Topology and its Applications 160 (2013) 2523–2530

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Topology and its Applications

www.elsevier.com/locate/topol

Homogeneity and h-homogeneity

S.V. MedvedevFaculty of Mechanics and Mathematics, South Ural State University, pr. Lenina, 76, Chelyabinsk 454080,Russia

a r t i c l e i n f o a b s t r a c t

MSC:54F5054D30

Keywords:Homogeneous spaceh-Homogeneous spaceZero-dimensionalPseudocompactParacompact

The relations between zero-dimensional homogeneous spaces and h-homogeneousones are investigated. We suggest an indication of h-homogeneity for a homogeneouszero-dimensional paracompact space and its modification for a topological group.We describe some cases when the product X×Y is an h-homogeneous space providedX is h-homogeneous and Y is homogeneous.

© 2013 Elsevier B.V. All rights reserved.

We shall investigate the relations between zero-dimensional homogeneous spaces and h-homogeneous ones.For this we revise some basic statements about such spaces and suggest its improved versions. At first glancethe properties of zero-dimensional homogeneous spaces and h-homogeneous ones are similar, but this isnot so. For example, the Čech–Stone compactification of the rationals gives an example of an h-homogeneousspace that is not homogeneous. On the other hand, a zero-dimensional homogeneous compact space whichis not h-homogeneous was constructed by van Douwen [1]. There is also the Motorov example [2] of ahomogeneous zero-dimensional compact space which is divisible by 3 but not divisible by 2.

The main tool for checking homogeneity of an h-homogeneous first-countable space is Theorem 2. Butoutside the class of first-countable spaces it does not work. The Homogeneity Lemma, which is due tovan Mill [3], allows us to construct a number of examples of zero-dimensional homogeneous spaces withinteresting properties. We obtain a generalization of the lemma (see Theorem 3). Note that two methodsfor constructing h-homogeneous metric spaces were described in [4] and [5].

Every zero-dimensional h-homogeneous space must be weight-homogeneous. To establish h-homogeneityof a space X we introduce a cardinal function h(X) similar to the extent of X (see Section 3). We suggestan indication of h-homogeneity for a homogeneous paracompact space (see Theorem 4). Using it, we willshow that the notions of homogeneity and h-homogeneity are equivalent for a weight-homogeneous metricspace X with indX = 0 and cf(w(X)) > ω (see Corollary 3). For separable spaces the last statement isnot valid. In fact, van Douwen [1] constructed the separable metric zero-dimensional topological group H

E-mail address: medv@is74.ru.

0166-8641/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.topol.2013.07.046

2524 S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530

which is not h-homogeneous. Note that the group H is precompact. In contrast, every separable metriz-able zero-dimensional topological group, which is not locally precompact, is an h-homogeneous space (seeTheorem 8).

In Section 5 we describe some cases when the product X × Y is an h-homogeneous space if X

is h-homogeneous and Y is homogeneous. Terada [6] and Medini [7] studied the product X × Y forh-homogeneous Tychonoff spaces X and Y . Motorov [8] announced that the product X × Y of two zero-dimensional first countable compacta X and Y will be h-homogeneous provided X is h-homogeneous andY is homogeneous. We show (see Theorem 11) that in the last statement the condition “X and Y are firstcountable” can be dropped. Corollary 6 states that the product of a homogeneous compact space and theCantor set is a homogeneous, h-homogeneous space.

1. Notation

For all undefined terms and notation see [9].X ≈ Y means that X and Y are homeomorphic spaces. We identify cardinals with initial ordinals;

in particular, ω = {0, 1, 2, . . .}.A family B consisting of non-empty open subsets of a space X is a π-base if for every non-empty open

subset U ⊆ X there exists V ∈ B such that V ⊆ U .The smallest cardinal number k such that every open cover of a space X has an open refinement of

cardinality � k is called the Lindelöf number of the space X and is denoted by l(X).A clopen set is a set which is both closed and open. A space X is called zero-dimensional if X is a

non-empty T1-space and has a base consisting of clopen sets. Note that every zero-dimensional space is aTychonoff space. We say that a space X is disconnected if X can be represented in the form X1⊕X2, where X1

and X2 are non-empty subspaces of X. A space X is homogeneous if for any two points x, y ∈ X there existsa homeomorphism f : X → X with f(x) = y. A space X is called h-homogeneous if every non-empty clopensubset of X is homeomorphic to X. We do not suppose that an h-homogeneous space is zero-dimensional.A connected space is a trivial example of an h-homogeneous space because only the whole space is anon-empty clopen set in it; this case is not interesting for us. A space X is called weight-homogeneous if allnon-empty open subspaces of X have the same weight.

To verify h-homogeneity of a space we will often use the following theorem. Theorem 1 was proved byTerada [6] for a zero-dimensional space X. But Medini [7, Theorem 2] showed that the last assumption isredundant.

Theorem 1. Let X be a non-pseudocompact Tychonoff space. If X has a π-base consisting of clopen subsetsthat are homeomorphic to X, then X is h-homogeneous.

Example 1. Let Z ⊂ R2 be the union of the square [0, 1]× [0, 1] and the interval [1, 2]×{0}. Take the Cantor

set C. Clearly, the product Z × C is neither homogeneous nor zero-dimensional. Nevertheless, Z × C is anh-homogeneous space.

2. Indication of homogeneity

We have a unique tool (namely, Theorem 2) which allows us to check the homogeneity of anh-homogeneous space. Van Mill [10] refers to Theorem 2 as a well-known statement (see also [1]).

Theorem 2. Let X be a zero-dimensional h-homogeneous first countable space. Then X is homogeneous.

S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530 2525

In this section we obtain a slight generalization of the Homogeneity Lemma from [3]. Van Mill provedTheorem 3 for a zero-dimensional separable metric space and noted that it holds for a zero-dimensional firstcountable space. Proving Theorem 3, we use the van Mill technique [3].

Theorem 2 can be also deduced from Theorem 3 and the definition of an h-homogeneous space.We begin with a simple lemma.

Lemma 1. Let V be a clopen subset of a space X and a ∈ V . Suppose that g : V → g(V ) is a homeomorphismsuch that g(V ) is a clopen subset of X, b = g(a), and V ∩ g(V ) = ∅. Then there is a homeomorphismf : X → X such that f |V = g|V and f(b) = a.

Proof. The required map is defined by the rule

f(x) =

⎧⎪⎨⎪⎩

g(x) if x ∈ V,

g−1(x) if x ∈ g(V ),x if x /∈ V ∪ g(V ). �

Theorem 3. Let X be a Hausdorff space. Let {Vn: n ∈ ω} be a decreasing clopen base at a point a ∈ X and{Un: n ∈ ω} be a decreasing clopen base at a point b ∈ X such that Un is homeomorphic to Vn for eachn ∈ ω. Then there is a homeomorphism f : X → X such that f(a) = b, f(b) = a, and f ◦ f = idX .

Proof. If a = b, put f = idX . Below we will assume that a �= b.Without loss of generality, V0 ∩ U0 = ∅ because X is a Hausdorff space. Hence, Vn ∩ Un = ∅ for each

n ∈ ω.Denote by gn a homeomorphism of Vn onto Un. If there exists a number n with gn(a) = b, the theorem

follows from Lemma 1.Consider the case when gn(a) �= b for each n. Since X is a Hausdorff space and {gn(Vi): i � n} forms

a base at the point gn(a), for each n there is a j > n such that Uj ∩ gn(Vj) = ∅. To simplify the writing,we take j = n + 1, i.e., we will assume that gn(Vn+1) ∩ Un+1 = ∅. Then Vn+1 ∩ g−1

n (Un+1) = ∅.Define the mapping f0 : X → X by the rule

f0(x) =

⎧⎪⎨⎪⎩

g0(x) if x ∈ V1 ∪ g−10 (U1),

g−10 (x) if x ∈ U1 ∪ g0(V1),x otherwise.

Since V1 ∪ g−10 (U1) ⊆ V0, U1 ∪ g0(V1) ⊆ U0, and V0 ∩ U0 = ∅, f0 is well defined. One can check that f0 is a

homeomorphism.By induction on n, for each n � 1 we construct the homeomorphism fn : X → X by the rule

fn(x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

gn(x) if x ∈ Vn+1 ∪ g−1n (Un+1),

g−1n (x) if x ∈ Un+1 ∪ gn(Vn+1),f−1n−1 ◦ gn ◦ f−1

n−1(x) if x ∈ fn−1(Vn+1 ∪ g−1n (Un+1)),

fn−1 ◦ g−1n ◦ fn−1(x) if x ∈ f−1

n−1(Un+1 ∪ gn(Vn+1)),fn−1(x) otherwise.

By construction, for each n ∈ ω we have(a1) fn(Vn+1) = gn(Vn+1) ⊂ Un and fn(Un+1) = g−1

n (Un+1) ⊂ Vn,(a2) (Un+1 ∪ Vn+1) ∩ fn(Un+1 ∪ Vn+1) = ∅,(a3) fn ◦ fn = idX .

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Put f = lim{fn : n → ∞}. In particular, f(a) = b and f(b) = a. Let us show that the mapping f : X → X

is well defined. Take a point x ∈ X \{a, b}. Then x /∈ Vn∪Un for some n. From (a1) and (a2) it follows thatfn(x) /∈ Vn+1∪Un+1 and x /∈ fn(Vn+1∪Un+1). Hence, fn+1(x) = fn(x). So, f(x) = fn(x) ∈ X \{a, b}. Next,for y = f−1

n (x) one can verify that y /∈ Vn+1 ∪Un+1 and f(y) = x. Hence, f maps X onto X. If z /∈ Vn ∪Un

and z �= x, then f(x) = fn(x) �= fn(z) = f(z) because fn : X → X is a homeomorphism. Thus, f isone-to-one.

As noted above, if x /∈ Vn ∪ Un for some n, then f(x) = fn(x). Since ∩{Un ∪ Vn: n ∈ ω} = {a, b}, f andf−1 are continuous at every point x ∈ X \ {a, b} because each fn is a homeomorphism.

By construction, fn+1(Vn+1) ⊂ fn(Vn+1) for each n. Using (a1), we obtain f(Vn+1) ⊂ Un. Under theconditions of the theorem, {Un: n ∈ ω} is a clopen base at the point b. Since Vn+1 is a neighborhood of thepoint a, we conclude that f is continuous at a. Similarly, f is continuous at b. f−1 is the same. Thus, f isa homeomorphism.

From (a3) it follows that f ◦ f = idX . �3. Indications of h-homogeneity

Denote by h(X) the index of h-homogeneity of a space X. By definition, h(X) = k if k is the largestcardinal satisfying the following condition: for every non-empty clopen subset U of X there are a non-emptyclopen subset V ⊂ U and a discrete family V = {Vα: α ∈ k} consisting of clopen subsets of X such that⋃{Vα: α ∈ k} ⊆ U and Vα ≈ V for every α ∈ k. We will say that every such family V is subordinated to

the set U by means of V .Note that we have to be firmly convinced that the family V exists. Hence, h(X) might not be defined for

a space X. For example, the cardinal h(C) is not defined for the Cantor set C. For our purposes here, theuse of the notion of sup is not suitable.

Clearly, the index of h-homogeneity, if it exists, is less or equal to the Lindelöf number for every space.

Theorem 4. Let X be a homogeneous zero-dimensional paracompact space such that l(X) = h(X) = k, wherek � ω. Then X is an h-homogeneous space.

Proof. Fix a non-empty clopen subset U of X and a family {Vα: α ∈ k} subordinating to U by means of V .Take a point a ∈ V . From homogeneity of X it follows that for every point x ∈ X there exists a

homeomorphism fx : X → X such that fx(a) = x. Since the Lindelöf number l(X) = k, the clopen cover{fx(V ): x ∈ X} of X has an open locally finite refinement W = {Wα: α ∈ k1} of cardinality k1 � k. Using[11, Lemma 7.6.12], we may assume that W is a disjoint clopen cover of X. By construction, every Wα

is homeomorphic to a clopen subset V ∗α of Vα. Then the space X =

⋃{Wα: α ∈ k1} is homeomorphic to

U∗ =⋃{V ∗

α : α ∈ k1}. Clearly, U∗ ⊆ U and U∗ is a clopen subset of X.Choose a clopen base B for X. The family {U∗: U ∈ B} forms a clopen π-base for X such that every

U∗ ≈ X. One can verify that the space X is not pseudocompact. By virtue of Theorem 1, the space X ish-homogeneous. �Corollary 1. Let X be a homogeneous zero-dimensional metrizable space such that w(X) = h(X). Then X

is an h-homogeneous space.

Proof. The corollary follows from Theorem 4 because for a metrizable space X we have l(X) = w(X). �Suppose that {Xα: α ∈ A} is a family of topological spaces, X =

∏{Xα: α ∈ A}, and b = {bα: α ∈ A}

is a point in X. Then the σ-product of the family {Xα: α ∈ A} with the basic point b is the subspace of Xconsisting of all points x ∈ X that differs from the point b only in finitely many coordinates.

S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530 2527

Theorem 5. Let {Xα: α ∈ A} be a family of homogeneous zero-dimensional spaces, X =∏{Xα: α ∈ A},

and Y ⊂ X be the corresponding σ-product with center at b ∈ X. Suppose that the following conditions aresatisfied:

(1) for every finite J ⊂ A, the product XJ =∏{Xj : j ∈ J} is a paracompact space with l(XJ) � k,

(2) there is a β ∈ A such that h(Xβ) = k.Then Y is an h-homogeneous space.

Proof. Since every Xα is a homogeneous space, Y is the same. By [9, Theorem 6.2.14], we have indX = 0;hence, indY = 0. According to [12, Theorem 6], Y is a paracompact space. From [11, Corollary 1.6.45]it follows that l(Y ) � k. The condition (2) implies that l(Y ) � l(Xβ) � h(Xβ) = k because Y contains aclosed subset homeomorphic to Xβ . Then l(Y ) = k.

Put X∗ =∏{Xα: α ∈ A \ {β}} and b∗ = {bα: α ∈ A \ {β}} if b = {bα: α ∈ A}. Let Y ∗ ⊂ X∗ be the

corresponding σ-product with center at b∗ ∈ X∗. Then Y ≈ Xβ × Y ∗. Hence, h(Y ) � h(Xβ) = k. Sincel(Y ) = k and l(Y ) � h(Y ), we obtain h(Y ) = k. By virtue of Theorem 4, Y is an h-homogeneous space. �

The last theorem yields the following corollary.

Corollary 2. Suppose that X is an h-homogeneous zero-dimensional metrizable non-compact space of weightk and τ � ω. Let Y ⊂ Xτ be the σ-product with center at b ∈ Xτ . Then Y is an h-homogeneous space.

In the following theorem we describe the case when every clopen subset U ⊆ X has a subordinatedfamily V. In [5] Theorem 6 was proved for a metric space X with dimX = 0.

Theorem 6. Let X be a homogeneous, weight-homogeneous metrizable space such that indX = 0 andcf(w(X)) > ω. Then X is h-homogeneous.

Proof. Let w(X) = k. Fix a metric � on X. Choose a point a ∈ X and a base {On: n ∈ ω} at the point a

consisting of clopen subsets of X.Consider a non-empty clopen subset U ⊆ X. Then w(U) = k because X is weight-homogeneous. Since

cf(k) > ω, there is a closed ε-metrically discrete subspace Z ⊂ U of cardinality k for some ε > 0. Fromhomogeneity of X it follows that each point z ∈ Z has a clopen neighborhood Vz ⊂ U such that diam(Vz) <ε/2 and Vz is homeomorphic to On for some n. Denote by kn the cardinality of the family Vn = {Vz: z ∈ Z

and Vz ≈ On}. Then sup{kn: n ∈ ω} = k. There is a j with kj = k because cf(k) > ω. Hence, the familyVj is subordinated to U by means of any Vz ∈ Vj . It remains to apply Corollary 1. �

Theorems 6 and 2 yield

Corollary 3. Let X be a metrizable space such that indX = 0 and cf(w(X)) > ω. Then X is h-homogeneousif and only if X is homogeneous and weight-homogeneous.

4. h-Homogeneity of topological groups

Let W be a neighborhood of the neutral element e of a topological group G. A subset A of G is calledleft W -disjoint if b /∈ aW for any distinct a, b ∈ A. In a similar way we can define a right W -disjoint subsetA of G.

The following theorem is proved for a left W -disjoint subset A. Of course, its conclusion holds for a rightW -disjoint subset A too.

2528 S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530

Theorem 7. Let G be a paracompact zero-dimensional topological group with the Lindelöf number l(G) = k.Suppose that for every neighborhood U of the neutral element e of G there are a neighborhood W of e anda left W -disjoint subset A ⊂ U with |A| = k. Then the space G is h-homogeneous.

Proof. Take a neighborhood U of e satisfying U2 ⊆ U . Under the condition of the theorem, there are aneighborhood W of e and a left W -disjoint subset A ⊂ U with |A| = k. Choose a clopen neighborhood V ofe such that V 4 ⊂ W , V ⊂ U , and V −1 = V . According to Lemma 1.4.22 from [11], the family {aV : a ∈ A}is discrete in G. Hence, the union U∗ =

⋃{aV : a ∈ A} is a clopen subset of G. Clearly, U∗ ⊆ UV ⊆ U .

Then h(G) = k. By Theorem 4, G is an h-homogeneous space. �A topological group G is called locally precompact if there exists a neighborhood U of the neutral element

e of G such that U can be covered by finitely many left and right translates of each neighborhood of e in G.The following theorem was proved in [13].

Theorem 8. Let G be a separable zero-dimensional metrizable topological group which is not locally precom-pact. Then the space G is h-homogeneous.

Proof. Consider a neighborhood U of e. Suppose that U cannot be covered by finitely many left translates ofsome neighborhood V of e in G; the case for right translates is similar. According to the Birkhoff–Kakutanitheorem (see [11, Corollary 3.3.13]), the group G admits a left-invariant metric � generating the originaltopology of G. Since G is not locally precompact, there exists an infinite m−1-metrically discrete set A ={an: n ∈ ω} ⊆ U for some m ∈ ω. Take a neighborhood W of e such that W ⊆ V and diam(W ) < m−1.Then the set A is left W -disjoint. Since l(G) = ω, Theorem 7 implies that the space G is h-homogeneous. �5. h-Homogeneity of the product X × Y

A space X is divisible by a space F if there exists Y such that F × Y ≈ X. We say that a space X isstrongly k-divisible (or strongly divisible by k) if X ≈ Dk×X, where Dk is the discrete space of cardinality k.

Lemma 2. (1) Let a space X be strongly k-divisible for an infinite cardinal k. Then X is strongly k1-divisiblefor any cardinal k1 < k.

(2) Let X be an h-homogeneous disconnected space. Then X is strongly n-divisible for each positiveinteger n.

(3) Let X be an ω-divisible space. Then X is non-pseudocompact.

Proof. (1) Consider the cardinal k as a discrete space. Then X ≈ k×X ≈ (k1×k)×X ≈ k1×(k×X) ≈ k1×X.(2) Fix n > 1. As a disconnected space, X = X1 ⊕X2, where X1 and X2 are non-empty subspaces of X.

Since X is h-homogeneous, X ≈ X1. Then X1 = X11 ⊕X12. Repeating the division, we conclude that X

contains n non-empty pairwise disjoint clopen subsets U1, . . . , Un each of which is homeomorphic to X. So,X = U1 ⊕ · · · ⊕ Un ≈ n×X.

(3) Let f : X → ω × Y be a homeomorphism for a space Y . Take the projection p : ω × Y → ω. Thenthe composition of f and p is continuous and unbounded. Hence, X is a non-pseudocompact space. �Theorem 9. Let a strongly k-divisible Tychonoff space X has a π-base consisting of clopen subsets thatare homeomorphic to X, where k � ω. Let Y be a homogeneous paracompact zero-dimensional space withl(Y ) � k. Then X × Y is an h-homogeneous space.

Proof. From Lemma 2 and Theorem 1 it follows that X is an h-homogeneous space. Take a non-empty opensubset W of X × Y . Then W contains a non-empty product U × V , where U is a clopen subset of X and

S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530 2529

V is a clopen subset of Y . Fix a point a ∈ V . From homogeneity of Y it follows that for every point y ∈ Y

there exists a homeomorphism fy : Y → Y such that fy(a) = y. Since l(Y ) � k, the cover {fy(V ): y ∈ Y }of Y has an open refinement V of cardinality � k. We may assume that V is locally finite because Y isparacompact. According to [11, Lemma 7.6.12], the cover V has a discrete clopen refinement {V ∗

α : α ∈ k1}for some k1 � k. Then Y =

⊕{V ∗

α : α ∈ k1}. By construction, each V ∗α is homeomorphic to a non-empty

clopen subset Vα of V . According to Lemma 2, we have X ≈ k1 ×X. From h-homogeneity of X it followsthat U =

⊕{Uα: α ∈ k1}, where each Uα ≈ X. Hence,

X × Y ≈ X ×(⊕

{Vα: α ∈ k1})≈

⊕{X × Vα: α ∈ k1}

≈⊕

{Uα × Vα: α ∈ k1} = W ∗.

By construction, the set W ∗ ⊆ U × V ⊆ W . Clearly, W ∗ is a clopen subset of X × Y .From the above it follows that the product X × Y has a π-base consisting of clopen subsets that are

homeomorphic to X × Y . Since k � ω, the space X × Y is non-pseudocompact. Theorem 1 implies thatX × Y is an h-homogeneous space. �Corollary 4. Let X be an h-homogeneous non-compact metrizable space and Y be a homogeneous metrizablespace such that w(Y ) � w(X) and indX = indY = 0. Then X × Y is an h-homogeneous space.

Proof. The corollary follows from Theorem 9 because X is a strongly k-divisible space, where k = w(X). �Theorem 10. Let X be an h-homogeneous non-pseudocompact zero-dimensional space and Y be a homoge-neous zero-dimensional Lindelöf space. Then X × Y is an h-homogeneous space.

Proof. The theorem is a particular case of Theorem 9 when k = ω. One can verify that X is stronglyω-divisible. �Corollary 5. Let Y be a homogeneous zero-dimensional Lindelöf space. Then Q × Y is a homogeneous,h-homogeneous space, where Q is the space of rationals.

Theorem 11. Let X be an h-homogeneous disconnected pseudocompact space and Y be a homogeneous com-pact space. Then X × Y is an h-homogeneous space.

Proof. Take a non-empty clopen subset V of the space Y . Fix a point a ∈ V . From homogeneity of Y itfollows that for every point y ∈ Y there exists a homeomorphism fy : Y → Y such that fy(a) = y. Since Y

is compact, the cover {fy(V ): y ∈ Y } of Y has a finite subcover {V ∗i : 1 � i � n} for some n, where each

V ∗i is clopen in Y . Put V1 = V ∗

1 and Vi = V ∗i \

⋃{Vj : j < i} for i > 1. Without loss of generality, V1 = V

and each Vi is non-empty. Then Y =⋃{Vi: i � n}, where Vi ∩ Vj = ∅ whenever i �= j. By construction,

each Vi is homeomorphic to a clopen subset Vi of V .Consider a non-empty clopen subset U of the space X. By Lemma 2, we have X ≈ n × X. From

h-homogeneity of X it follows that U =⋃{Ui: 1 � i � n}, where each Ui is clopen in X, Ui ≈ X, and

Ui ∩ Uj = ∅ whenever i �= j. Put Z =⋃{Ui × Vi: i � n}. Then

X × Y ≈ X ×(⊕

{Vi: i � n})≈

⊕{X × Vi: i � n}

≈⋃

{Ui × Vi: i � n} = Z.

One can check that Z is a clopen subset of U × V ; hence, Z is clopen in X × Y .

2530 S.V. Medvedev / Topology and its Applications 160 (2013) 2523–2530

Let A be the Boolean algebra over V generated by the clopen sets V1, . . . , Vn. Since A is finite, it mustbe atomic. Let W1, . . . ,Wm be the atoms of A. Denote by nj the number of sets V1, . . . , Vn containing Wj

for j � m. Note that each nj � 1 because V1 = V =⋃{Wj : j � m}. Then

Z =⋃{

Ui ×⋃

{Wj : Wj ⊆ Vi}: i � n}≈

⊕{X ×

⋃{Wj : Wj ⊆ Vi}: i � n

}

≈⊕

{X × nj ×Wj : j � m} ≈⊕

{X ×Wj : j � m}

≈ X ×(⋃

{Wj : j � m})≈ X × V1 ≈ U × V.

Thus, X × Y ≈ U × V for every non-empty clopen subsets U and V of the spaces X and Y , respectively.By virtue of [9, Corollary 3.10.27], the product X×Y is pseudocompact. Take a non-empty clopen subset

C of X×Y . Proposition 6 and Lemma 7 from [7] imply that C can be written as the union of finitely manypairwise disjoint clopen rectangles. Say, C is the union of l such rectangles. Then C ≈ l×X × Y ≈ X × Y .Hence, X × Y is h-homogeneous. �Example 2. Let us show that the Cantor cube 2k for k > ω is an h-homogeneous space. Indeed, 2k ≈ 2ω×2k.It is well known that the Cantor set 2ω is an h-homogeneous disconnected compact space and the Cantorcube 2k is a homogeneous compact space. It remains to apply Theorem 11.

Corollary 6. Let Y be a homogeneous compact space. Then:(1) the product Z = C × Y is a homogeneous, h-homogeneous compact space, where C is the Cantor set;(2) Y is an open retract of a homogeneous, h-homogeneous compact space Z with w(Z) = w(Y ).

Proof. The former statement follows from Theorem 11. Next, Y is an open retract of C × Y . �Acknowledgements

The author is grateful to Jan van Mill for offering information about the van Douwen paper [1]. Theauthor would like to thank the referee for a number of helpful suggestions for improvement in the article.

References

[1] E.K. van Douwen, A compact space with a measure that knows which sets are homeomorphic, Adv. in Math. 52 (1) (1984)1–33.

[2] D.B. Motorov, A homogeneous zero-dimensional bicompactum which is not divisible by 2, in: V.V. Fedorčuk (Ed.), GeneralTopology. Spaces, Mapping and Functors, MSU, Moscow, 1992, pp. 65–73 (in Russian).

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