Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

THE MAXIMAL G-COMPACTIFICATIONS OF G-SPACES WITH SPECIAL ACTIONS

V. A. CHATYRKO AND K. L. KOZLOV

Abstract. An action on a G-space induces uniformities on the phase space. It is shown when the maximal G-compactification of a G-space can be obtained as a completion of the phase space with respect to one of these uniformities. Structure of G-spaces with special actions is investigated.

This paper is the continuation of the previous work of the authors [2] and is partially supported by Kungliga Vetenskapademian, project 12529. Besides old results which are now proved using another techniques the new ones are presented.

All spaces are assumed to be Tychonoff and mappings are continuous mappings of spaces. Let R denote the real numbers, and nbd is an abridged notation for neighbourhood.

Let G be a topological group. By a G-space X we mean a Tychonoff space X (phase space) with a continuous action of group G. If for a G-space X there exist a compact G-space bX and an equivariant dense embedding of X into bX then we call bX a G-compactification (see, for example, [3]). If a G-space has a G-compactification then (see, for example, [1]) there is the largest element β_{G}X among all G-compactifications which is called the maximal G-compactification.

Uniform structures are introduced by coverings [4], and we say that the uniformity U_{1} is finer than the uniformity U_{2} if U_{2} ⊂ U_{1}. If the topology of a topological space and the topology induced by a uniformity on it are the same then we say that the uniformity is compatible with the topology of the space. If X is a G-space then the uniformity U on X is called invariant if for any g ∈ G and γ ∈ U gγ ∈ U.

2000 Mathematics Subject Classification. 54D35.

Key words and phrases. G-space, uniformity.

The second author is supported by RFFI, project 99-01-00128.

V. A. Chatyrko and K. L. Kozlov, The maximal G-compactifications of G-spaces with special actions, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 15–21, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204118.

In 1975 J. de Vries [3] introduced the notion of a bounded action (an action on a space X is bounded if there exists a uniformity U on X compatible with its topology such that for any u ∈ U there is a nbd O of identity in G such that the pair of points x and gx belong to one element of u for any x ∈ X and any g ∈ O) and proved that a G-space X has a G-compactification iff the action is bounded.

In 1984 M. G. Megrelishvili [5] introduced the concept of an equiuniformity (uniformity is equiuniformity if it is compatible with the topology of the phase space, invariant and the action is bounded by it) and proved the following theorem.

Theorem A. (M. G. Megrelishvili [5]) If U is an equiuniformity on a G space X then its completion X̃ with respect to U is a G space. Besides if f : X → Y is an equivariant uniformly continuous mapping to a complete uniform space Y then there exists the unique equivariant continuous mapping f̃ : X̃ → Y such that f̃∘ i = f where i is a natural embedding of X into X̃.

The following statements are evident.

Proposition 1. Let U_{α}, α ∈ A, be the family of equiuniformities on a G space X. Then its least upper bound is an equiuniformity.

Corollary 1. Among all equiuniformities there is a maximal one.

Proposition 2. If U is an equiuniformity on a G space X then the set of all coverings which can be refined by a finite covering from U is an equiuniformity.

Corollary 2. Among all equiuniformities there is a maximal totally bounded one.

Let A be the family of all open nbds of the identity in G. Every O ∈ A sets two coverings of a G-space X:

γ_{O} = {Ox : x ∈ X} and γ̄_{O} = {cl(Ox) : x ∈ X}.

Denote by U_{G} ( Ū_{G} ) the family of all coverings of X which have a refinement of the form γ_{O} (γ̄_{O}), O ∈ A. It may be easily checked that the family U_{G} is a uniformity on X (not nessesary compatible with the topology of X).

Remark 1. If Ū_{G} is a uniformity on X then the uniformity U_{G} is finer than Ū_{G}, but they may not be compatible with the topology of the phase space.

Now we shall reformulate J. de Vries’s criterion mentioned above.

Theorem B. (J. de Vries [3]) A G-space X has a G-compactification iff there is a uniformity U on X compatible with its topology such that U_{G} is finer than U.

Let U^{∗} be the totally bounded uniformity on the space X compatible with its topology such that any bounded continuous function on X is uniformly continuous with respect to it. It is the maximal totally bounded uniformity on X.

MAXIMAL G-COMPACTIFICATIONS OF G-SPACES 17

Theorem 1. Let X be a G-space. If the uniformity U_{G} is finer than U^{∗} then

β_{G}X = βX.

Proof. It is easy to see that U^{∗} is an equiuniformity and the rest follows from Corollary 2 and Theorem A. □

Lemma 1. Let U be a uniformity on a G-space X compatible with its topology. If Ū_{G} is a uniformity on X then the following conditions are equivalent:

(1) U_{G} is finer than U;

(2) Ū_{G} is finer than U.

Moreover, if the uniformity U_{G} is compatible with the topology of X then uniformities U_{G} and Ū_{G} are the same.

Proof. Let the uniformity U be generated by the family of coverings. In order to show (1) ⇒ (2) for any v ∈ U take v′ ∈ U such that v′ is a star refinement of v. Then the covering [v′] ∈ U which consists of closures of elements of v′ is a refinement of v. Take γ_{O} ∈ U_{G} such that γ_{O} is a refinement of v′. Then γ̄_{O} ∈ Ū_{G} is a refinement of [v′]. From this it follows that γ̄_{O} is a refinement of v and hence Ū_{G} is finer than U.

The implication (2) ⇒ (1) follows from Remark 1.

From Remark 1 it follows that U_{G} is finer than Ū_{G}. If U_{G} is compatible with the topology of X then instead of U we can take U_{G} in our lemma. Then Ū_{G} is finer U_{G} also. Hence U_{G} and Ū_{G} are the same. □

Proposition 3. If Ū_{G} is a uniformity compatible with the topology of X then it is a maximal equiuniformity.

Proof. Since the uniformity U_{G} is finer than Ū_{G}, it follows from Theorem B that the action is bounded by the uniformity Ū_{G}.

In order to prove that the uniformity Ū_{G} is invariant it is sufficient to show that for any γ̄_{O} = {cl(Ox) : x ∈ X}, O ∈ A, gγ̄_{O} ∈ Ū_{G}. Since for any g ∈ G the mapping g : X → X, x → gx is a homeomorphism it follows that g(cl(Ox)) = cl((gO)x). Take U = gOg^{−1}. Then U ∈ A and (gO)x = (Ug)x for any x ∈ X. Thus gγ̄_{O} = γ̄_{U}.

Hence Ū_{G} is an equiuniformity. Its maximality follows from Lemma 1. □

The proof of the following theorem immediately follows from Proposition 3, Corollary 2 and Theorem A.

Theorem 2. Let X be a G-space. If Ū_{G} is a uniformity compatible with the topology of X then

β_{G}X is the Samuel compactification of X with respect to Ū_{G}.

The next example shows that the usage of uniformity Ū_{G} gives us more opportunities in finding maximal G-compactifications.

Example 1. Let S = {z ∈ C : |z| = 1} be a unit circle on the complex plain, and a be such an element of S that a^{n} ≠ 1, n ∈ N. Let us put G = {a^{n} : n ∈ Z} (it is a group with a natural multiplication), X = S \ G and the action of G on X is induced by multiplication in C.

The uniformity U_{G} is not compatible with the topology of X because the group G is countable and the cardinality of each nonempty open set of X is uncountable and the uniformity Ū_{G} is compatible because G is a dense subset of S.

Remark 2. Earlier Theorems 1 and 2 were proved in another way in [2] using results of J. de Vries [3] and Yu. M. Smirnov [1].

We can characterize the case when Ū_{G} is the uniformity compatible with the topology of the phase space.

Theorem 3. Let X be a G-space. The family Ū_{G} is a uniformity compatible with the topology of X iff the action has the property:

(a) for any x ∈ X and any nbd O ∈ A there exists y ∈ X such that x ∈ int cl(Oy).

Proof. First of all let us notice that property (a) is equivalent to the following one:

the family {int cl(Ox) : x ∈ X} is a covering of X for any nbd O ∈ A.

Since the universal uniformity is finer than Ū_{G} an open covering of X may be refined in any covering {cl(Ox) : x ∈ X} from Ū_{G}. From this necessity immediately follows.

In order to prove sufficiency we must first of all check that Ū_{G} is the uniformity (see, for example, [4, page 524]). Recall that A is the family of all open nbds of identity in G.

1. Right from the definition of Ū_{G} it follows that if γ ∈ Ū_{G} and γ is a refinement of a covering β of X then β ∈ Ū_{G}.

2. It is evident that if β_{1} and β_{2} ∈ Ū_{G} be such that γ̄_{V} and γ̄_{W} are refined in β_{1} and β_{2} for some V, W ∈ A respectively, then for O ∈ A such that O ⊂ V ∩ W we have that γ̄_{O} is refined both in γ̄_{V} and γ̄_{W} and hence in β_{1} and β_{2}.

3. For β ∈ Ū_{G} let V ∈ A be such that γ̄_{V} is refined in β. Take O ∈ A such that O = O^{−1} and O^{3} ⊂ V. We shall prove that γ̄_{O} is a barycentric refinement of γ̄_{V}.

Let us show that O cl(W x) ⊂ cl(OW x) for any nbds O and W of identity in G. If a ∈ O cl(W x) then a = ht, where h ∈ O and t ∈ cl(W x). Since the action is continuous for any nbd V_{a} of a there are a nbd V_{t} of t such that hV_{t} ⊂ V_{a}. Thus there is t′ ∈ V_{t} ∩ W x such that ht′ ∈ V_{a}. Hence, a ∈ cl(OW x).

For any x ∈ X there exists z ∈ X such that x ∈ int cl(Oz). Now if x ∈ cl(Oy) then cl(Oz) ∩ Oy ≠ ∅ since int cl(Oz) is a nbd of x. From this it follows that

y ∈ O^{−1} cl(Oz) ⊂ cl(O^{2}z) and Oy ⊂ O cl(O^{2}) ⊂ cl(O^{3}z) ⊂ cl(V z).

Thus cl(Oy) ⊂ cl(V z) and so st(x, γ̄_{O}) ⊂ cl(V_{z}). Hence, γ̄_{O} is a barycentric refinement of γ̄_{V}.

Using the same process, we can find a barycentric refinement of γ̄_{O} which would be the star refinement of γ̄_{V} [4, Lemma 5.1.15].

4. Let x, y be a pair of distinct points of X. Since the action is continuous and X is a Tychonoff space there are nbd O ∈ A, O^{−1} = O and nbds W_{x}, W_{y} of x and y respectively, such that cl(OW_{x})∩cl(OW_{y}) = ∅. Let us show that no element of the cover γ̄_{O} contains both x and y. Indeed, if x ∈ cl(Oz) and y ∈ cl(Oz) for some z ∈ X then W_{x} ∩Oz ≠ ∅ and W_{y} ∩Oz ≠ ∅. From this it follows that z ∈ O^{−1}W_{x} and z ∈ O^{−1}W_{y} and, hence, OW_{x}∩OW_{y} ≠ ∅. This is a contradiction with the choice of nbds O, W_{x} and W_{y}.

So all conditions for the uniformity Ū_{G} are fulfilled.

Since for any x ∈ X and any nbd O of identity in G there exists z ∈ X such that x ∈ int cl(Oz) then an open covering can be refined in any covering from Ū_{G}. So every open set in topology induced by uniformity Ū_{G} is open in X. If W is open in X and x ∈ W then there exist O ∈ A and a nbd V of x such that O = O^{−1} and cl(O^{2}V) ⊂ W. If x ∈ cl(Oy) and z ∈ cl(Oy) then there exist x_{1} ∈ V and h ∈ O such that x_{1} = hy. From this it follows that y ∈ Ox_{1} and z ∈ cl(O^{2}x_{1}) ⊂ cl(O^{2}V) ⊂ W. Hence, st(x, γ̄_{O}) ⊂ W and so W is open in the topology induced be the uniformity. □

Proposition 4. Consider the following properties for a G-space X.

(a) for any x ∈ X and any nbd O ∈ A there exists y ∈ X such that x ∈ int cl(Oy).

(b) for any x ∈ X and any nbd O ∈ A x ∈ int cl(Ox),

(c) for any x ∈ X and any nbd O ∈ A x ∈ int(Ox),

Then (c) ⟹ (b) ⟹ (a) and the inverse implications are not valid.

Proof. The implications (c) ⟹ (b) ⟹ (a) are evident.

(c) ⇍= (b). Consider the following example. Let Q be the set of rational numbers of the interval I = (0, 1). There is a natural linear order on Q. Let G be a group of all order preserving homeomorphisms of Q [4, page 18] with the topology of uniform convergence [4, page 329]. Using Theorem 2 one may show that β_{G}Q = [0, 1] and X = I is an invariant subset of β_{G}Q. Now it is easy to see that a G-space X satisfies (b) but not (c).

(b) ⇍= (a). Consider the following example. Put X = β_{G}Q where Q and G as above. It is easy to see that a G-space X satisfies (a) but not (b) because the action has fixed points. □

Remark 3. G-spaces with property (b) were examined by V. V. Uspenskiǐ in [6].

Below we shall describe G-spaces with properties listed above.

Lemma 2. If a G-space X satisfies property (a) then for any points x, y ∈ X we have either

int cl(Gx) = int cl(Gy) or int cl(Gx) ∩ int cl(Gy) = ∅.

Proof. For the proof it is sufficient to show that

if int cl(Gx) ∩ int cl(Gy) ≠ ∅ then int cl(Gx) ⊂ cl(Gy).

Let z ∈ int cl(Gx) and O_{z} is an arbitrary nbd of z. We may take z′ ∈ O_{z} ∩ Gx, z′′ ∈ int cl(Gy) ∩ Gx and g ∈ G such that gz′′ = z′. From the continuity of the action it follows that there exists a nbd O_{z′′} such that gO_{z′′} ⊂ O_{z}. If we take y′ ∈ O_{z′′} ∩ Gy then gy′ ∈ O_{z} and hence Gy ∩ O_{z} ≠ ∅. Since O_{z} is an arbitrary nbd of z it follows that z ∈ cl(Gy) and so int cl(Gx) ⊂ cl(Gy). □

Corollary 3. If a G-space X satisfies property (a) then either

int cl(Gx) = cl(Gx) or int cl(Gx) = ∅

for any point x ∈ X.

Theorem 4.

A) If a G-space X satisfies property (a) then X is a disjoint union of clopen sets and each clopen set from this union is the closure of an orbit of some point and thus it contains the continuous one-to-one image of some quotient space of group G.

B) If a G-space X satisfies property (b) then in addition to A) each such clopen set is the closure of orbit of its any point. But orbits of different points from the common clopen set may not be homeomorphic.

C) If a G-space X satisfies property (c) then each such clopen set is homeomorphic to the orbit of its any point.

Proof. Proof of the statements A) and B) follows from Lemma 2 and Corollary 3. The second example from Proposition 4 shows that in case of action with property (a) not orbit of any point may be taken (there may be fixed points). In case of action with property (b) the first example from Proposition 4 shows that different orbits (rationals and irrationals) may not be homeomorphic.

In case of action with property (c) we have for any x ∈ X an open mapping g → gx of G into X and hence a homeomorphism of some quotient space of G onto its orbit. □

The following questions are not yet known to authors.

Question 1. When does the family Ū_{G} generate uniformity on X (not nessesary compatible with the topology of X)?

Question 2. Let a G-space X satisfy property (a). Does there exist a dense invariant subspace X′ of X such that the restriction of action on it has property (b)?

Question 3. Let a G-space X satisfy property (b). Does there exist a dense invariant subspace X′ of X such that the restriction of action on it has property (c)?

Question 4. Can every compactification of a Tychonoff space be obtained as a G-compactification for some acting group G on X?

References

[1] S. A. Antonjan and Ju. M. Smirnov, Universal objects and bicompact extensions for topological groups of transformations, Dokl. Akad. Nauk SSSR 257 (1981), no. 3, 521– 526. MR 82i:54072

[2] V. A. Chatyrko and K. L. Kozlov, Dimension of maximal equivariant compact extensions, Preprint LiTH-MAT-R-2001-11, Linköping University, 2001.

[3] Jan de Vries, On the existence of G-compactifications, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 3, 275–280. MR 58 #31002

[4] Ryszard Engelking, General topology, PWN—Polish Scientific Publishers, Warsaw, 1977. MR 58 #18316b

[5] M. G. Megrelishvili, Equivariant completions and compact extensions, Soobshch. Akad. Nauk Gruzin. SSR 115 (1984), no. 1, 21–24. MR 86m:54054

[6] V. V. Uspenskiĭ, Topological groups and Dugundji compact spaces, Mat. Sb. 180 (1989), no. 8, 1092–1118, 1151, English translation in Math. USSR-Sb. 67 (1990), no. 2, 555– 580. MR 91a:54064

Department of Mathematics, Linköping University, 581 83 Linköping, Sweden

   E-mail address: vitja@mai.liu.se

Department of Mechanics and Mathematics, Moscow State University, 117234 Moscow, Russia