The Uniform Effros Property and Local Homogeneity

Remark 2.1

Let Z be a Tychonoff space and let 𝔘 be a uniformity of Z that induces its topology. If V ∈ 𝔘, then we define the cover of Z, 𝕮(V ) = {B(z, V ) | z ∈ Z}.

Remark 2.2

Note that by [3: 8.3.13], for every compactum Z, there exists a unique uniformity 𝔘 _Z on Z that induces the original topology of Z.

We need the following result [3: 8.3.G]:

Theorem 2.3

Let Z be a compactum and let 𝔘 _Z (Remark 2.2) be the unique uniformity of Z that induces its topology. Then for every open cover 𝒲 of Z, there exists V ∈ 𝔘 _Z such that 𝕮(V ) refines 𝒲.

To know more about uniformities see [3: Chapter 8].

Let Z be a topological space and let g : Z → Z be a map. Then the graph of g is the set Γ(g) = {(z, g(z)) | z ∈ Z}.

Remark 2.4

Let Z be a Tychonoff space, let 𝔘 be a uniformity that induces the topology of Z, let U ∈ 𝔘 and let g : Z → Z be a map. Then the fact that Γ(g) ⊂ U is equivalent to the fact that ρ _Z (z, g(z)) <U for all points z of Z.

Notation 2.5

Let Z be a continuum and let 𝓗(Z) be its group of homeomorphisms with compactopen topology. If z is an element of Z, then define γ _z : 𝓗(Z) → Z by γ _z (h) = h(z).

A homogeneous continuum Z is said to be an Effros continuum if γ _z is an open map for all points z of Z (Notation 2.5).

Let Z be a Tychonoff space and let 𝔘 be a uniformity that induces the topology of Z. Then Z has the uniform property of Effros with respect to 𝔘, provided that for each U ∈ 𝔘, there exists V ∈ U such that if z ₁ and z ₂ are two points of Z with ρ _Z (z ₁ , z ₂) < V , there exists a homeomorphism h : Z ↠ Z such that h(z ₁) = z ₂ and ρ _Z (z, h(z)) <U for all elements z of Z. Note that this last statement is equivalent to the fact that Γ(h) ⊂ U (Remark 2.4). The entourage V is called an Effros entourage for U . A homeomorphism h : Z ↠ Z satisfying ρ _Z (z, h(z)) < U , for all points z of Z, is called a U-homeomorphism [10: Definition 1.4.58].

The next theorem is [10: Theorem 1.4.59].

Theorem 2.6

Let Z be a connected Tychonoff space and let 𝔘 be a uniformity that induces the topology of Z. If Z has the uniform property of Effros with respect to 𝔘, then Z is a homogeneous space.

Throughout the paper, all the spaces are Tychonoff spaces.

Lemma 3.1

Let Z be a compactum. For each element z of Z, let W _z be an open subset of Z containing z. Suppose that U belongs to 𝔘 _Z ( Remark 2.2 ) and satisfies that given z′ ∈ W _z , for some z in Z, there exists a U-homeomorphism h : Z ↠ Z such that h(z) = z′. If V ∈ 𝔘 _Z and 𝕮(V ) refines the open cover 𝔚 = {W _z | z ∈ Z}, then the following is true:

(*) If z ₁ and z ₂ belong to Z and ρ _Z (z ₁ , z ₂) < V , then there exists a homeomorphism h : Z ↠ Z such that h(z ₁) = z ₂ and ρ _Z (z, h(z)) < 2U, for all points z in Z.

Proof

Since Z is a compactum, by Theorem 2.3, there exists V ∈ 𝔘 _Z (Remark 2.2) such that 𝕮(V ) refines 𝔚. Let z ₁ and z ₂ be two elements of Z such that ρ _Z (z ₁ , z ₂) < V . Since 𝕮(V ) refines 𝔚, there exists an element z ₀ of Z such that {z ₁ , z ₂} ⊂ W _z 0 . By our assumption, there exist two U -homeomorphisms h ₂ , h ₁ : Z ↠ Z such that h ₁(z ₀) = z ₁, h ₂(z ₀) = z ₂. Let h=h2∘h1−1 . Then h : Z ↠ Z is a homeomorphism and h(z ₁) = z ₂. Let z be an element of Z. Since ρZ(z,h(z))=ρZ(z,h2∘h1−1(z)) , ρZ(z,h1−1(z))=ρZ(h1∘h1−1(z),h1−1(z))<U and ρZ(h1−1(z),h2∘h1−1(z))<U , we have that ρ _Z (z, h(z)) < 2U .

We start with the definition of the uniform local homogeneity given by Ungar [13]. A Tychonoff space Z is uniformly locally homogeneous with respect to 𝔘 (a uniformity that induces the topology of Z), provided that for each element z of Z and every U ∈ 𝔘, there exists an open subset O _U,z with z ∈ O _U,z such that if z′ ∈ O _U,z , there exists a homeomorphism h : Z ↠ Z such that h(z) = z′ and Γ(h) ⊂ U .

Theorem 3.2

Let Z be a Tychonoff space and let 𝔘 be a uniformity that induces the topology of Z. If Z has the uniform property of Effros with respect to 𝔘, then Z is uniformly locally homogeneous with respect to 𝔘.

Proof

Let z be a point of Z and let U ∈ 𝔘. Since Z has the uniform property of Effros with respect to 𝔘, there exists an Effros entourage V for U . Note that Int _Z (B(z, V )) is an open subset of Z containing z. Let z′ ∈ Int _Z (B(z, V )). Then ρ _Z (z, z′) < V . Thus, there exists a U -homeomorphism h : Z ↠ Z such that h(z) = z′. Since h is a U -homeomorphism, we have that Γ(h) ⊂ U (Remark 2.4). Therefore, Z is uniformly locally homogeneous with respect to 𝔘.

Theorem 3.3

Let Z be a compactum. If Z is uniformly locally homogeneous, then Z has the uniform property of Effros.

Proof

Let U and U′ be elements of 𝔘 _Z (Remark 2.2) such that 2U′ ⊂ U . Since Z is uniformly locally homogeneous, for each z in Z, there exists an open subset O _U',z of Z such that if z′ ∈ O _U',z , then there exists a homeomorphism h : Z ↠ Z such that h(z) = z′ and Γ(h) ⊂ U ′. Observe that 𝔒 = {O _{U',z | z} ∈ Z} is an open cover of Z. Since Z is a compactum, by Theorem 2.3, there exists V ∈ 𝔘 _Z such that 𝕮(V ) refines 𝔒. Let z ₁ and z ₂ be two elements of Z such that ρ _Z (z ₁ , z ₂) < V . By Lemma 3.1 and Remark 2.4, there exists a homeomorphism h : Z ↠ Z such that h(z ₁) = z ₂ and ρ _Z (z, h(z)) < 2U ′ for all elements z of Z. Since 2U ′ ⊂ U , we obtain that ρ _Z (z, h(z)) < U , for every point z in Z. Therefore, V is an Effros entourage for U and Z has the uniform property of Effros.

Question 3.4

Is Theorem 3.3 true if Z is not a compactum?

A topological space Z is locally homogeneous _H , according to Ho [6], provided that for each element z in Z and every open subset W of Z containing z, there exists an open subset W ′ of Z satisfying that z ∈ W ′ ⊂ W and that for each element z′ of W ′, there exists a homeomorphism h : Z ↠ Z such that h(W ′) ⊂ W and h(z) = z′.

Theorem 3.5

Let Z be a Tychonoff space and let 𝔘 be a uniformity that induces the topology of Z. If Z has the uniform property of Effros with respect to 𝔘, then Z is locally homogeneous _H .

Proof

Let z be a point of Z and let W be an open subset of Z containing z. Then there exists U ∈ 𝔘 such that B(z, U ) ⊂ W . Let U ′ ∈ 𝔘 be such that 2U ′ ⊂ U . Since Z has the uniform property of Effros with respect to 𝔘, there exists an Effros entourage V for U’. Without loss of generality, we assume that V ⊂ U ′. Let z′ ∈ Int _Z (B(z, V )). Then ρ _Z (z, z′) < V . Hence, there exists a U ′-homeomorphism h : Z ↠ Z such that h(z) = z′. Since h is a U ′-homeomorphism, for each z″ ∈ Int _Z (B(z, V )), we have that ρ _Z (z″, h(z″)) < U ′. Also, since ρ _Z (z, z″) < V and V ⊂ U ′, we obtain that ρ _Z (z, h(z″)) < 2U ′. Since 2U ′ ⊂ U , we have that ρ _Z (z, h(z″)) < U . Hence, h(Int _Z (Bz, V )) ⊂ B(z, U ) ⊂ W . Therefore, Z is locally homogeneous _H .

Theorem 3.6

Let Z be a compactum. If Z is locally homogeneous _H , then Z has the uniform property of Effros.

Proof

Let U and U ′ be elements of 𝔘 _Z (Remark 2.2) such that 2U ′ U . For each point z of Z, we consider the open subset Int _Z (B(z, U ′)) of Z. Since Z is locally homogeneous _H , for each point z of Z, there exists an open subset O _z such that z ∈ O _z ⊂ Int _Z (B(z, U ′)) and that for each element z′ of O _z , there exists a homeomorphism h : Z ↠ Z such that h(O _z ) ⊂ Int _Z (B(z, U ′)) and h(z) = z′. Note that 𝔒 = {O _z | z ∈ Z} is an open cover of Z. By Theorem 2.3, there exists V ∊ 𝔘 _Z such that 𝕮(V ) refines 𝔒. Let z ₁ and z ₂ be two elements of Z such that ρ _Z (z ₁ , z ₂) < V . By Lemma 3.1, there exists a homeomorphism h : Z ↠ Z such that h(z ₁) = z ₂ and ρ _Z (z, h(z)) < 2U ′ for all elements z of Z. Since 2U′ ⊂ U , we obtain that ρ _Z (z, h(z)) < U , for every point z in Z. Therefore, V is an Effros entourage for U and Z has the uniform property of Effros.

Example 3.7

Let X be the set of integers, ℤ, with the cofinite topology. By [11: p. 133], X is homogeneous _H but not uniformly locally homogenous. Thus, by Theorem 3.2, X does not have the uniform property of Effros. Hence, the converse of Theorem 3.6 is not true if Z is not a compactum.

From Theorems 3.2, 3.3, 3.5, and 3.6, we obtain:

Theorem 3.8

Let Z be a compactum. Then the following are equivalent:

1. Z has the uniform property of Effros.

2. Z is uniformly locally homogeneous.

3. Z is locally homogeneous _H .

A topological space Z is locally homogeneous _F , according to Fora [4], provided that for each point z of Z, there exists an open subset W of Z containing z such that for every element z′ of W , there exists a homeomorphism h : Z ↠ Z such that h(z) = z′.

Theorem 3.9

Let Z be a Tychonoff space and let 𝔘 be a uniformity that induces the topology of Z. If Z has the uniform property of Effros with respect to 𝔘, then Z is locally homogeneous _F .

Proof

Suppose Z has the uniform property of Effros with respect to 𝔘. By Theorem 3.5, we have that Z is locally homogeneous _H . By [11: Theorem 4], we obtain that Z is locally homogeneous _F .

Remark 3.10

The converse of Theorem 3.9 is false. By [11: p. 132], we have that local homogeneity _H does not follow from local homogeneity _F . Thus, by Theorem 3.5, the uniform property of Effros does not follow from the local homogeneity _F .

The next theorem gives us properties of Effros continua.

Theorem 3.11

Let Z be an Effros continuum. Then:

1. Z has the uniform property of Effros.

2. Z uniformly locally homogeneous.

3. Z is locally homogeneous _H .

4. Z is locally homogeneous _F .

Proof

Suppose Z is an Effros continuum. By [10: Theorem 1.4.60], Z has the uniform property of Effros. Now, the theorem follows from Theorems 3.2, 3.5 and 3.9.

Theorem 3.12

Let Z be a compactum. If Z is strongly locally homogeneous, then Z has the uniform property of Effros.

Proof

Let U and U ′ be elements of 𝔘 _Z (Remark 2.2) such that 4U′ ⊂ U . For each point z of Z, we consider the open subset Int _Z (B(z, U ′)) of Z. Since Z is strongly locally homogeneous, for each z in Z, there exists an open subset O _U',z of Z such that z ∈ O _U',z ⊂ Int _Z (B(z, U ′)) and if z′ ∈ O _U,z , then there exists a homeomorphism h : Z ↠ Z such that h(z) = z′ and h| _{Z\O U',z} = 1 _{Z\O U',z} . Observe that 𝔒 = {O _U',z | z ∈ Z} is an open cover of Z. Since Z is a compactum, by Theorem 2.3, there exists V ∈ 𝔘 _Z such that 𝕮(V ) refines 𝕺. Let z ₁ and z ₂ be two elements of Z such that ρ _Z (z ₁ , z ₂) < V . Since 𝕮(V ) refines 𝕺, there exists an element z ₀ of Z such that {z ₁ , z ₂} ⊂ O _{U',z 0} . By our assumption, there exist two homeomorphisms h ₂ , h ₁ : Z ↠ Z such that h ₁(z ₀) = z ₁, h ₂(z ₀) = z ₂,

Let h=h2∘h1−1 . Then h : Z ↠ Z is a homeomorphism and h(z ₁) = z ₂. Let z be an element of

O _{U',z 0} . Since {z,h1−1(z)}={h1∘h1−1(z),h1−1(z)}⊂OU′,z0⊂IntZ⁡(B(z0,U′)) and {h1−1(z),h2∘h1−1(z)}⊂OU′,z0⊂IntZ⁡(B(z0,U′)) , we have that ρZ(z,h1−1(z))=ρZ(h1∘h1−1(z),h1−1(z))<2U′ , ρZ(h1−1(z),h2∘h1−1(z))<2U′ and ρ _Z (z, h(z)) < 4U ′. Since 4U ′ ⊂ U , we obtain that ρ _Z (z, h(z)) <U . If z ∈ Z \ O _{U',z 0} , then z = h(z) and, in particular, ρ _Z (z, h(z)) < U . Therefore, V is an Effros entourage for U and Z has the uniform property of Effros.

Corollary 3.13

If Z is a strongly locally homogeneous continuum, then Z is a homogenous continuum.

A topological space Z is closed-homogeneous, according to Fora [4], provided that for each pair of points z and z′ of Z and a closed subset K of Z, with K ⊂ Z \{z, z′}, there exists a homeomorphism h : Z ↠ Z such that h(z) = z′ and h| _K = 1 _K .

Theorem 3.14

Let Z be a compactum. If Z is closed-homogeneous, then Z has the uniform property of Effros.

Proof

Suppose Z is closed-homogeneous compactum, by [11: Theorem 10], Z is strongly locally homogeneous. Hence, by Theorem 3.12, Z has the uniform property of Effros.

Remark 3.15

Note that, by [11: Theorem 9], each closed-homogeneous space is homogeneous.

A continuum Z is almost connected im kleinen at a point z of Z, provided that for each open subset W of Z containing z, there exists a subcontinuum K of Z such that Int _Z (K) ≠ ∅ and K ⊂ W . The continuum Z is connected im kleinen at a point z of Z if for each open subset W of Z containing z, there exists a subcontinuum K of Z such that z ∈ Int _Z (K) ⊂ K ⊂ W .

Theorem 3.16

If Z is a continuum that is either uniformly locally homogeneous, locally homo- geneous _H , strongly locally homogeneous or closed-homogeneous, then the following are equivalent:

1. Z is locally connected.

2. Z is locally connected at some point.

3. Z is connected im kleinen at some point.

4. Z is almost connected im kleinen at every point.

5. Z is almost connected im kleinen at some point.

Proof

If Z is uniformly locally homogeneous or locally homogeneous _H , by Theorem 3.8, Z has the uniform property of Effros. If Z is strongly locally homogeneous, by Theorem 3.12, Z has the uniform property of Effros. If Z is closed-homogenoeus, by Theorem 3.14, Z has the uniform property of Effros. Now, the theorem follows from [10: Theorem 1.4.61].

Notation 4.1

Given a continuum Z, we define Jones’ set function 𝒯 as follows: if A is a subset of Z, then

A thorough study of Jones’ set function 𝒯 is given in [10].

The following theorem is Jones’ Aposyndetic Decomposition Theorem [10: Theorem 3.3.8], with the addition that the quotient space also has the uniform property of Effros.

Theorem 4.2

Let Z be a decomposable continuum with the uniform property of Effros. If 𝒯 ({z}) | z ∈ Z}, then the following hold:

1. 𝓖 is a continuous, monotone and terminal decomposition of Z.

2. The elements of 𝓖 are cell-like, acyclic, homogeneous and mutually homeomorphic continua.

3. The quotient map q : Z ↠ Z/𝓖 is uniformly completely regular and atomic.

4. The quotient space Z/𝓖 is an aposyndetic homogeneous continuum with the uniform property of Effros and it does not contain nondegenerate proper terminal subcontinua.

Proof

We show that Z/𝓖 has the uniform property of Effros. Let 𝒰 ∈ 𝔘 _Z/𝓖. Since the quotient map, q, is continuous, we have that (q × q)⁻¹(𝒰) ∈ 𝔘 _Z . Since Z has the uniform property of Effros, there exists an Effros entourage V for (q × q)⁻¹(𝒰). Since q is an open map [10: Corollary 1.1.24], we have that (q × q)(V ) ∈ 𝔘 _Z/𝓖. We prove that (q × q)(V ) is an Effros entourage for 𝒰. Let χ ₁ and χ ₂ be two elements of Z/𝓖 such that ρ _Z/𝓖(χ ₁ , χ ₂) < (q × q)(V ). Then there exist two elements z ₁ and z ₂ in Z such that q(z ₁) = χ ₁, q(z ₂) = χ ₂ and ρ _Z (z ₁ , z ₂) < V . Since V is an Effros entourage for (q × q)⁻¹(𝒰), there exists a (q × q)⁻¹(𝒰)-homeomorphism h : Z ↠ Z such that h(z ₁) = z ₂. As in the proof of [10: Theorem 3.3.6], it can be seen that the map ζ : Z/𝓖 ↠ Z/𝓖 given by ζ(χ) = qoh(q ⁻¹(χ)) is a homeomorphism and ζ(χ ₁) = χ ₂. We prove that ζ is a 𝒰-homeomorphism. Let χ be a point of Z/𝓖 and let z ∈ q ⁻¹(χ). Since h is a (q × q)⁻¹(𝒰)-homeomorphism, we have that ρ _Z (z, h(z)) < (q × q)⁻¹(𝒰); i.e., (z, h(z)) ∈ (q × q)⁻¹(𝒰). Hence, (q × q)(z, h(z)) ∈ 𝒰. Thus, since q(z) = χ, we obtain that ρ _Z/𝓖(χ, ζ(χ)) < 𝒰. Therefore, ζ is 𝒰-homeomorphism. Hence, (q × q)(V ) is an Effros entourage for 𝒰 and Z/𝓖 has the uniform property of Effros. The theorem now follows from [10: Theorem 3.3.8].

Notation 4.3

Let Z be a continuum and let z be an element of Z. Then

The following theorem is Prajs’ Mutual Aposyndetic Decomposition Theorem [10: Theorem 3.4.9], with the addition that the quotient space also has the uniform property of Effros. The proof of the part that the quotient space has the uniform property of Effros is similar to the one given for Theorem 4.2.

Theorem 4.4

Let Z be a decomposable continuum with the uniform property of Effros. If 𝓠 = {Q _z | z ∈ Z}, then the following hold:

1. 𝓠 is a continuous decomposition of Z.

2. The elements of 𝓠 are homogeneous mutually homeomorphic closed subsets of Z.

3. The quotient map q : Z ↠ Z/𝓠 is uniformly completely regular.

4. The quotient space Z/𝓠 is a mutually aposyndetic homogeneous continuum with the uniform property of Effros.

Regarding Jones’ Aposyndetic Decomposition Theorem, we have:

Theorem 4.5

Let Z be a continuum and let 𝓖 = {𝒯 ({z}) | z ∈ Z}. If Z is either uniformly locally homogeneous, locally homogeneous _H , strongly locally homogeneous or closed-homogeneous, then the following hold:

1. 𝓖 is a continuous, monotone and terminal decomposition of Z.

2. The elements of 𝓖 are cell-like, acyclic, homogeneous and mutually homeomorphic continua.

3. The quotient map q : Z ↠ Z/𝓖 is uniformly completely regular and atomic.

4. The quotient space Z/𝓖 is an aposyndetic homogeneous continuum with the uniform property of Effros and it does not contain nondegenerate proper terminal subcontinua.

Proof

The proof of the fact that in each case Z has the uniform property of Effros is done in Theorem 3.16. Now the theorem follows from Theorem 4.2.

Theorem 4.6

Let Z be a continuum and let 𝓠 = {Q _z | z ∈ Z}. If Z is either uniformly locally homogeneous, locally homogeneous _H , strongly locally homogeneous or closed-homogeneous, then the following hold:

1. 𝓠 is a continuous decomposition of Z.

2. The elements of 𝓠 are homogeneous mutually homeomorphic closed subsets of Z.

3. The quotient map q : Z ↠ Z/𝓠 is uniformly completely regular.

4. The quotient space Z/𝓠 is a mutually aposyndetic homogeneous continuum with the uniform property of Effros.

The proof of Theorem 4.6 is similar to the one given for Theorem 4.5 except that we use Theorem 4.4 instead of Theorem 4.2.