Equivariant Completions

Michael Megrelishvili (Levy)* Department of Mathematics and Computer Science Bar-Ilan University, 52900 Ramat-Gan, Israel

Abstract. An important consequence of a result of Katetov and Morita states that every metrizable space is contained in a complete metrizable space of the same dimension. We give an equivariant version of this fact in the case of a locally compact σ acting group.

Introduction.

Let α : G × X → X be a continuous action of a topological group G on a uniform space (X, μ). We give a sufficient condition for the existence of a continuous extension α̂ : Ĝ × X̂ → X̂ where Ĝ is the sup-completion (i.e., the completion with respect to its two-sided uniformity) and (X̂, μ̂) is the completion of (X, μ). Our sufficient condition is necessary in the following important situation: Ĝ is Baire, μ is metrizable and for every g ∈ Ĝ the g-transition X̂ → X̂ is μ̂-uniformly continuous. As an application of a general equivariant completion theorem we unify the verification of the sup-completeness property for some natural groups.

An important consequence of a result of Katetov [10] and Morita [14] states that every metrizable space is contained in a complete metrizable space of the same dimension (see Engelking [6, 7.4.17]). Using the G-factorization theorem [13] we obtain an equivariant generalization of the last fact in the case of locally compact σ-compact acting group G. This generalization, at the same time, improves some “equivariant results” of de Groot [7] and de Vries [16].

A sufficient condition for the existence of G-completions in the case of locally compact G was obtained by Bronstein [3, 20.3].

Typeset by AMS-TEX

*Partially supported by the Israel Ministry of Sciences, Grant No. 3505.

1991 Mathematics Subject Classification. 54H15, 22A05.

Key words and phrases. equivariant completion, factorization, dimension.

1. Conventions and known results.

All spaces are assumed to be Tychonoff. The filter of all neighborhoods (nbd’s) of an element x of a space X is denoted by N_{x}(X). If μ is a compatible entourage uniformity on a topological space X then for every ε ∈ μ and x ∈ X denote by ε(x) the nbd {y ∈ X : (x, y) ∈ ε}. The greatest compatible uniformity is denoted by μ_{max}. If G is a topological group then G_{d} denotes the topological group with the same underlying group as in G, but provided with the discrete topology. The left, right and two-sided uniformity is denoted by U_{L}, U_{R} and U_{LR} respectively. The neutral element is always denoted by e.

If α : G × X → X is an action then the g-transition of X is the mapping α^{g} : X → X, α^{g}(x) = α(g, x). As usual, instead of α(g, x), we will write gx. For x ∈ X the x-orbit mapping is α_{x} : G → X, α_{x}(g) = gx. If α is continuous then X is called a G-space. Let μ be a uniformity on X. Then the system ⟨G, (X, μ), a⟩ (or simpler: α) is called saturated if each g-transition is μ-uniformly continuous. For any such system there exists a canonical action α̂_{0} of G on the completion X̂. We will say that (X̂, μ̂) is a G-completion if α̂_{0} is continuous. The following natural questions are central in the paper:

(a) When does the continuity of α : G × X → X imply the continuity of the canonical action α̂_{0} : G × X̂ → X̂?

(b) Let (X, μ) be complete. Under what conditions does there exist a continuous action α̂ : Ĝ × X → X which extends α?

(c) When does a metrizable G-space admit metric G-completions (of the same dimension)?

Examples 1.2 and 3.5 for (a), 3.2 for (b), 3.11 and 3.12 for (c) will show that these questions are non-trivial.

If a G-completion X̂ is compact, then we get a compact G-extension of X. Due to J. de Vries, a G-space X is said to by G-Tychonoff if X admits compact Gextensions. Denote by Tych^{G} the class of all G-Tychonoff triples ⟨G, X, α⟩. Recall [18] that if G is locally compact then Tych^{G} coincides with the class of all Tychonoff G-spaces.

Example 1.1. [12] There exists a continuous action α of a separable complete metrizable group G on J(ℵ_{0}) (the so-called hedgehog space of spininess ℵ_{0} [6]) such that ⟨G, J(ℵ_{0}), α⟩ has no compact G-extensions.

For every topological group G the left translations define a triple ⟨G, G, α_{L}⟩ which always is G-Tychonoff (see Brook [4]).

Example 1.2. [4],[17, p.147] Let G be a locally compact group. Consider the triple ⟨G, G, α_{L}⟩. If G is non-compact and non-discrete then the canonical action G × βG → βG is not continuous.

A system ⟨G, (X, μ), α⟩ is called

(a) quasibounded [13] if for every ε ∈ μ there exists a pair (δ, U) ∈ μ × N_{e}(G) such that (gx, gy) ∈ ε whenever (x, y) ∈ δ and g ∈ U;

(b) bounded [18] if for every ε ∈ μ there exists U ∈ N_{e}(G) satisfying (x, gx) ∈ ε for every (g, x) ∈ U × X.

Denote by Unif^{G} the class of all quasibounded saturated systems ⟨G, (X, μ), a⟩ with continuous α and by Comp^{G} the class of all compact G-spaces. Since every compact G-space is bounded with respect to its unique uniformity (see [4]) then we obtain

Lemma 1.3. Comp^{G} ⊂ Unif^{G} .

The following result directly follows from the definitions.

Lemma 1.4. If ⟨G, (X, μ), α⟩ is uniformly equicontinuous and α is continuous, then ⟨G, (X, μ), α⟩ ∈ Unif^{G} .

Theorem 1.5. Let α : G × X → X be a continous action. The statements are equivalent:

(i) ⟨G, X, α⟩ ∈ Tych^{G}; (ii) ⟨G, (X, μ), α⟩ ∈ Unif^{G} for a certain compatible μ; (iii) ⟨G, (X, ξ), α⟩ is quasibounded for a certain compatible ξ.

Proof. Since the quasiboundedness is hereditary by 1.3 we obtain (i) ⇒ (ii). Trivially (ii) ⇒ (iii). An argument for (iii) ⇒ (i), see [13],[11]. □

We will use the following known results.

Theorem 1.6. [13] Let X be a topogical group and α : G × X → X be a continous action of a topological group G on X by automorphisms. Then ⟨G, (X, μ), α⟩ ∈ Unif^{G} whenever μ ∈ {U_{L}, U_{R}, U_{LR}}.

Theorem 1.7. [13] Suppose that G is Baire and α : G × X → X is a d-saturated action on a metrizable uniform space (X, d) such that for a certain dense subset Y ⊂ X orbit mappings α_{y} : G → X, where y ∈ Y are continuous. Then

(a) α is continuous;

(b) ⟨G, (X, d), α⟩ ∈ Unif^{G};

(c) X is G-Tychonoff.

2. Inherited actions of dense subgroups.

Lemma 2.1. Let α : G × X → X be an action with continuous g-transitions, H be a dense subgroup of G and Y be a dense H-subspace of X such that the orbit mapping α_{y} : G → X is continuous for every y ∈ Y. If μ is a compatible uniformity on X and ⟨H, (Y, μ |_{Y} ), α |_{H×Y} ⟩ is quasibounded, then

(a) ⟨G, (X, μ), α⟩ is quasibounded;

(b) α is continuous;

(c) If ⟨H, (Y, μ |_{Y} ), α |_{H×Y} ⟩ is saturated, then ⟨G, (X, μ), α⟩ ∈ Unif^{G} .

Proof. (a) Given any ε_{0} ∈ μ, choose a symmetric entourage ε_{1} ∈ μ such that ε_{1}^{5} ⊂ ε_{0}. Since α |_{H×Y} is quasibounded with respect to μ |_{Y} , there exists a pair (δ, U) ∈ μ × N_{e}(H) such that

(1)  (gp, gq) ∈ ε_{1} for every g ∈ U and every (p, q) ∈ δ ∩ (Y × Y).

Take δ_{1} ∈ μ with the property δ_{1}^{2} ⊂ δ. Since H is a dense subgroup of G then the closure cl_{G}(U) belongs to N_{e}(G). Therefore, it suffices to show that (gp, gq) ∈ ε_{0} whenever g ∈ cl_{G}(U) and (p, q) ∈ δ_{1}. Assuming the contrary take g_{0} ∈ cl_{G}(U) \ U and (x_{1}, x_{2}) ∈ δ_{1} such that

(2)  (g_{0}x_{1}, g_{0}x_{2}) ∉ ε_{0}.

Since α^{g_{0}} : X → X is continuous and Y is dense then a certain pair (y_{1}, y_{2}) ∈ δ ∩ (Y × Y) satisfies

(g_{0}x_{1}, g_{0}y_{1}) ∈ ε_{1}, (g_{0}x_{2}, g_{0}y_{2}) ∈ ε_{1}.

Using the continuity of the orbit mappings α_{g_{0}y_{1}}, α_{g_{0}y_{2}} we pick V ∈ N_{e}(G) such that

(3)  (g_{0}x_{1}, gg_{0}y_{1}) ∈ ε_{1}^{2}, (g_{0}x_{2}, gg_{0}y_{2}) ∈ ε_{1}^{2} for every g ∈ V.

Since ε_{1}^{5} ⊂ ε_{0} it follows from (2) and (3) that (gg_{0}y_{1}, gg_{0}y_{2}) ∉ ε_{1} for every g ∈ V. By our assumption, g_{0} ∈ cl_{G}(U). Therefore, V g_{0} ∩ U is not empty. Hence, hg_{0} ∈ U for a certain h ∈ V. Then we get (hg_{0}y_{1}, hg_{0}y_{2}) ∉ ε_{1} which contradicts (1).

(b) Since all g-transitions are continuous, it suffices to check the continuity of α at (e, x_{0}) for an arbitrary x_{0} ∈ X. For a given ε_{0} ∈ μ take a symmetric ε_{1} ∈ μ such that ε_{1}^{4} ⊂ ε_{0}. According to (a) choose a symmetric δ ∈ μ and U_{1} ∈ N_{e}(G) satisfying (gp, gq) ∈ ε_{1} for every g ∈ U_{1} and (p, q) ∈ δ. Fix an element y_{0} ∈ Y ∩ δ(x_{0}). Since α_{y_{0}} : G → X is continuous there exists U_{2} ∈ N_{e}(G) such that (y_{0}, gy_{0}) ∈ ε_{1} for every g ∈ U_{2}. Now, if x ∈ δ(x_{0}) and g ∈ U_{1} ∩ U_{2} then gx ∈ ε_{0}(x_{0}). This proves the continuity of α.

(c) According to the definition of Unif^{G}, we have only to show that gε ∈ μ for any (g, ε) ∈ G × μ. Using (a) we pick a pair (δ, U) ∈ μ × N_{e}(G) such that δ ⊂ tε for every t ∈ U. Clearly, g = ht for a certain pair (h, t) ∈ H × U. Therefore, hδ ⊂ htε = gε. By our hypothesis, α^{h} |_{Y} is μ |_{Y} -uniformly continuous. Since α^{h} : X → X is a homeomorphism and Y is dense, then α^{h} is μ-uniformly continuous. Therefore, hδ ∈ μ, which yields gε ∈ μ. □

Proposition 2.2. Let H be a dense subgroup of G. Then

⟨G, X, α⟩ ∈ Tych^{G} iff ⟨H, X, α |_{H×X} ⟩ ∈ Tych^{G} .

Proof. Necessity is trivial. The converse follows from Theorem 1.5 and Lemma 2.1(a). □

Combining 1.1 and 2.2 we get

Example 2.3. There exists a continuous action α : H × J(ℵ_{0}) → J(ℵ_{0}) of a metrizable countable group H such that ⟨H, J(ℵ_{0}), α⟩ is not H-Tychonoff.

Question 2.4. Let G be a monothetic group. Is it true that every Tychonoff G-space is G-Tychonoff?

3. Main results.

Theorem 3.1. Let ⟨G, (X, μ), α⟩ ∈ Unif^{G} . Then there exists a continuous action α̂ : Ĝ × X̂ → X̂ which extends α and satisfies ⟨Ĝ, (X̂, μ̂), α̂⟩ ∈ Unif^{G} .

Proof. Let φ and γ be Cauchy filters in (G, U_{LR}) and (X, μ), respectively. Denote by φγ the system {AB : A ∈ φ, B ∈ γ}. An essential step in our proof is the following Claim. φγ is a μ-Cauchy filter basis. Proof of Claim. For a given ε ∈ μ choose a pair (δ_{1}, U) ∈ μ × N_{e}(G) such that

(1)  (gx, gy) ∈ ε for every (x, y) ∈ δ_{1} and g ∈ U.

The inclusion U_{R} ⊂ U_{LR} implies that φ is U_{R}-Cauchy. Therefore, there exists g_{0} ∈ G such that Ug_{0} ∈ φ. Since α^{g_{0}} is μ-uniformly continuous one can choose δ_{2} ∈ μ with the property

(2)  (g_{0}x, g_{0}y) ∈ δ_{1} for every (x, y) ∈ δ_{2}.

There exists a symmetric entourage δ_{3} ∈ μ such that δ_{3}^{3} ⊂ δ_{2}. Using the quasiboundedness we pick a symmetric δ_{4} ∈ μ and V_{1} ∈ N_{e}(G) satisfying

(3)  (vx, vy) ∈ δ_{3} whenever (x, y) ∈ δ_{4} and v ∈ V_{1}.

Since γ is μ-Cauchy, then δ_{4}(x_{0}) ∈ γ for a certain x_{0} ∈ X. The continuity of α_{x_{0}} : G → X implies the existence of V_{2} ∈ N_{e}(G) such that

(4)  (x_{0}, vx_{0}) ∈ δ_{4} for v ∈ V_{2}.

It follows from (3),(4) and the inclusions δ_{4} ⊂ δ_{3}, δ_{3}^{3} ⊂ δ_{2} that

(5)  (x_{1}, vx_{2}) ∈ δ_{2} for x_{1}, x_{2} ∈ δ_{4}(x_{0}) and v ∈ V_{1} ∩ V_{2} = V.

For the next argument observe that φ is U_{L}-Cauchy. Choose A ∈ φ such that g_{1}^{−1}g_{2} ∈ V for every g_{1}, g_{2} ∈ A. Since Ug_{0} ∈ φ then without restriction of generality we may suppose that A ⊂ Ug_{0}. Now, if g_{1}, g_{2} ∈ A then g_{1} = t_{1}g_{0}, g_{2} = t_{2}g_{0} for certain t_{1}, t_{2} ∈ U. Since g_{1}^{−1}g_{2} ∈ V then by (5) we have (x_{1}, g_{0}^{−1}t_{1}^{−1}t_{2}g_{0}x_{2}) ∈ δ_{2} for every x_{1}, x_{2} ∈ δ_{4}(x_{0}). Using (2) we obtain (g_{0}x_{1}, t_{1}^{−1}t_{2}g_{0}x_{2}) ∈ δ_{1}. By (1) we get (t_{1}g_{0}x_{1}, t_{2}g_{0}x_{2}) ∈ ε. Therefore, (g_{1}x_{1}, g_{2}x_{2}) ∈ ε for every x_{1}, x_{2} ∈ δ_{4}(x_{0}) and every g_{1}, g_{2} ∈ A. Since A ∈ φ and δ_{4}(x_{0}) ∈ γ, this means that φγ is μ-Cauchy.

Now we are ready for the construction of α̂. Denote by i : X → X̂, the canonical embedding and consider the composition f = i ∘ α : G × X → X̂. Let ξ be a Cauchy filter in the uniform product (G, U_{LR}) × (X, μ). There exists a U_{LR}-Cauchy filter φ and a μ-Cauchy filter γ such that φ × γ is a filter basis of ξ. Since i(φγ) is a subfilter of f(ξ) in X̂, the above claim implies that f(ξ) generates a convergent filter in X̂. Therefore, we can use the well-known extension theorem [1, ch.I, sec. 8.5, Th.1]. Taking into account that the completion of the product (G, U_{LR}) × (X, μ) is canonically equivalent to the uniform product (Ĝ, Û_{LR}) × (X̂, μ̂), we get the continuous mapping α̂ : Ĝ × X̂ → X̂ which extends α. By the principle of the extension of identities [1, Ch.I, Sec.8.1, Cor.1] α̂ is an action. Finally, from Lemma 2.1 it directly follows that ⟨Ĝ, (X̂, μ̂), α̂⟩ ∈ UnifĜ . □

The paper [11] (which can be regarded as a preprint) contains a slightly different proof of Theorem 3.1. The weaker result (the continuity of the canonical completion α̂_{0} : G × X̂ → X̂) can be found in [13].

By Example 1.2 it is clear that the quasiboundedness in Theorem 3.1 is essential. The following example shows that this condition cannot be dropped even for complete uniformity μ.

Example 3.2. Let ℚ be the topological group of all rational numbers. Consider the system ⟨ℚ, (ℚ, μ_{max}), α_{L}⟩. Then μ_{max} is complete but there is no continuous non-trivial action of ℚ̂ = ℝ on ℚ.

Results in section 1 (and Corollary 3.4 and Proposition 3.7 below) show that Theorem 3.1 is often applicable. There is an important case where our sufficient condition, at the same time, is necessary.

Proposition 3.3. Let α : G × X → X be a continuous action on a metrizable uniform space (X, μ). If Ĝ is Baire then the following statements are equivalent:

(a) There exists a μ̂-saturated continuous extension α̂ : Ĝ × X̂ → X̂;

(b) ⟨G, (X, μ), α⟩ ∈ Unif^{G} .

Proof. (b) ⇒ (a) is true by Theorem 3.1. For the converse use Theorem 1.7. □

Corollary 3.4. [13] If G is Baire then every metrizable G_{d}-completion of a G-space is a G-completion.

Example 1.2 emphasizes the importance of the metrizability condition in Corollary 3.4 and Theorem 1.7. Now we show that the assumption concerning G also is not superfluous.

Proposition 3.5. Let G be a countable non-trivial metrizable group such that Ĝ is connected. Consider the triple ⟨G, X := G, α_{L}⟩. Then there exists a compact metrizable G_{d}-extension of X which is not G-extension.

Proof. Clearly, dim X = 0. Since G_{d} is discrete and countable, it follows from a result of de Groot and Mcdowell [8, Cor.2.5] that there exists a zero-dimensional compact G_{d}-extension cX of X. Suppose that this is a G-extension. Then the corresponding action α_{c} : G × cX → cX is continuous. Since cX is a compact G-space, by Lemma 1.3 and Theorem 3.1 there exists a continuous extension α̂_{c} : Ĝ × cX → cX. By our assumption Ĝ is connected. Therefore, α̂_{c}(Ĝ, e) = e. This implies that G is trivial. □

An application for topological groups. Theorem 3.1 makes possible an easy and unified verification of the sup-completeness property in some natural cases.

Proposition 3.6. In each case listed below G is sup-complete:

(a) [2, Ch.X, §3, Ex.16] G = Unif(X, μ) — the group of all unimorphisms of a complete uniform space (X, μ) endowed with the topology of uniform convergence.

(b) [2, Ch.X,§3, Ex.19] G = Is(X, d) — the group of all isometries of a complete metric space (X, d) endowed with the topology of pointwise convergence.

(c) ( [15] or [5, 7.8.9]) G = Aut X — the group of all topoligical automorphisms of a locally compact group X endowed with the Birkhoff topology.

Proof. (a) Let α : G × X → X be a natural action. Clearly, α is μ-saturated and μbounded; in particular, ⟨G, (X, μ), α⟩ ∈ Unif^{G} . By Theorem 3.1, Ĝ ⊂ Unif(X, μ) = G. Therefore, Ĝ = G. (b),(c) Combine 3.1 with 1.4 and 1.6 respectively. The continuity of α̂ : Ĝ×X → X

implies

(1) For every g ∈ Ĝ the g-transition is a homeomorphism of X;

(2) The topology of Ĝ contains the topology of pointwise convergence. ¿From (1) and (2) it easily follows that g ∈ Is(X, d) and g ∈ Aut X, respectively. □ Proposition 3.7. [11] Let α : G × X → X be a continuous action of a locally compact group G. Then ⟨G, (X, μ_{max}), α⟩ ∈ Unif^{G} . Proof. Let B be a compact nbd of the identity and let a system {d_{i} : i ∈ I} of pseudometrics generate μ_{max}. Denote by θ the uniformity on X induced by the system {d_{i}^{(n)} : i ∈ I, n ∈ ℕ} where

d_{i}^{(n)}(x, y) = sup{d_{i}(gx, gy) : g ∈ B^{n}}.

Since {α^{g} : g ∈ B^{n}} is d_{i}-equicontinuous, θ is compatible with the original topology. Evidently ⟨G, (X, θ), a⟩ ∈ Unif^{G} and μ_{max} ⊂ θ. Finally, observe that the maximality of μ_{max} implies μ_{max} = θ. □

By Theorem 3.1 and Example 3.2 it is clear that the assumption “locally compact” in Proposition 3.7 cannot be dropped. Theorem 3.8. Let α : G × X → X be a continuous action of a locally compact σ-compact group G on a metrizable space X. Then there exists a metric uniformity d on X such that the following conditions hold:

(a) Each α^{g} : X → X (g ∈ G) is d-uniformly continuous; (b) The canonical action α̂ : G×(X̂, d̂) → (X̂, d̂) is continuous and ⟨G, (X̂, d̂), α̂⟩ ∈

Unif^{G};

(c) dim X̂ = dim X.

Proof. Let ρ be any compatible metric uniformity on X and consider the identity Id_{X} : (X, μ_{max}) → (X, ρ). By Proposition 3.7, ⟨G, (X, μ_{max}), α⟩ ∈ Unif^{G} . Therefore, we can apply the G-factorization theorem [13, Th.2.6]. Taking into account the σ-compactness of G we can choose a metric uniformity μ on X such that ⟨G, (X, μ), α⟩ ∈ Unif^{G} and the uniform dimension [9] ΔdμX is not greater than Δdμ_{max}X. By Theorem 3.1, ⟨G, (X̂, μ̂), α̂⟩ ∈ Unif^{G} . Since (X, μ) is uniformly dense in (X̂, μ̂), then ΔdμX = Δdμ̂X̂ [9, p.78]. On the other hand, Δdμ_{max}X = dim X [9, p.147]. Therefore, Δdμ̂X̂ ≤ dim X. For every metrizable space Y, the number dim Y is the minimum of Δdθ for all compatible metric uniformities θ on Y [9, p.153]. Thus, dim X̂ ≤ Δdμ̂X̂. This establishes dim X̂ ≤ dim X. Since X̂ is perfectly normal, the inequality dim X ≤ dim X̂ follows from Čech’s monotonicity theorem. □

Remark 3.9. Part (a) of Theorem 3.8 is contained in de Groot [7]. Part (b) improves a result of de Vries [16, Th.4.7].

Definition 3.10. Let α : G×X → X be a continuous action. We say that a G-space X is (weakly) G-metrizable if there exists a metric uniformity d on X such that (⟨G, (X, d), α⟩ is saturated) ⟨G, (X, d), α⟩ ∈ Unif^{G} .

If ⟨H, J(ℵ_{0}), α⟩ is as in Example 2.3, then by Theorem 1.5, J(ℵ_{0}) is not Hmetrizable. On the other hand, since H is countable, J(ℵ_{0}) is weakly H-metrizable. Indeed, for every metric d on J(ℵ_{0}) with the bounded diameter we may consider a new metric d^{∗}(x, y) = sup{2^{−n}d(g_{n}x, g_{n}y) : n ∈ ℕ, g_{n} ∈ H} (see [8, Lemma 2.2]). In contrast to this example, Theorem 1.7 implies that if G is Baire then a G-space X is weakly G-metrizable iff X is G-metrizable. The last fact explains the following Example 3.11. The polish transformation group ⟨G, J(ℵ_{0}), α⟩ from Example 1.1 is

not weakly G-metrizable.

Example 3.12. [8, Ex.2.10] Let X = ℚ be the space of rational numbers and G the

group of all autohomeomorphisms of ℚ endowed with the discrete topology. Then

X is not weakly G-metrizable.

As Examples 3.11 and 3.12 show, local compactness and σ-compactness are

essential in Theorem 3.8(a).

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