The character of free topological groups II

@Applied General Topology c Universidad Polit´ecnica de Valencia Volume 6, No. 1, 2005 pp. 43-56 The character of free topological groups II Peter Nickolas and Mikhail Tkachenko 1 Abstract. A systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. In the first case, it is shown that the characters of the free and the free abelian topological groups on Xare both equal to the “small cardinal” dif Xis compact and metrizable, but also, more generally, if Xis a nondiscrete kω-space all of whose compact subsets are metrizable, or if Xis a non-discrete Polish space. An example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. In the case of a compact space X, an explicit formula is derived for the character of the free topological group on X involving no cardinal invariant of Xother than its weight; in particular the character is fully determined by the weight in the compact case. This paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions. 2000 AMS Classification: Primary 22A05, 54H11, 54A25; Secondary 54D30, 54D45 Keywords: Free (abelian) topological group, entourage of the diagonal, character, 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. 44 P. Nickolas and M. Tkachenko 1. Introduction In a previous paper [11], we investigated the topological character of free and free abelian topological groups. The results obtained were for the free groups on uniform spaces, with applications to the free groups on topological spaces deduced as appropriate. Also, the principal results were obtained without the imposition of strong uniform or topological conditions on the given spaces, though numerous corollaries were derived at various points for metrizable spaces, compact spaces and other classes of spaces. In this sequel to [11], we specifically investigate the characters of the free and free abelian topological groups on metrizable spaces and on compact spaces, and on certain closely related spaces, obtaining more detailed information in both cases than was available in [11]. In the metrizable case, we show that the equality χ(A(X)) = χ(F(X)) = d holds if Xis a compact metrizable space (as was already observed in [11]), but also if Xis a non-discrete kω-space all compact subsets of which are metrizable (Theorem 2.9), or if Xis a non-discrete Polish space (Corollary 2.12). On the other hand, there exists a zero-dimensional separable metric space Xsuch that χ(A(X)) = χ(F(X)) = c(Example 2.18). If Xis a metrizable space in which the subset of all non-isolated points is compact and non-empty, then χ(A(X)) = d(Theorem 2.7), but under the same hypotheses the character χ(F(X)) may be arbitrarily large (Example 2.8). In the case of a compact space X, our main result gives an explicit formula for χ(F(X)) involving no cardinal invariant of Xother than the weight (Theorem 3.5), showing in particular that the character is fully determined by the weight in the compact case. If the weight w(X) of Xis at least c, then our result implies that χ(A(X)) = χ(F(X)) = w(X)ℵ0. Our notation and terminology here are as in [11]. From time to time, results from [11] will be used here, and again the reader is referred to the source for these, though on occasion we quote them here for convenience. 2. Free groups on metrizable spaces As usual, w(X, U) denotes the weight of the Hausdorff uniform space (X, U), where by the weight we mean in all cases the least cardinal of a base of U, so that w(X, U) = ℵ0implies in particular that the family of all uniform entourages of the diagonal in X2does not have a minimal element. (A similar convention applies to our usage of other cardinal invariants.) The principal results of [11] on the characters of the free groups on (pseudo) metrizable spaces are the following three (see Theorem 2.21 and Corollaries 2.22 and 3.16, respectively). Theorem 2.1. Let (X, U)be an arbitrary uniform space with w(X, U) = ℵ0. Then χ(A(X, U)) = d. Corollary 2.2. If Xis an infinite compact metrizable space, then χ(A(X)) = d. The character of free topological groups II 45 Following [11], we call a space X ω-narrow (equivalently, pseudo-ω1-compact) if every locally finite family of open sets in Xis countable. It is clear that all Lindel¨of and all separable spaces are ω-narrow. Corollary 2.3. χ(A(X)) = χ(F(X)) for every ω-narrow space X. From Corollaries 2.2 and 2.3, we have: Corollary 2.4. The equalities χ(F(X)) = χ(A(X)) = dhold for every infinite compact metrizable space X. Brief comments were made in [11] about the character of a free topological group when equipped with the Graev topology rather than the free topology. We make one further such observation. Recall that if Xis a topological space, then FG(X) denotes the abstract free group Fa(X) over Xtopologized with Graev’s topology, that is, the finest invariant group topology on Fa(X) coarser than the topology of F(X). From Corollary 2.6 of [11] it follows that χ(FG(X)) = χ(A(X)), for every space X. This equality combined with our Corollary 2.3 implies the following result. Theorem 2.5. If Xis an ω-narrow space, then χ(FG(X)) = χ(F(X)). A further result from [11] is the following (Theorem 2.23). In it, we use χ∆(X) to denote the character of the diagonal ∆ in X×X. Theorem 2.6. If a Tychonoff space Xsatisfies χ∆(X)≤ ℵ0, then either X and A(X)are discrete or χ(A(X)) = d. Starting from the above results, we develop here a sequence of new results which give more detailed information on the characters of the free and free abelian topological groups on metrizable spaces and certain spaces closely related to metrizable spaces. Theorem 2.7. If Xis a metrizable space and the set X′of all non-isolated points of Xis compact and non-empty, then χ(A(X)) = d. Proof. We claim that χ∆(X) = ℵ0. Indeed, let dbe a metric on Xwhich induces the topology of X. For every x∈Xand ε > 0, denote by B(x, ε) the open ball with center at xand radius εwith respect to d. If n∈N, we put Un= ∆ ∪[{B(x, 1/n)×B(x, 1/n) : x∈X′}, where ∆ is the diagonal in X×X. It is easy to see that the sets Unform a base at the diagonal ∆ in X×X, which proves our claim. Now the desired conclusion follows from Theorem 2.6.  It is interesting to note that the non-abelian analog of Theorem 2.7 fails, as the next example shows. Example 2.8. The character of the free topological group F(X) on a metrizable space Xwith a single non-isolated point can be arbitrarily large (while the character of A(X) is equal to dby Theorem 2.7). 46 P. Nickolas and M. Tkachenko Indeed, let X=C⊕Dbe the topological sum of a non-trivial convergent sequence Cwith limit point x0∈Cand a discrete space Dof an infinite cardinality τ. For every a∈D, put Ca=a−1x−1 0Ca, and consider the subspace Y=Sa∈DCaof F(X). As is shown in [1], Yis homeomorphic to the Fr´echet–Urysohn fan V(τ) of cardinality τwith vertex at the identity eof the group F(X). A straightforward diagonal argument shows that τ < χ(e, Y )≤χ(F(X)). It turns out that Corollary 2.4 remains valid in a more general case. Let us say that a space Xwith a kω-decomposition X=Sn∈ωXnis a kmω-space if each Xnis metrizable. Equivalently, a kmω-space is a kω-space all compact subsets of which are metrizable. Theorem 2.9. Let Xbe a non-discrete kmω-space. Then χ(A(X)) = d= χ(F(X)). Proof. By assumption, there exists a kω-decomposition X=Sn∈ωXn, where each Xnis compact and metrizable. Clearly Xnis non-discrete for some n∈ω, for otherwise Xwould be discrete. Therefore, Xcontains infinite compact subsets (convergent sequences), so Corollary 2.18 of [11] and our Corollary 2.3 together imply that d≤χ(A(X)) = χ(F(X)). It remains to verify that χ(F(X)) ≤d. Denote by Ythe one-point compactification of the topological sum X′= ⊕n∈ωX′ n, where X′ n=Xn× {n}for each n∈ω. It is easy to see that the infinite compact space Yis metrizable, and so we have χ(F(Y)) = dby Corollary 2.4. Choose an element a∈Yand put Z=Sn∈ωan·X′ n⊆F(Y). Then Z∩Fn+1(Y) = Sn k=0 ak·X′ k, so the intersection Z∩Fn+1(Y) is closed in Fn+1(Y) for each n∈ω. By Graev’s theorem in [4], Sn∈ωFn(Y) is a kωdecomposition of the group F(Y), so Zis closed in F(Y). Therefore, F(Y) contains a subgroup topologically isomorphic to F(Z) (see [8, Th. 1]), and hence χ(F(Z)) ≤χ(F(Y)). Let f:X′→Xbe the mapping defined by f(y, n) = y for all y∈Xn,n∈ω. Since X=Sn∈ωXnis a kω-decomposition of X, the mapping fis quotient. Clearly, f(X′) = X. Define a mapping g:X′→Zby g(x) = anxfor each x∈X′ n,n∈ω. It is easy to see that gis a homeomorphism, so that the mapping h=f◦g−1:Z→Xis a quotient. Hence the extension of hto a homomorphism b h:F(Z)→F(X) is continuous and open. We therefore conclude that χ(F(X)) ≤χ(F(Z)) ≤χ(F(Y)) = d. By Corollary 2.3, χ(F(X)) = χ(A(X)) for each separable metrizable space X. Our next task is to calculate the values χ(A(Q)) = χ(F(Q)) and χ(A(Rω)) = χ(F(Rω)). Here we show that all these cardinals are equal to d. This will follow from a more general result: If Xis a non-discrete separable metrizable space which is absolutely Gδ,Fσor Gδσ, then χ(A(X)) = χ(F(X)) = d(see Theorem 2.11). In particular, χ(A(X)) = χ(F(X)) = dfor every non-discrete Polish space X. Lemma 2.10. Let Xbe a non-discrete separable metrizable space such that X×ω∼ =X. Then χ(A(X)) = χ(F(X)) = χ∆(X). The character of free topological groups II 47 Proof. Since Xis Lindel¨of and X×ω∼ =X, Corollary 3.21 of [11] gives us χ(F(X)) = χ(A(X)) = χ∆(X×ω) = χ∆(X), as required.  We recall that a separable metrizable space Xis called absolutely Gδ,Fσor Gδσ if Xis of type Gδ,Fσor Gδσ, respectively, in some (equivalently, every) metrizable compactification of X. Theorem 2.11. Let Xbe a non-discrete separable metrizable space. If Xis absolutely Gδ,Fσor Gδσ, then χ(A(X)) = χ(F(X)) = d. Proof. Since χ(F(X)) = χ(A(X)) by Corollary 2.3, it suffices to verify that χ(A(X)) = d. If Xis absolutely Gδ, then it is a perfect image of a closed subspace Kof the irrationals P∼ =Nω, by (C) on page 144 of [2]. Let f:K→X be the corresponding perfect mapping. Then fextends to a continuous open homomorphism b f:A(K)→A(X), so that χ(A(X)) ≤χ(A(K)). Since Kis closed in the separable metrizable space P, every continuous (pseudo)metric on Kextends to a continuous (pseudo)metric on P, and Theorem 1.2.9 of [10] and Lemma 4 of [14] imply that A(K) is topologically isomorphic to a subgroup of A(P). Hence χ(A(K)) ≤χ(A(P)). Note that P∼ =P×ω. Since Pis absolutely Gδ, Theorem 8.13 of [2] implies that χ∆(P) = d, and hence χ(A(P)) = dby Lemma 2.10. Since Xis non-discrete (hence contains infinite compact subsets), from Corollary 2.18 of [11] it follows that d≤χ(A(X)) ≤χ(A(K)) ≤χ(A(P)) = d. Similarly, if Xis absolutely Fσor Gδσ, then so is the product X×ω, and [2, Th. 8.13] implies that χ∆(X×ω) = d. Since X×ω×ω∼ =X×ω, we can apply Lemma 2.10 to conclude that χ(A(X×ω)) = d. Clearly, Xis a continuous open image of X×ω, which immediately implies that χ(A(X)) ≤χ(A(X×ω)) = d. Since Xis non-discrete, an application of Corollary 2.18 of [11] finishes the proof.  The above theorem implies, in particular, that χ(F(Q)) = χ(F(K×Q)) = d for every compact metrizable space K. Since complete separable metrizable (≡Polish) spaces are absolutely Gδ, we obtain the following. Corollary 2.12. If Xis a non-discrete Polish space, then d=χ(A(X)) = χ(F(X)). We therefore have, for example, χ(F(P)) = χ(F(Rω)) = d. However, our results leave the following open problems. Problem 2.13. Does the inequality χ(F(X)) ≤dhold for any absolutely Borel (analytic)separable metrizable space X? Problem 2.14. Let Xbe an ω-narrow space such that w(X, U)≤d, where U is the fine uniformity of X. Is then χ(F(X)) ≤d? What if X×ω∼ =X? Since the group A(X) on a separable metrizable space Xis separable, its character does not exceed c. On the other hand, χ(A(X)) ≥dfor a non-discrete metrizable space Xby Corollary 2.18 of [11]. Our aim is to show that there 48 P. Nickolas and M. Tkachenko exists in ZFC a separable metrizable space Xsatisfying χ(A(X)) = χ(F(X)) = c. First, we study the character of Xin its ˇ Cech–Stone compactification βX, and relate it with the character of the diagonal ∆Xin the product X×X. The straightforward proof of the next lemma is left to the reader. Lemma 2.15. Let bX be an arbitrary compactification of a Tychonoff space X. Then χ(X, bX) = χ(X, βX). The following result generalizes Corollary 15 of [12]. Lemma 2.16. If a space Xis paracompact, then χ(X, βX)≤χ∆(X). Proof. Let Bbe a base for the diagonal ∆Xin X2such that |B| =χ∆(X). For every U∈ B, choose an open cover γ=γUof Xsuch that [{V×V:V∈γ} ⊆ U. For every open set Vin X, put e V=βX \clβX(X\V). It is clear that e V is open in βX and e V∩X=V. In particular, Vis dense in e V, and hence e V⊆clβX (V). If U∈ B, consider the family eγU={e V:V∈γU}and the set WU=SeγU. Then WUis open in βX and X⊆WUfor each U∈ B. We claim that the family λ={WU:U∈ B} is a base for Xin βX. Let Wbe an arbitrary open neighborhood of Xin βX. Put F=βX \W and consider the closed subset P=X×Fof X×βX. Denote by ∆βX the diagonal in (βX)2. Evidently, ∆X= (X×βX)∩∆βX is closed in X×βX and P∩∆X=∅. Since the product X×βX is normal (see [3, Th. 5.1.38]), we can find disjoint open sets Oand O′in X×βX such that ∆X⊆Oand P⊆O′. Then there exists U∈ B such that U⊆O∩(X×X). Take an arbitrary element V∈γUand pick a point x∈V. Since V×V⊆U⊆O, we have {x} × V⊆O, and hence {x} × clβX(V)⊆clX×βX (O). By our choice, the sets Oand O′are disjoint and P=X×F⊆O′. Therefore, clβX (V)∩F=∅. Since e V⊆clβX (V) and F=βX \W, we conclude that e V⊆W. This inclusion holds for each V∈γU, so WU=SeγU⊆W. This proves our claim. Finally, from our definition of λit follows follows that |λ| ≤ |B|, and hence χ(X, βX)≤χ∆(X).  Let Xbe a paracompact space. Then every open neighborhood of the diagonal ∆Xin X2belongs to the fine uniformity Uon X. This implies, in our notation, that w(X, U) = χ∆(X). Since the free abelian topological group on Xis precisely the free abelian topological group on the uniform space (X, U), Corollary 2.11 of [11] implies the following result. Corollary 2.17. Let Xbe a paracompact space. Then χ∆(X)≤χ(A(X)). By Corollary 2.4, the character of the groups F(X) and A(X) on every infinite compact metrizable space Xis equal to the cardinal d, which is consistently less than c(see [2, 15]). The equalities χ(F(X)) = χ(A(X)) = dremain valid for every non-discrete Polish space X(see Corollary 2.12). In the general case, the situation is different. The character of free topological groups II 49 Example 2.18. There exists a zero-dimensional separable metric space Xsuch that the groups F(X) and A(X) both have character equal to c. Indeed, let Xand Ybe disjoint Bernstein subsets of the real line Rsuch that R=X∪Yand |X|=|Y|=c. Then compact subsets of Xand Yare at most countable and both Xand Yare dense in R. Hence Xis a zero-dimensional separable metric space. Clearly, the groups F(X) and A(X) are also separable, so their respective characters do not exceed c. Since χ(A(X)) = χ(F(X)), by Corollary 2.3, it suffices to verify that χ(A(X)) = c. Denote by Zthe one-point compactification of R. Then Zis also a compactification of X. Therefore, χ(X, Z) = χ(X, βX)≤χ∆(X), by Lemmas 2.15 and 2.16. Since Yis a Bernstein subset of R, we have |R\U| ≤ ℵ0for every open set Uin Rcontaining X. In addition, the cardinality of Yis equal to c, so we have c≤ψ(X, R) = ψ(X, Z)≤χ(X, Z)≤χ∆(X), where we use ψto denote the pseudocharacter. This chain of inequalities and Corollary 2.17 enable us to conclude that c≤χ∆(X)≤χ(A(X)) ≤c. Thus we have χ(F(X)) = χ(A(X)) = c. It may be worth remarking that if U′is the natural metric uniformity on Xinherited from R, then (X, U′) is an ω-narrow uniform space of countably infinite weight, and Theorem 2.1 together with Corollary 2.3 imply that χ(F(X, U′)) = χ(A(X, U′)) = d. 3. Free groups on compact spaces Let Xbe a compact Hausdorff space. Then by combining Corollary 2.12 of [11] and our Corollary 2.3, we obtain the inequality w(X)≤χ(A(X)) = χ(F(X)) ≤w(X)ℵ0, which constitutes one of the main facts about the characters of the free groups on compact Hausdorff spaces derived in [11]. In this section, we apply different methods to derive a great deal more detailed information about these characters. As just noted, the difference between F(X) and A(X) mentioned in Example 2.8 disappears in the case when Xis compact. In fact, we will show that the character of these groups depends only on the weight of Xin the compact case. Our proof of this fact requires several auxiliary results. First, we recall some definitions and notation used in [11]. 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 apartially 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 50 P. Nickolas and M. Tkachenko 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, ≥). For a space X, denote by MXthe family of all continuous mappings of X onto separable metrizable spaces. Equivalently, since every separable metrizable space is homeomorphic to a subspace of Iω, where I= [0,1], we can consider MXas a family of continuous mappings of Xto Iω. If f:X→Y and g:X→Zare elements of MX, we say that frefines gor, in symbols, f≺gif there exists a continuous mapping ϕ:Y→Zsuch that g=ϕ◦f. Also, following [11], denote by PXthe family of all continuous pseudometrics on the space Xbounded by 1. For d1, d2∈ PX, we write d1≤d2if d1(x, y)≤d2(x, y) for all x, y ∈X. This gives us the quasi-ordered set (MX,≺) and the partially ordered set (PX,≤). Lemma 3.1. The equality D(PX,≤) = d·d(MX,≺)is valid for every infinite compact Hausdorff space X. Proof. By Corollary 2.4 of [11] and our Corollary 2.2, we have D(PY,≤) = χ(A(Y)) ≤dfor every compact metrizable space Y. Let dbe a continuous pseudometric on a given compact space X, with d≤1. There exists a continuous mapping f:X→Yonto a compact metrizable space Yand a continuous metric on Ysuch that d(x, y) = (f(x), f(y)) for all x, y ∈X. Since D(PY,≤)≤d, we can find a dominating family Dfin (PY,≤) satisfying |Df| ≤ d. For every κ∈Df, define a continuous pseudometric eκon Xby eκ(x, y) = κ(f(x), f(y)) for all x, y ∈X. Then e Df={eκ:κ∈Df} ⊆ PX for each f∈ MX. Let Nbe a dense subset of (MX,≺) satisfying |N | = d(MX,≺). It is easy to see that the family D=S{e Df:f∈ N} is dominating in (PX,≤), so that D(PX,≤)≤ |D| ≤ d· |N| =d·d(MX,≺). Conversely, let Dbe a dominating family in PXsuch that |D| =D(PX,≤). Since Xis compact, for every d∈ D we can find a continuous mapping f=fd of Xonto a compact metrizable space Yand a continuous metric on Ysuch that d(x, y) = (f(x), f(y)) for all x, y ∈X. Then the set {fd:d∈ D} is dense in (MX,≺). Indeed, let g∈ MXbe arbitrary. Then the image Z=g(X) is a compact metrizable space. Choose a metric ∈ PZwhich generates the topology of Zand define a continuous pseudometric eon X by e(x, y) = (g(x), g(y)) for all x, y ∈X. Clearly e∈ PX, so there exists d∈ D such that e≤d. An easy verification shows that fd≺g, and hence the family {fd:d∈ D} is dense in (MX,≺). We conclude therefore that d(MX,≺)≤ |D| =D(PX,≤). Finally, since d≤χ(A(X)) = D(PX,≤) by Corollaries 2.4 and 2.18 of [11], we apply the inequalities just proved to deduce that D(PX,≤)≤d·d(MX,≺)≤d·D(PX,≤) = D(PX,≤), which implies the required equality.  Now we present a theorem which summarizes several results established earlier. The character of free topological groups II 51 Theorem 3.2. d≤χ(A(X)) = χ(F(X)) = D(PX,≤) = d·d(MX,≺)for every infinite compact space X. Proof. Combining Corollaries 2.4 and 2.18 of [11] and Lemma 3.1 of the present paper, we obtain d≤χ(A(X)) = D(PX,≤) = d·d(MX,≺). Since χ(A(X)) = χ(F(X)) by Corollary 2.3, this proves the theorem.  On occasion, the exact calculation of D(PX,≤) or d(MX,≺) for a compact space Xcan be a non-trivial task. In Theorem 3.5, we give an explicit value for the character of the groups F(X) and A(X) on an infinite compact space Xwhich avoids any reference to the quasi-ordered sets (PX,≤) or (MX,≺). Nevertheless, our proof of Theorem 3.5 will involve the set (MX,≺) in an essential way, as well as the family CZ(X) of all cozero-sets in X. As usual, we denote by [CZ(X)]≤ωthe collection of all countable subfamilies of CZ(X). Given γ, λ ∈[CZ(X)]≤ω, we write γ≪λif every element of λis the union of a subfamily of γ. This gives rise to the quasi-ordered set ([CZ(X)]≤ω,≪). Lemma 3.3. If Xis compact Hausdorff, then d([CZ(X)]≤ω,≪) = d(MX,≺). Proof. If w(X)≤ ℵ0, then Xis metrizable, so that d([CZ(X)]≤ω,≪) = 1 = d(MX,≺). Suppose therefore that w(X)>ℵ0. For every U∈CZ(X), fix a continuous function fU:X→Isuch that X\U=f−1 U(0). Given a countably infinite subfamily γ={Un:n∈ω}of CZ(X), we consider the corresponding diagonal product fγ=△n∈ωfUn:X→ Iω. We also consider the analogously defined diagonal product into Ikfor some k∈Ncorresponding to any given finite subfamily of CZ(X). This correspondence defines a mapping Φ from [CZ(X)]≤ωto MX, where Φ is defined by Φ(γ) = fγ, if we agree to restrict the range space of fγto fγ(X). Choose a dense set Γ in ([CZ(X)]≤ω,≪) of the minimal cardinality. We claim that the image Φ(Γ) is dense in the quasi-ordered set (MX,≺). Indeed, let gbe a continuous mapping of Xonto a second countable space Y. Choose a countable base Bfor Yand put λ={g−1(V) : V∈ B}. Then λ∈[CZ(X)]≤ω, so we can find γ∈Γ with γ≪λ. Let us show that fγ= Φ(γ) satisfies fγ≺g. Suppose that x, y ∈Xand g(x)6=g(y). Then g(x)∈V6∋ g(y) for some V∈ B. Since x∈g−1(V)∈λand γ≪λ, there exists U∈γsuch that x∈U⊆g−1(V). Then y /∈U. Clearly fU(x)6=fU(y), and from U∈γit follows that fγ(x)6=fγ(y). Therefore, fγ(x) = fγ(y) always implies g(x) = g(y). This fact enables us to define a mapping h:fγ(X)→Ysuch that g=h◦fγ. Since g, fγare continuous mappings and fγis closed, we conclude that his also continuous. Hence fγ≺g. This proves our claim, and hence d(MX,≺)≤ |Φ(Γ)| ≤ |Γ|=d([CZ(X)]≤ω,≪). Conversely, let Nbe a dense set in (MX,≺) of the minimal cardinality. Choose a countable base Bfor Iωand put γf={f−1(V) : V∈ B} for each f∈ MX. Evidently, γf∈[CZ(X)]≤ω. Let us verify that the set Γ = {γf:f∈ N} is dense in ([CZ(X)]≤ω,≪). Consider an arbitrary element λ={Un:n∈ω}