Fundamenta Mathematicae (to appear)

THE EQUIVARIANT UNIVERSALITY AND COUNIVERSALITY OF THE CANTOR CUBE

MICHAEL G. MEGRELISHVILI AND TZVI SCARR

Abstract. Let ⟨G, X, α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let ⟨H({0, 1}^{ℵ_{0}}), {0, 1}^{ℵ_{0}}, τ⟩ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then

(1) there exists a topological group embedding φ : G ↪ H({0, 1}^{ℵ_{0}});

(2) there exists an embedding ψ : X ↪ {0, 1}^{ℵ_{0}}, equivariant with respect to φ, such that ψ(X) is an equivariant retract of {0, 1}^{ℵ_{0}} with respect to φ and ψ.

1. Introduction

The Cantor cube C = {0, 1}^{ℵ_{0}} is a universal space in the class of zerodimensional, separable, metrizable spaces, that is, every such space can be topologically embedded into C. In particular, every compact, zerodimensional, metrizable space is homeomorphic to a closed subset of C. Sierpiński [15] showed that every non-empty closed subset of C is a retract of C. This gives us the following well-known fact.

Fact 1.1. Every non-empty, compact, zero-dimensional, metrizable space is homeomorphic to a retract of C.

Our Main Theorem, formulated in the abstract above, is an equivariant generalization of Fact 1.1 for non-Archimedean acting groups. A topological group is non-Archimedean if it has a local base at the identity consisting of open subgroups. The class of non-Archimedean groups includes:

• the prodiscrete (in particular, the profinite) groups;

• the groups arising in non-Archimedean functional analysis [14] (for example, the additive groups of the fields of p−adic numbers);

• the group Is(X, d) of all isometries of an ultrametric space (X, d), with the topology of pointwise convergence; • the locally compact, totally disconnected groups [3];

2000 Mathematical Subject Classification. 54H15, 22A99.

Some of this work is part of the second author’s Ph.D. dissertation at Bar-Ilan University, ISRAEL under the supervision of Professor Hillel Furstenberg.

Keywords: Cantor cube, G-compactification, universal space, non-Archimedean group.

• the symmetric group S_{∞} on a countably infinite set, with the topology of pointwise convergence;

• the full group H(X) of autohomeomorphisms of X, with the compactopen topology, where X is a compact Hausdorff zero-dimensional space (see Lemma 3.2 below).

In fact, a topological group G is non-Archimedean iff G is a topological subgroup of H(X) for some appropriate compact Hausdorff zero-dimensional space X. This complete characterization of the non-Archimedean groups is a part of Theorem 3.3 below. It is easy to show that the class of all non-Archimedean groups is a variety in the sense of [12]. That is, this class is closed under the formation of topological subgroups, products and quotient groups.

Note that the transformation groups having zero-dimensional (in particular, ultrametric) phase spaces have many applications in descriptive set theory [1, 6, 7].

2. Preliminaries and Conventions

All topological spaces in this paper are assumed to be Hausdorff. The neutral element of a group G is denoted by e_{G}. The weight w(X) of a topological space X is defined to be τ(X) · ℵ_{0}, where τ(X) denotes the minimal cardinality of a base for X.

For information on uniform spaces, we refer the reader to [4]. If μ is a uniformity for X, then the collection of elements of μ which are finite coverings of X forms a base for a uniformity for X which we denote by μ_{fin}. If (X, μ) is a uniform space, the uniform completion (X̂, μ̂_{fin}) of (X, μ_{fin}) is a compact uniform space known as the Samuel compactification of (X, μ). A partition of a set X is a covering of X consisting of pairwise disjoint subsets of X. Following [14], we say that a uniform space (X, μ) is non-Archimedean if it has a base consisting of partitions of X. Equivalently, μ is generated by a system {d_{i}} of ultrapseudometrics, that is, pseudometrics, each of which satisfies the strong triangle inequality d_{i}(x, z) ≤ max{d_{i}(x, y), d_{i}(y, z)}. Clearly, a non-Archimedean uniform space is zero-dimensional in the uniform topology. A topological group is non-Archimedean iff its right uniformity is non-Archimedean.

The following result is well known (see, for example, [4] and [5]).

Lemma 2.1. Let (X, μ) be a non-Archimedean uniform space. Then both (X, μ_{fin}) and the uniform completion (X̂, μ̂) of (X, μ) are non-Archimedean uniform spaces.

A topological transformation group, or G-space, is a triple ⟨G, X, α⟩, where G is a topological group (called the acting group), X is a topological space (called the phase space), and α : G × X → X is a continuous action. For each g ∈ G, the g-transition map is the function α^{g} : X → X, α^{g}(x) = gx.

Definition 2.2. Let ⟨G_{1}, X_{1}, α_{1}⟩ be a G_{1}-space, and let ⟨G_{2}, X_{2}, α_{2}⟩ be a G_{2}-space. Suppose that φ : G_{1} ↪ G_{2} is a topological group embedding.

(1) A continuous function ψ : X_{1} → X_{2} is equivariant with respect to φ (or, simply, equivariant, if φ is clear from the context) if, for all g ∈ G_{1} and x ∈ X_{1}, ψ(gx) = φ(g)ψ(x).

(2) Let ψ : X_{1} → X_{2} be an equivariant embedding with respect to φ. We say that ψ(X_{1}) is an equivariant retract of X_{2} (with respect to φ and ψ) if there is a continuous retraction r : X_{2} → ψ(X_{1}) which is equivariant with respect to φ^{−1} : φ(G_{1}) → G_{1}.

Let ⟨G, X, α⟩ be a G-space. If ⟨G, Y, γ⟩ is a compact Hausdorff G-space and ψ : X → Y is equivariant, then Y is called a G-compactification of X. If, in addition, ψ is a topological embedding, then Y is a proper Gcompactification of X. A G-space ⟨G, X, α⟩ is G-Tychonoff if it has a proper G-compactification. Not every Tychonoff G-space is G-Tychonoff [8]. De Vries [19] proved that if G is locally compact, then every Tychonoff G-space is G-Tychonoff. For every G-space X there exists a (possibly improper) maximal G-compactification β_{G}X [18]. For more information on G-compactifications, as well as for a general method of constructing Tychonoff G-spaces which are not G-Tychonoff, see [11].

Let G be a topological group. Recall [2] that the collection of coverings {Ux|x ∈ G}, where U is a neighborhood of e_{G}, forms a base for the right uniformity μ_{R} for G. In 1957, Teleman [16] proved that for arbitrary Hausdorff G, the Samuel compactification Ĝ of G with respect to its right uniformity is a proper G-compactification of the G-space ⟨G, G, α_{L}⟩, where α_{L} is the usual left action of G on itself. In fact, Ĝ is isomorphic to β_{G}G and is called the greatest ambit (see, for example, [20]). β_{G}G is the maximal proper G-compactification of ⟨G, G, α_{L}⟩.

To the best of our knowledge, very little is known about the dimension of β_{G}X. Some special results can be found in [8, 9]. The dimension of the greatest ambit β_{G}G may be greater than the topological dimension of G (simply take a cyclic dense subgroup G of the circle group; then dim G = 0 and dim β_{G}G = 1). However, in the case of the Euclidean group G = ℝ^{n}, we have dim β_{G}G = dim G. This follows from Theorem 5.12 in [4]. By a result of Pestov [13], one has dim β_{G}G = 0 iff G is non-Archimedean. An alternative proof of this will be given in Theorem 3.3.

3. Proof of the Main Results

Fact 3.1. ([9]) Every G-Tychonoff G-space X has a proper G-compactification Y such that w(Y) ≤ w(X) · w(G) and dim Y ≤ dim β_{G}X.

Lemma 3.2. If X is a compact Hausdorff zero-dimensional space, then H(X) is a non-Archimedean group.

Proof. For each two-element compact clopen partition {K_{1}, K_{2}} of X, define

B(K_{1}, K_{2}) = {φ ∈ H(X) : φ(K_{1}) = K_{1}, φ(K_{2}) = K_{2}}.

Let B = {B(K_{1}, K_{2}) : {K_{1}, K_{2}} is a compact clopen partition of X}. Then B is a local base at e_{H(X)} consisting of clopen subgroups, and, hence, H(X) is non-Archimedean. □

The following theorem provides a useful characterization of non-Archimedean groups. (As noted before, the equivalence of (i) and (ii) was established by Pestov [13].)

Theorem 3.3. The following assertions are equivalent:

(i) G is a non-Archimedean topological group; (ii) dim β_{G}G = 0; (iii) G is a topological subgroup of H(X) for some compact Hausdorff zero-

dimensional space X such that w(X) = w(G).

Proof. (i) ⇒ (ii) Suppose G is non-Archimedean. Then the right uniformity μ_{R} for G is a non-Archimedean uniformity. By Lemma 2.1, the precompact uniformity (μ_{R})_{fin} for G is also a non-Archimedean uniformity. Let (Ĝ, μ̂) be the uniform completion of (G, (μ_{R})_{fin}). Then, again by Lemma 2.1, μ̂ is a non-Archimedean uniformity, and, hence, Ĝ is zero-dimensional. But (Ĝ, μ̂) is exactly β_{G}G.

(ii) ⇒ (iii) By Fact 3.1, there exists a zero-dimensional proper Gcompactification ⟨G, X, α_{L}^{∗}⟩ of ⟨G, G, α_{L}⟩ such that w(X) = w(G). Let ψ : G → X be the corresponding equivariant embedding.

We will show that the map φ : G → H(X) defined by φ(g) = (α_{L}^{∗})^{g} is a topological group embedding. Observe that φ is one-to-one because α_{L}^{∗} extends the action α_{L}. To prove the continuity of φ, suppose α^{g} ∈ O = {f ∈ H(X) : f(K) ⊆ U}, where K ⊆ X is compact and U ⊆ X is open. Using the compactness of K and the continuity of α_{L}^{∗}, we can find a neighborhood V of g such that φ(V) ⊆ O. Hence, φ is continuous.

It remains to show that if O ⊆ G is open, then φ(O) is open in φ(G). Let O ⊆ G be open. Then ψ(O) is open in ψ(G). Let W ⊆ X be open such that ψ(O) = W ∩ ψ(G). Define B = {f ∈ H(X) : f(ψ(e_{G})) ∈ W }. Then B is open in H(X) and φ(O) = B ∩ φ(G). Hence, φ(O) is open in φ(G).

(iii) ⇒ (i) Follows directly by Lemma 3.2. □

Fact 3.4. (Brouwer) The Cantor cube {0, 1}^{ℵ_{0}} is the unique (up to homeomorphism) non-empty, compact, metrizable, zero-dimensional, perfect space. □

Now we are ready to prove our main result.

Theorem 3.5. Let G be a non-Archimedean and second countable group, and let X be a compact, metrizable, zero-dimensional G-space. Then

(1) there exists a topological group embedding φ : G ↪ H(C);

(2) there exists an embedding ψ : X ↪ C, equivariant with respect to φ, such that ψ(X) is an equivariant retract of C with respect to φ and ψ.

Proof. By Theorem 3.3, there exists a compact, second countable (and thus metrizable) zero-dimensional space Y such that H(Y) contains G as a topological subgroup. We may as well assume that all homeomorphisms of Y corresponding to elements of G transform a certain base point y_{0} ∈ Y onto itself (if not, replace Y with a disjoint union Y ∪ {y_{0}} and redefine those homeomorphisms in an obvious way).

Let us identify the action of G on X with a homomorphism w : G → H(X), and let D be a copy of the Cantor set. By Brouwer’s theorem, the space C = X × Y × D is homeomorphic to the Cantor set, and, clearly, the map φ : G → H(C),

g ↦ (w(g), g, id_{D}) ∈ H(X) × H(Y) × H(D) ⊆ H(C),

is a continuous homomorphism, thus turning C into a G-space. This homomorphism is also an embedding, for its composition with the projection onto H(Y) is the identity mapping, so it is one-to-one and the inverse is continuous.

We define ψ : X → C by x ↦ (x, y_{0}, d_{0}), where d_{0} ∈ D is any base point, and the retraction r : C → ψ(X) by r(x, y, d) = (x, y_{0}, d_{0}). Then ψ and r are equivariant, and the proof is complete. □

Theorem 3.6. H(C) is universal in the class of all non-Archimedean, second countable groups, that is, every such group is topologically isomorphic to a subgroup of H(C).

Final Remarks.

(1) By Theorem 1.5.1 of [1], the group S_{∞} is also universal in this class.

(2) The group H(I^{ℵ_{0}}) is universal in the class of all second countable topological groups, where I is the closed interval [0, 1] (see [17]). Moreover, by [10], the topological transformation group ⟨H(I^{ℵ_{0}}), I^{ℵ_{0}}⟩ is universal in the class of all compact, metrizable G-spaces with second countable acting group G.

(3) The action on C which we defined in the proof of Theorem 3.5 intrinsically depends on the original action of G on X, as the following example shows.

Example 3.7. Let α : S_{∞} × C → C be the natural “permutation of coordinates” action

α(g, (x_{n})) = (x_{g(n)}).

Let \overline{0} and \overline{1} denote the two constant sequences of C. Let H = \{\overline{0}, \overline{1}\} ⊆ C. Consider H as an S_{∞}-subspace of C.

Claim. H is not an equivariant retract of C with respect to φ = id_{S_{∞}} and ψ = id_{H}. Proof. The Cantor cube C is an S_{∞}-ambit under the action α, that is, it contains a point whose orbit is dense in C. In fact, all points which contain infinitely many 0’s and infinitely many 1’s have dense orbits. Hence, every image of C under an equivariant map is also an S_{∞}-ambit. However, H is not an S_{∞}-ambit.

Acknowledgments. The authors would like to thank the referees for many constructive suggestions and for a very substantial shortening of the proof of the Main Theorem.

References

[1] H. Becker and A. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, Cambridge, 1996.

[2] R. Engelking, General Topology, PWN, Warsaw, 1977.

[3] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Volume One, Springer-Verlag, 1963.

[4] J. Isbell, Uniform Spaces, American Mathematical Society, Providence, 1964.

[5] J. Isbell, Zero-dimensional Spaces, Tohoku Mat. J. (2) 7(1955), 1-8.

[6] A. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995.

[7] A. Kechris, Descriptive Dynamics, to appear in Descriptive Set Theory and Dynamical Systems, London Math. Soc. Lecture Note Series, Cambridge University Press, 2000.

[8] M. Megrelishvili, A Tychonoff G-space which has no compact G-extensions and Glinearizations, Russ. Math. Surv. 43 (1988), 145-146.

[9] M. Megrelishvili, Compactification and Factorization in the Category of G-spaces in Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, J. Adámek and S.MacLane, editors, World Scientific, Singapore (1989), 220-237.

[10] M. Megrelishvili, Free Topological G-groups, New Zealand J. of Math. 25(1996), 59-72.

[11] M. Megrelishvili and T. Scarr, Constructing Tychonoff G-spaces which are not G-Tychonoff, Top. Appl. 86(1998), 69-81.

[12] S.A. Morris, Varieties of topological groups (a survey), Colloq. Math., vol. XLVI (1982), 147-165.

[13] V.G. Pestov, On free actions, minimal flows, and a problem by Ellis, Trans. Amer. Math. Soc. 350:761(1998), 4149-4175.

[14] A.C.M. van Rooij, Non-Archimedean Functional Analysis, Pure and Applied Mathematics, Marcel Dekker, 1978.

[15] W. Sierpiński, Sur les projections des ensembles complémentaires aux ensembles (A), Fund. Math. 11(1928), 117-122.

[16] S. Teleman, Sur la représentation linéaire des groupes topologiques, Ann. Sci. Ecole Norm. Sup. 74(1957), 319-339.

[17] V.V. Uspenskij, A universal topological group with countable base, Funct. Anal. Appl. 20(1986), 160-161.

[18] J. de Vries, Topological Transformation Groups I, Mathematical Centre Tracts 65, Mathematisch Centrum, Amsterdam, 1975.

[19] J. de Vries, On the existence of G-compactifications, Bull. Acad. Polon. Sci. Ser. Math. 26(1978), 275-280.

[20] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers, Dordrect-Boston-London, 1993.

Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel

  E-mail address: megereli@macs.biu.ac.il
  URL: http://www.cs.biu.ac.il/ megereli

Department of Applied Mathematics, Jerusalem College of Technology, 21 Havaad Haleumi St., Jerusalem, ISRAEL 91160

  E-mail address: scarr@avoda.jct.ac.il