. If we assume inductively that bg∈V, we also have g2g1∈Vby the normality of V, and then we have g−1 1gg1=xεi ixεi+1 i+1 g2g1∈V, from which we have g∈V, again by normality. It follows by induction that W⊆V, as required.  This allows us to extend Theorem 2.13 to the non-abelian case, assuming additionally that our uniform space is ℵ0-narrow. Theorem 3.13. If (X, U)is an ℵ0-narrow uniform P-space, then χ(F(X, U)) = w(X, U). Proof. By Lemma 3.12, the topology of F(X, U) is generated by Graev’s extensions of the uniformly continuous pseudometrics on (X, U). It follows, therefore, that χ(F(X, U)) ≤D(P(X, U),≤) = w(X, U). However, Theorem 2.3 and Lemma 3.1 together imply that D(P(X, U),≤) = χ(A(X, U)) ≤χ(F(X, U)). Combining these inequalities, we obtain the required conclusion.  In contrast to Theorem 2.13, the assumption of ℵ0-narrowness cannot be removed in Theorem 3.13, as is shown by the following example. Example 3.14. For every cardinal τ > ℵ1, there exists a Hausdorff uniform P-space (X, U) such that w(X, U) = ℵ1< τ < χ(F(X, U)). Indeed, let X=L⊕Dbe the topological sum of the one-point Lindel¨ofication Lof a discrete space Ywith |Y|=ℵ1and a discrete space Dof cardinality τ > ℵ1. Denote by x0the unique non-isolated point of L(and of X). Then a base of open neighborhoods of x0in L(and in X) consists of the sets L\C, where Cis an arbitrary countable subset of Y. Since |Y|=ℵ1, it is easy to see that χ(x0, L) = χ(x0, X) = ℵ1. Let Ube the fine uniformity of the space X. Then a basic entourage of the diagonal ∆ in X×Xhas the form UC={(x, y)∈L×L:x, y ∈L\C} ∪ ∆, where C⊆Yis countable. Therefore, w(X, U) = ℵ1. Let us show that χ(F(X, U)) ≥τ. For every a∈D, put La=a−1x−1 0La, and consider the subspace Z= Sa∈DLaof F(X, U)∼ =F(X). Apply an argument similar to that in [1, Prop. 3.2] to show that Zis homeomorphic to the fan V(τ) obtained from the topological sum Tof τcopies of Lby identifying to a point the set T′ of all non-isolated points of T. Since each of the τdistinct spines of the fan V(τ) is homeomorphic to L, a straightforward diagonal argument implies that τ < χ(e, Z)≤χ(F(X)). The character of free topological groups I 39 Finally, we have the main result of this section. Theorem 3.15. The equality χ(A(X, U)) = χ(F(X, U)) holds for every ℵ0- narrow uniform space (X, U). Proof. If (X, U) is a uniform P-space, then the required equality is given by Theorems 2.13 and 3.13, while if (X, U) is not a uniform P-space, then the equality follows from Theorems 2.15 and 3.10 and Lemma 3.1.  The above theorem has several applications; the following four are immediate. Corollary 3.16. χ(A(X)) = χ(F(X)) for every ℵ0-narrow space X. Using Corollary 2.22, we have in particular: Corollary 3.17. If Xis an infinite compact metrizable space, then χ(F(X)) = d. Corollary 3.18. Suppose that a space Xcontains a dense Lindel¨of subspace. Then χ(A(X))=χ(F(X)). Proof. It is easy to see that the space Xis ℵ0-narrow. Indeed, let Ybe a dense Lindel¨of subspace of X. If γis a discrete family of non-empty open sets in X, then the family µ={U∩Y:U∈γ}is a discrete family of non-empty open subsets of Y. However, every such family of subsets of Yis countable, so |γ|=|µ| ≤ ℵ0. Now the necessary conclusion follows from Corollary 3.16.  Clearly, every space of countable cellularity is ℵ0-narrow. Therefore, we have the following. Corollary 3.19. If a space Xhas countable cellularity, then χ(A(X)) = χ(F(X)). We believe that Corollary 3.21 below compared with Theorem 2.3 or Corollary 2.4 gives a more comprehensive expression for the character of the groups F(X) and A(X) on a Lindel¨of space X. It is based on a simple relation between dense subsets of (ωUX,≤) and the character of the diagonal in (X×ω)2, where UXis the fine uniformity of the space Xand the set ωcarries the discrete topology. We recall that if s={Un:n∈ω}and t={Vn:n∈ω}are elements of ωU, where Uis a uniformity on some set, then s≤tmeans that Un⊆Vnfor each n∈ω. Lemma 3.20. If Xis a paracompact topological space and UXis the fine uniformity on X, then d(ωUX,≤) = χ∆(X×ω). Proof. We can assume that Xis not discrete—otherwise, the equality becomes trivial. Denote by UYthe fine uniformity on Y≡X×ω. The paracompactness of Ximplies that every neighborhood of the diagonal ∆Xin X2belongs to the fine uniformity UX, and similarly for the diagonal ∆Yin Y2. The family of all neighborhoods of ∆Yin Y2contains a base which can be naturally identified