Constructing Tychonoff G-spaces which are not G-Tychonoff

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

Abstract. Jan de Vries’ compactification problem is whether every Tychonoff Gspace can be equivariantly embedded in a compact G-space. In such a case, we say that G is a V -group. De Vries showed that every locally compact group G is a V -group. The first example of a non-V -group was constructed in 1988 by the first author. Until now, this was the only known counterexample. In this paper, we give a systematic method of constructing non-compactificable G-spaces. We show that the class of non-V -groups is large and contains all ℵ_{0}-bounded non-locally precompact groups. This establishes the existence of monothetic (even cyclic) non-V -groups, answering a question of the first author. As a related result, we obtain a characterization of locally compact groups in terms of “G-normality.”

1. Introduction.

A topological transformation group, or a G-space, is a triple ⟨G, X, α⟩, where G is a topological group, X is a topological space, and α: G × X → X is a continuous action. For basic information on G-spaces, see de Vries [16,21]. If X is Tychonoff (respectively, normal, compact, etc.), then ⟨G, X, α⟩ (or just X, for short) is called a Tychonoff (respectively, normal, compact, etc.) G-space. A G-space is G-Tychonoff if it can be equivariantly embedded into a compact Hausdorff G-space. We call a group G a V -group if every Tychonoff G-space is G-Tychonoff.

In [17], Jan de Vries posed the “compactification problem” in its full generality, which in our terms becomes: is every topological group G a V -group? See Carlson [4] for the case G = ℝ. In [3], R. B. Brook investigated a more general class of groups and showed that for any topological group G, the G-space ⟨G, G, α_{L}⟩ (α_{L}(g_{1}, g_{2}) = g_{1}g_{2}) is G-Tychonoff.

Recall de Vries’ well-known result [20] which states that every locally compact group is a V -group. (For the case where G is a compact Lie group, see [12]. If G is compact, see [1,18]). Recall some G-spaces which are G-Tychonoff:

(a) [17] Every coset G-space ⟨G, G/H, α_{L}⟩. (If H = {e}, this gives Brook’s result mentioned above).

(b) [7] Every G-space X under an equicontinuous action.

(c) [10] Every G-group X (and, hence, every linear G-space X).

(d) [10] Every metric G-space (X, d), where G is second category and α^{g} : X → X is d-uniformly continuous for every g ∈ G.

(e) [15] Every Baire G-space X, where G is ℵ_{0}-bounded and acts transitively on X.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 54H15, 22A99.

Key words and phrases. G-Tychonoff, G-normal, α-uniform function, Ascoli-Arzela theorem, ℵ_{0}-bounded group.

In 1988, Megrelishvili [9] answered de Vries’ question negatively. He found a continuous action α of a separable, complete metrizable group G on J(ℵ_{0}), the so-called metrizable hedgehog of spininess ℵ_{0}, such that ⟨G, J(ℵ_{0}), α⟩ is not G-Tychonoff. Therefore, the group G of this example is not a V -group. By [11], no dense subgroup of G is a V -group.

It is an open question whether there are non-locally compact V -groups. In this paper, we prove the following

Main Theorem. If G is an ℵ_{0}-bounded topological group which is not locally precompact, then G is not a V -group.

In particular, this provides an example of a cyclic group G which is not a V - group, answering a question of Megrelishvili about monothetic groups [11]. As a related result, we characterize locally compact groups in terms of G-normality.

The compactification problem is still open in the case of a (monothetic) precompact group, even for a dense cyclic subgroup of the circle group 𝕋. It is also an intriguing question whether the additive group ℚ of rational numbers is a V -group.

Our approach is to adapt techniques for constructing regular spaces which are not completely regular. For example, I. Tree [14] has constructed a machine which turns any regular topological space which is not normal into a regular space which is not completely regular. We have adapted his technique to the context of G-spaces, and we have constructed a machine which turns a normal G-space which is not G-normal in some sense into a Tychonoff G-space which is not G-Tychonoff.

Some of the results of this paper were presented by the second author at the Eighth Prague Topological Symposium in August, 1996.

2. Preliminaries and Conventions.

All spaces are Tychonoff, and all cardinals are assumed to be infinite. The filter of all neighborhoods of an element x of a space X is denoted by N_{x}(X). The neutral element of a group is denoted by e. A group G is locally precompact if it is a subgroup of a locally compact group, or, equivalently, if its sup-completion (the completion with respect to its two-sided uniformity) is locally compact. Recall [19,20,1] that a continuous function f : X → ℝ defined on a G-space X is called α-uniform if for every ε > 0 there exists U ∈ N_{e}(G) such that for all g ∈ U, x ∈ X, we have |f(x) − f(gx)| < ε. Denote by C_{α}(X) (respectively, C(X)) the set of all bounded α-uniform (respectively, continuous bounded) functions on X.

Lemma 2.1.

(i) C_{α}(X) is a closed sub-algebra of C(X).

(ii) If f is in C_{α}(X) and r ∈ ℝ, then the functions μ(x): = min{f(x), r} and h(x): = |f(x)|

are also in C_{α}(X). □

The compactification of X which corresponds to the algebra C_{α}(X) is the maximal G-compactification [19] and is denoted by β_{G}X.

Fact 2.2 [19,20]. A Tychonoff G-space ⟨G, X, α⟩ is G-Tychonoff iff β_{G}X is a proper G-compactification of X iff C_{α}(X) separates the points and the closed subsets of X. □

As mentioned above, the G-space of all left translations ⟨G, G, α_{L}⟩ is G-Tychonoff. In this case, the maximal (proper) G-compactification β_{G}G is called the greatest ambit. (For further information, see [21]).

We say that subsets A and B of a G-space ⟨G, X, α⟩ are α-disjoint if there exists U ∈ N_{e}(G) such that UA ∩ UB = ∅. Two disjoint invariant subsets are obviously α-disjoint. Moreover, if A and B are separated by an α-uniform function, then A and B are α-disjoint. In fact, we have the following stronger

Lemma 2.3. Let ⟨G, X, α⟩ be a G-space. Let C and D be subsets of X which are separated by an α-uniform function. Then there are sequences {U_{n}}, {O_{n}} and {O′_{n}} of nbds of e, C and D, respectively, such that for all n ∈ ℕ,

cl(U_{n}O_{n}) ⊆ O_{n+1}, cl(U_{n}O′_{n}) ⊆ O′_{n+1}, and O_{n} ∩ O′_{n} = ∅.
□

We say that a G-space X is G-normal (equivariantly normal in the terminology of [8]) if every pair of α-disjoint closed subsets of X has α-disjoint nbds. Equivalently, X is G-normal iff C_{α}(X) separates the closed α-disjoint subsets of X. The continuity of the action α guarantees that for every closed subset F of X and every point x ∈ X \ F, the subsets F and {x} are α-disjoint. Therefore, every G-normal G-space is G-Tychonoff. Every coset G-space ⟨G, G/H, α_{L}⟩ is G-normal for arbitrary G. If G is locally compact, then every normal G-space is G-normal [8]. For the converse, see Theorem 5.2 below.

Definition 2.4. A G-space ⟨G, X, α⟩ is called weakly G-normal if C_{α}(X) separates the closed invariant subsets of X.

Proposition 2.5. Let G be an arbitrary topological group which is not supcomplete. Then there exists a normal G-space X of weight w(X) = w(G) which is not G-normal.

Proof. Denote by Ĝ the sup-completion of G. Let 𝔹 be a base of nbds of e in G. We may suppose that the cardinality of 𝔹 is not greater than w(G). By Brook’s theorem [3] the Ĝ-space ⟨Ĝ, Ĝ, α_{L}⟩ is Ĝ-Tychonoff. By [11], there is a compact Ĝextension Y of ⟨Ĝ, Ĝ, α_{L}⟩ such that w(Y) ≤ w(Ĝ) · w(Ĝ) = w(Ĝ) = w(G). We now form the topological Ĝ-sum X = ⊕{Y_{U} : U ∈ 𝔹}, where each Y_{U} := Y × {U} is a copy of Y. Let α denote the action of Ĝ on X restricted to G. Clearly, ⟨G, X, α⟩ is a normal G-space and w(X) = w(G). In order to show that it is not G-normal, we will construct two closed α-disjoint subsets C and D of X which are not separated by C_{α}(X).

Let U ∈ 𝔹, and let clĜ(U) be the closure of U in Ĝ. Since G is not sup-complete, we can pick g_{U} ∈ clĜ(U)\G. Set C := {(g_{U}, U) : U ∈ 𝔹} and D := {(e, U) : U ∈ 𝔹}. Clearly, C and D are closed α-disjoint subsets of X. It is easy to show, however, that for all V ∈ 𝔹

(g_{V} , V) ∈ cl_{X}(V D) ∩ C.

By lemma 2.3, C and D are not separated by C_{α}(X). □

For a compact space X, H(X) denotes the (topological) group of all homeomorphisms of X, endowed with the compact-open topology. Let ⟨G, (X, μ), α⟩ be an arbitrary G-space, where μ is a compatible uniformity for X, and let A ⊆ G. We say that A acts μ-uniformly equicontinuously if for every ε ∈ μ there exists δ ∈ μ such that for all g ∈ A and for all x, y ∈ X, (x, y) ∈ δ implies (gx, gy) ∈ ε. We say that G acts locally uniformly equicontinuously if there exists U ∈ N_{e}(G) such that U acts μ-uniformly equicontinuously.

Lemma 2.6. Let (G, τ) be a topological group. For g ∈ G, let g̃ : β_{G}G → β_{G}G be the extension of the transition map α^{g} : G → G. Then the map φ : G → H(β_{G}G) defined by φ(g) = g̃ is a topological group embedding.

Proof. Straightforward. □

Lemma 2.7. The action of G on the greatest ambit β_{G}G is locally uniformly equicontinuous iff G is locally precompact.

Proof. Suppose the action α̃ : G×β_{G}G → β_{G}G is locally uniformly equicontinuous. That is, there is a neighborhood U of e which acts μ-uniformly equicontinuously, where μ is the unique uniformity for β_{G}G. Then, the Ascoli-Arzela theorem implies that the closure of φ(U) in H(β_{G}G) is compact. Therefore, the closure of φ(G) in H(β_{G}G) is a locally compact group containing the group G = φ(G) (Lemma 2.6).

Conversely, suppose G is locally precompact. By the equivariant completion theorem [11], there is a continuous extending action of the sup-completion Ĝ of G on β_{G}G. By our assumption, there exists a compact neighborhood V of e in Ĝ. Easy compactness arguments now imply that V acts μ-uniformly equicontinuously on β_{G}G. Hence the neighborhood V ∩ G of e in G also acts μ-uniformly equicontinuously on β_{G}G. □

Recall that a topological group G is called ℵ_{0}-bounded [2,6] if for every V ∈ N_{e}(G) there exists a countable subset S of G such that SV = G. Guran [6] proved that G is ℵ_{0}-bounded iff G is a topological subgroup of a product of second countable topological groups. If G is separable, Lindelöf, or ccc, then G is ℵ_{0}-bounded.

Fact 2.8 (Equivariant Approximation Theorem) [10]. Let G be an ℵ_{0}- bounded group, and let X be a compact G-space. Then X can be represented as a G-limit of an inverse G-system {⟨G, X_{i}, α_{i}⟩ : i ∈ I} of compact metrizable Gspaces X_{i} such that dim X_{i} ≤ dim X.

3. From non-G-normality to non-G-Tychonoff-ness.

We now adapt the original construction of Tree [14] to the context of G-spaces. Let ⟨G, X, α⟩ be a regular G-space. We describe the construction of a related Gspace ⟨G, X^{+}, α^{+}⟩. We assume that there are disjoint closed G-subspaces C, D ⊆ X.

Let ω = ℕ ∪ {0} in the discrete topology. Let Y be the quotient space formed from X × ω by identifying the pairs (c, 2i + 1) and (c, 2i + 2) for c ∈ C, i ∈ ω, and the pairs (d, 2i) and (d, 2i + 1) for d ∈ D, i ∈ ω. Let p: X × ω → Y be the quotient map. For n ∈ ω, let i_{n}: X → X × ω be the canonical injection x ↦ (x, n).

Fix a point a ∉ Y, and let X^{+} = Y ∪ {a}. Topologize X^{+} by setting Y to be an open subset with its quotient topology and the nth basic nbd of a to be

N_{n}(a) = {a} ∪ p(i_{2n}(X \ C)) ∪ ⋃{p(i_{m}(X)): m > 2n}.

It is trivial to check that this generates a topology τ on X^{+}. We remark only that if O is an open subset of Y, then O ∩ N_{n}(a) is also open in Y. To see this, it is enough, by definition of the quotient topology, to show that p^{−1}(O∩N_{n}(a)) is open. Observe that p^{−1}(O) is open by continuity of p, and p^{−1}(N_{n}(a)) = N_{n}(a) \ {a} is open by definition of N_{n}(a).

We now define a function α^{+} : G×X^{+} → X^{+}. For any g ∈ G, set α^{+}(g, a) = a. For p((x, n)) ∈ Y, set α^{+}(g, p((x, n))) = p(i_{n}(α(g, x))). Note that α^{+} is well-defined because C and D are invariant.

Claim 3.1. ⟨G, X^{+}, α^{+}⟩ is a G-space.

Proof. It is easy to see that α^{+} is an action on X^{+}. It remains to show that α^{+} is continuous. Continuity of α^{+} at points (g, y) for y ∈ Y is easily proved using the continuity of α. For continuity at the point (g, a), note that for any n ∈ ℕ, GN_{n}(a) = N_{n}(a).

Theorem 3.2. Let ⟨G, X, α⟩ be a normal G-space with closed disjoint invariant subsets C, D ⊆ X which are not separated by C_{α}(X). (Hence X is a non-weakly Gnormal G-space). Then ⟨G, X^{+}, α^{+}⟩ is a normal (and, hence, Tychonoff) G-space which is not G-Tychonoff.

Proof. First we show that X^{+} is normal.

Claim 1. If A ⊆ Y, U is an open subset of X × ω, and p^{−1}(A) ⊆ U, then there is an open subset U′ of X × ω such that p^{−1}(A) ⊆ U′ ⊆ U, and p(U′) is open in the space Y.

Proof. For each (x, n) ∈ p^{−1}(A), we will define an open set U_{x,n} ⊆ X × {n} such that (x, n) ∈ U_{x,n} ⊆ U. If x ∉ C ∪ D (or if n = 0 and x ∉ D), then there is an open set U′ ⊆ X such that x ∈ U′ and U′ ∩ (C ∪ D) = ∅. Since (x, n) ∈ U, there is an open set U′′ ⊆ X × {n} such that (x, n) ∈ U′′ ⊆ U. Let U_{x,n} := (U′ × {n}) ∩ U′′. Then U_{x,n} is open, (x, n) ∈ U_{x,n}, U_{x,n} ⊆ U, and p(U_{x,n}) is open in Y, since p^{−1}(p(U_{x,n})) = U_{x,n}. If x ∈ C, then assume without loss of generality that n is odd. So both (x, n) and (x, n + 1) belong to p^{−1}(A). Since (x, n) ∈ U, there is an open set U_{n} ⊆ X × {n} such that (x, n) ∈ U_{n} ⊆ U. Similarly, there is an open set U_{n+1} ⊆ X × {n + 1} such that (x, n + 1) ∈ U_{n+1} ⊆ U. Let

U_{x,n} := {(y, n) ∈ U_{n}: (y, n + 1) ∈ U_{n+1}}
U_{x,n+1} := {(y, n + 1) ∈ U_{n+1} : (y, n) ∈ U_{n}}.

The case x ∈ D is handled similarly.

Finally, define U′ = ⋃_{(x,n)∈p^{−1}(A)} U_{x,n}. This proves claim 1.

Claim 2. If C ⊆ X^{+} is closed, and a ∉ C, then there exists an n ∈ ω such that C ⊆ ⋃_{k<n} p(X × {k}).

Proof. If not, then for every n ∈ ω, there are m ≥ n and x_{n} ∈ p(X × {m}) such that x_{n} ∈ C. Then x_{n} → a, so a ∈ C.

Claim 3. X^{+} is normal (and, hence, Tychonoff).

Proof. Let A, B ⊆ X^{+} be closed and disjoint. Case 1: a ∉ A ∪ B. In this case, A and B are closed subsets of Y. So p^{−1}(A) and p^{−1}(B) are closed disjoint subsets of X × w. Since X × ω is normal, there are disjoint open sets U, V ⊆ X×ω such that p^{−1}(A) ⊆ U and p^{−1}(B) ⊆ V. By Claim 1, let U′, V′ ⊆ X×ω be open sets such that p^{−1}(A) ⊆ U′ ⊆ U, p^{−1}(B) ⊆ V′ ⊆ V, and p(U′), p(V′) are open in Y. Note that U′ and V′ are disjoint. Now

A = p(p^{−1}(A)) ⊆ p(U′) and B = p(p^{−1}(B)) ⊆ p(V′).

Subclaim. p(U′) ∩ p(V′) = ∅.

Proof. Suppose z ∈ p(U′) ∩ p(V′). If z = p((x, n)), where x ∉ C ∪ D, then (x, n) ∈ U′ ∩ V′ = ∅. Contradiction. Hence we must have x ∈ C ∪ D. Without loss of generality, assume that x ∈ C, n is odd, and (x, n) ∈ U′, (x, n + 1) ∈ V′. From the construction of U′ in Claim 1, we see that there must c ∈ C such that (x, n) ∈ U_{c,n}. So (x, n + 1) ∈ U_{c,n+1}. Hence (x, n + 1) ∈ U′ ∩ V′. Contradiction.

Case 2: a ∈ A ∪ B. Without loss of generality, assume that a ∈ A \ B. So A ∩ Y = A \ {a} is closed in Y. So we apply Case 1 to separate A ∩ Y and B by disjoint open sets U and V, respectively. By Claim 2 there is n ∈ ω such that

B ⊆ ⋃_{k<n} p(X × {k}).

Assume without loss of generality that n is odd. Define V′ := V ∩ [(⋃_{k<n} p(X × {k})) ⋃ p((X \ C) × {n})]. Let U′ = U ∪ N_{n}(a). Then A ⊆ U′, B ⊆ V′, U′ ∩ V′ = ∅, and U′ and V′ are open.

This completes the proof that X^{+} is normal.

Next, we will show that ⟨G, X^{+}, α^{+}⟩ is not G-Tychonoff. Our proof follows closely the proof in [14]. For the sake of completeness, we have included many of the details.

Notation. For a subset A ⊆ X and n ∈ ω, let A_{n}^{+} = p(i_{n}(A)).

We claim that the point a and the set C_{0}^{+} are not separated by a bounded α^{+}-uniform function. The proof is by contradiction. Suppose f : X^{+} → ℝ is a bounded α^{+}-uniform function such that f(C_{0}^{+}) = 0 and f(a) = 1. As in Tree [14], there is an n ∈ ω such that for all m ≥ n and for all x ∈ C_{m}^{+}, |f(x)| ≥ \frac{1}{n}. Fix such

an n. Note that f(C_{n}^{+}) ∩ (−\frac{1}{n}, \frac{1}{n}) = ∅. Assume without loss of generality that n

is odd. Let n = 2r + 1. Define a function μ : X^{+} → ℝ by μ(x) = min{n|f(x)|, 1}. By lemma 2.1, μ is α^{+}-uniform. Note that μ(C_{0}^{+}) = 0 and μ(C_{n}^{+}) = 1. We now define n + 1 functions h_{0}, h_{1}, …, h_{n} : X → ℝ by restricting the domain of μ to X_{k}^{+} for k = 0, 1, …, n. More precisely,

h_{k}(x) = μ(p(i_{k}(x))).

Claim 4. Each h_{k} is α-uniform.

Proof. Fix ε > 0. Let U be a nbd of e in G such that if x ∈ X^{+} and g ∈ U, then |μ(x)−μ(gx)| < ε. Fix x ∈ X, g ∈ U and k ∈ {0, 1, …, n}. Then |h_{k}(x)−h_{k}(gx)| = |μ(p(i_{k}(x)))−μ(p(i_{k}(gx)))| = |μ(p(x, k))−μ(p(gx, k))| = |μ(p(x, k))−μ(gp(x, k))| < ε. This proves claim 4.

Observation.

(1) For i = 0, 1, …, r − 1,

h_{2i+1} ↾ C = h_{2i+2} ↾ C.

(2) For i = 0, 1, …, r,

h_{2_{i}} ↾ D = h_{2i+1} ↾ D.

(3) h_{0} ↾ C = 0.

(4) h_{n} ↾ C = 1.

Finally, define h : X → ℝ by

h(x) = ∑_{k=0}^{n}(−1)^{k+1}h_{k}(x).

By lemma 2.1, h is α-uniform. Now c ∈ C ⇒ h(c) = −h_{0}(c) + h_{n}(c) = 0 + 1 = 1, by (1), (3) and (4) above, while d ∈ D ⇒ h(d) = 0 by (2).

This contradicts the assumption that C and D cannot be separated by an α-uniform function. This completes the proof of theorem 3.2. □

4. Proof of the Main Theorem.

Definition 4.1. Let ⟨G, X, α⟩ be a G-space. Let S ⊆ G. Subsets A and B of X are said to be S-near if for every pair O_{1}, O_{2} of neighborhoods of A and B, respectively, there exist g_{1}, g_{2} ∈ S such that g_{1}O_{1} ∩ g_{2}O_{2} ≠ ∅, i.e., SO_{1} ∩ SO_{2} ≠ ∅.

Lemma 4.2. Let ⟨G, X, α⟩ be a G-space with a fixed point z. Let S ⊆ G. Let x ∈ X \ {z} be such that x and z are S-near. Set Y = X^{2} \ {(z, z)}, C = ({z} × X) \ {(z, z)}, and D = (X × {z}) \ {(z, z)}. Define α′ : G × Y → Y by the “onecoordinate” action α′(g, (x, y)) = (α(g, x), y). Then ⟨G, Y, α′⟩ is a G-space, and C and D are closed disjoint invariant S-near subsets of Y.

Proof. Only the statement about the S-nearness of C and D is non-trivial. Fix nbds O_{1} and O_{2} of C and D, respectively. Since (x, z) ∈ D, we have (x, z) ∈ O_{2}, and hence there are open nbds O_{x}(x), O_{z}(z) of x and z, respectively, such that

(x, z) ∈ (O_{x}(x) × O_{z}(z)) \ {(z, z)} ⊆ O_{2}.

It is easy to see that z is not isolated. Hence there exists y ∈ O_{z}(z) such that y ≠ z. Then (z, y) ∈ O_{1}, so there are open nbds O′_{z}(z), O_{y}(y) of z and y, respectively, such that

(z, y) ∈ (O′_{z}(z) × O_{y}(y)) \ {(z, z)} ⊆ O_{1}.

By hypothesis, x and z are S-near, so there are g_{1}, g_{2} ∈ S such that g_{1}O_{x}(x) ∩ g_{2}O′_{z}(z) ≠ ∅. Therefore,

g_{1}(O_{x} × O_{z}) ∩ g_{2}(O′_{z} × O_{y}) ≠ ∅.

This easily implies that g_{1}O_{1} ∩ g_{2}O_{2} ≠ ∅. □

Main Theorem 4.3. Let G be a non-locally precompact ℵ_{0}-bounded topological group. Then G is not a V -group, i.e., there is a Tychonoff G-space which is not G-Tychonoff.

Proof. Since G is not locally precompact, Lemma 2.7 implies that G does not act locally uniformly equicontinuously on X = β_{G}G.

We now build a G-space X_{U} for every U in a collection 𝔹 (of cardinality χ(G)) of basic nbds of e in G. By the approximation theorem 2.8, X is the G-limit of an inverse G-system of compact metrizable G-spaces X_{i} (i ∈ I). Let μ and μ_{i} denote the unique uniformity on X and X_{i}, respectively.

Let U ∈ 𝔹. Since U does not act μ-uniformly equicontinuously on β_{G}G, there exists an index i ∈ I such that U does not act μ_{i}-uniformly equicontinuously on (X_{i}, μ_{i}). Therefore, there is ε ∈ μ_{i} such that for every δ ∈ μ_{i} there exist (x_{δ}, y_{δ}) ∈ δ and g_{δ} ∈ U such that

(g_{δ}x_{δ}, g_{δ}y_{δ}) ∉ ε. (∗)

Thus we obtain nets ⟨x_{δ}⟩, ⟨y_{δ}⟩, ⟨g_{δ}x_{δ}⟩, and ⟨g_{δ}y_{δ}⟩ in (X_{i}, μ_{i}), indexed by the elements δ of μ_{i}. Passing to subnets if necessary, we may assume that there exist x^{U}, a^{U}, b^{U} ∈ X such that x_{δ} → x^{U}, y_{δ} → x^{U}, g_{δ}x_{δ} → a^{U}, and g_{δ}y_{δ} → b^{U}. By (∗) we have a^{U} ≠ b^{U}. Hence (x^{U}, x^{U}) ∈ Δ, and (a^{U}, b^{U}) ∉ Δ, where Δ = {(x, x)|x ∈ X_{i}}. Clearly, (x^{U}, x^{U}) and (a^{U}, b^{U}) are U-near in the G-space (X_{i}, μ_{i}) × (X_{i}, μ_{i}), with the natural “two-coordinate” action.

We now form the quotient G-space Y_{i} = (X_{i} × X_{i})/Δ. Consider the quotient G-map p : X_{i} × X_{i} → Y_{i}. Let z := p(x, x). Then z and p(a^{U}, b^{U}) are U-near in Y_{i}.

Since z is a fixed point of Y_{i}, we may apply Lemma 4.2, with X = Y_{i}, S = U, and x = p(a^{U}, b^{U}). That is, let X_{U} = (Y_{i} × Y_{i}) \ {(z, z)}, C^{U} = ({z} × Y_{i}) \ {(z, z)}, and D^{U} = (Y_{i}×{z})\{(z, z)}. Define the “one-coordinate” action α′ : G×X_{U} → X_{U} by

α′(g, (x, y)) = (α(g, x), y),

where α is the action of G on Y_{i}. Then, by lemma 4.2, ⟨G, X_{U}, α′⟩ is a G-space, and C^{U} and D^{U} are closed disjoint invariant U-near subsets of X_{U}. This completes the contruction of X_{U} from U.

Now form the topological G-sum S = ⊕{X_{U} : U ∈ 𝔹}. Let α^{∗} : G × S → S be the natural action. Define

C = ⋃_{U∈𝔹} C^{U}, D = ⋃_{U∈𝔹} D^{U}.

Then C and D are closed disjoint invariant subsets of S.

Claim 1. C and D are not separated by C_{α^{∗}}(S).

Proof. By lemma 2.3, it is enough to show that there do not exist α^{∗}-disjoint nbds of C and D. Let O_{C}, O_{D} be nbds of C and D, respectively. Let U ∈ N_{e}(G), U′ ⊆ U, with U′ ∈ 𝔹. Define O_{1} = O_{C} ∩ X_{U′}, O_{2} = O_{D} ∩ X_{U′}. Then U′O_{1} ∩ U′O_{2} ≠ ∅. Hence UO_{C} ∩ UO_{D} ≠ ∅.

Claim 2. S is metrizable.

Proof. It is sufficient to show that each X_{U} in the topological sum is metrizable. Each X_{i} is a metrizable compact space and Δ is closed in X_{i} × X_{i}. Clearly, Y_{i} = (X_{i} × X_{i})/Δ is metrizable and compact. Hence every X_{U} is metrizable, being a subspace of Y_{i} × Y_{i}.

By the above claims, ⟨G, S, α^{∗}⟩ is a normal G-space which is not weakly Gnormal. Therefore, by theorem 3.2, ⟨G, S^{+}, (α^{∗})^{+}⟩ is a Tychonoff G-space which is not G-Tychonoff. This completes the proof of the Main Theorem. □

We discuss here some topological properties of S^{+}. Observe that by our construction, w(S^{+}) = w(S) · χ(G). If G is second countable, then we can choose S such that w(S) = w(G). In this case, we get w(S^{+}) = w(G) = ℵ_{0}.

Note that the space S^{+} has only one point, namely a, of non-local compactness. This is interesting because any locally compact G-space is G-Tychonoff.

5. Further Results.

The following corollary to the Main Theorem provides a negative answer to Megrelishvili’s question [11] whether every monothetic group is a V -group.

Corollary 5.1. There exists a cyclic (metrizable) topological group G which is not a V -group.

Proof. Take, for example, a dense cyclic subgroup G of the monothetic non-locally precompact group from [13]. □

It is interesting to note that local compactness of the acting group G can be characterized in terms of G-normality.

Theorem 5.2. The following are equivalent for a topological group G: (i) G is locally compact. (ii) Every normal G-space is G-normal.

Proof. (i) ⇒ (ii) see [8].

(ii) ⇒ (i) First observe that by Proposition 2.5, G is sup-complete. Therefore, we may assume that G is not locally precompact. Then, by lemma 2.7, the action of G on X = β_{G}G is not locally uniformly equicontinuous. Let U ∈ N_{e}(G), and let μ be the unique uniformity for X. Then U does not act μ-uniformly equicontinuously. As in the proof of the Main Theorem, we obtain nets ⟨x_{δ}⟩, ⟨y_{δ}⟩ in X and a net ⟨g_{δ}⟩ in U, each indexed by the elements δ of μ. Without loss of generality, we may assume that there exist x^{U}, a^{U}, b^{U} ∈ X such that x_{δ} → x^{U}, y_{δ} → x^{U}, g_{δ}x_{δ} → a^{U}, g_{δ}y_{δ} → b^{U}, and a^{U} ≠ b^{U}. Form the topological G-sum S = ⊕{Y_{U} : U ∈ N_{e}(G)}, where each Y_{U} is a copy of X × X, with the “two-coordinate” action. Clearly, S is normal. Define C = {(x^{U}, x^{U})_{U} : U ∈ N_{e}(G)} and D = {(a^{U}, b^{U})_{U} : U ∈ N_{e}(G)}. Then C and D are closed and α-disjoint subsets of S. However, for every U ∈ N_{e}(G) and for every pair of nbds O_{1}, O_{2} of C and D, respectively, we have

cl(UO_{1}) ∩ cl(UO_{2}) ≠ ∅.

Now lemma 2.3 implies that C and D are not separated by C_{α}(X). This proves that S is not G-normal. □

In a topological space X, points a and b are called twins if f(a) = f(b) for every continuous function f : X → ℝ. In a G-space ⟨G, X, α⟩, points a and b are called α-twins if f(a) = f(b) for every α-uniform function f : X → ℝ.

In [14], Tree shows how to modify any regular space which is not completely regular in order to obtain a regular space with twins. We have adapted his technique to the context of G-spaces.

Let ⟨G, X, α⟩ be a regular G-space. We construct a related G-space ⟨G, X^{∗}, α^{∗}⟩. We assume that there is a closed invariant subset C of X. Let X^{∗} be the quotient space formed from X × {0, 1} by identifying the pairs (c, 0) and (c, 1) for c ∈ C.

Let p : X × {0, 1} → X^{∗} be the quotient map. Let i_{0}, i_{1} : X → X × {0, 1} be the canonical injections. Define α^{∗} : G × X^{∗} → X^{∗} by α^{∗}(g, p((x, n))) = p(i_{n}(α(g, x))).

The action α^{∗} is well-defined because C is invariant. It is easy to see that ⟨G, X^{∗}, α^{∗}⟩ is a G-space. The corresponding proof in [14] works here since C_{α}(X) is an algebra. Thus we obtain

Theorem 5.3. Let ⟨G, X, α⟩ be a normal G-space with a fixed point z and a closed invariant subset C which are not separated by C_{α}(X). Then ⟨G, X^{∗}, α^{∗}⟩ is a Tychonoff G-space with α^{∗}-twins. (In fact, X^{∗} is even normal.)

Combining the Main Theorem 4.3 and Theorem 5.3, we see that for every non-locally precompact, ℵ_{0}-bounded group G, there is a Tychonoff G-space with α-twins.

Questions.

(1) What can be said in the Main Theorem if G is locally precompact (nonlocally compact)? For example, is the additive group ℚ of rational numbers a V -group?

(2) In Theorem 5.2, can we replace “G-normal” by “weakly G-normal”?

(3) (Yu.Smirnov) Is there a Tychonoff G-space X on which every α-uniform function is constant?

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E-mail: megereli@macs.biu.ac.il scarr@macs.biu.ac.il