A Note on Almost Periodic Points and Minimal Sets in T₁- and T₂-Spaces

Abstract

We show that (1) there exist almost periodic orbits in 𝑇₂-spaces of which the closures are not minimal sets; (2) there exist minimal sets in locally compact 𝑇₁-spaces which are not compact; (3) there exist almost periodic orbits in 𝑇₂-spaces of which the closures contain not only almost periodic points. These give answers to the three problems given by Mai and Sun in (2007).

1. Introduction

Denote by ℝ, ℤ, ℤ₊, and ℕ the sets of real numbers, integers, nonnegative integers, and positive integers, respectively. For any topological space 𝑋, denote by 𝐶⁰(𝑋) the set of all continuous maps of 𝑋 into itself. For any 𝑓∈𝐶⁰(𝑋), let 𝑓⁰=𝑖𝑑_𝑋 be the identity map of 𝑋, and let 𝑓^𝑖=𝑓𝑓^𝑖−1 be the composition of 𝑓 and 𝑓^𝑖−1 (𝑖=1,2,3,…). 𝑓^𝑛 is called the 𝑛th iterate of 𝑓(𝑛∈ℤ₊).

The orbit of a point 𝑥∈𝑋 under 𝑓, denoted by 𝑂(𝑥,𝑓), is the set {𝑥,𝑓(𝑥),𝑓²(𝑥),…}. 𝑥∈𝑋 is called an almost periodic point of 𝑓 and 𝑂(𝑥,𝑓) is called an almost periodic orbit if for any neighborhood 𝑈 of 𝑥 there exists 𝑁∈𝑁 such that {𝑓^𝑛+𝑖(𝑥)∶𝑖=0,1,…,𝑁}∩𝑈≠∅ for all 𝑛∈𝑍₊.

A subset 𝑊 of 𝑋 is said to be invariant or 𝑓-invariant if 𝑓(𝑊)⊂𝑊, and 𝑊 is called a minimal set of 𝑓 if it is nonempty, closed, and 𝑓-invariant and if no proper subset of 𝑊 has these three properties.

The notion of 𝜔-regular space was introduced by Mai and Sun [1].

Definition 1.1 (See [1, Definition 2.1]). A topological space 𝑋 is called an 𝜔-regular space if for any closed set 𝑊⊂𝑋, any point 𝑥∈𝑋−𝑊 and any countable set 𝐴⊂𝑊, there exist disjoint open sets 𝑈 and 𝑉 such that 𝑥∈𝑈 and 𝐴⊂𝑉.

Mai and Sun [1] generalized several known results concerning almost periodic points and minimal sets of maps from regular spaces to 𝜔-regular spaces, and obtained the following theorems.

Theorem A (See [1, Theorem 2.3 ]). Let 𝑋 be an 𝜔-regular space, and 𝑓∈𝐶⁰(𝑋). Then the closure of every almost periodic orbit of 𝑓 is a minimal set.

Theorem B (See [1, Theorem 3.8]). Let 𝑋 be a locally compact topological space which is either Hausdorff or regular, and 𝑓∈𝐶⁰(𝑋). Then each minimal set of 𝑓 is compact.

Theorem C (See [1, Theorem 4.1 ]). Let 𝑋 be an 𝜔-regular space, and 𝑓∈𝐶⁰(𝑋). Then all points in the closure of any almost periodic orbit of 𝑓 are almost periodic.

Can these theorems be extended to more general topological spaces? [1, Example 2.4 and Remark 4.2] show that Theorems A and C cannot be extended to general 𝑇₁-spaces, respectively, and [1, Example 3.9] shows that Theorem B cannot be extended to general locally compact 𝑇₀-spaces. However, there remain three problems which have not been solved in [1].

Problem 1.2 (See [1, Problem 2.5 ]). Can the condition in [1, Theorem 2.3] that 𝑋 is an 𝜔-regular space be replaced by that 𝑋 is a 𝑇₂-space? In other words, need the closure of an almost periodic orbit in a 𝑇₂-space be a minimal set?

Problem 1.3 (See [1, Problem 3.10 ]). Let 𝑋 be a locally compact 𝑇₁-space. Is each minimal set of any 𝑓∈𝐶⁰(𝑋) compact?

Problem 1.4 (See [1, Problem 4.4 ]). Can the condition in [1, Theorem 4.1] that 𝑋 is an 𝜔-regular space be replaced by that 𝑋 is a 𝑇₂-space? In other words, does the closure of any almost periodic orbit in a 𝑇₂-space contain only almost periodic points?

In this note, we study the above three problems, and obtain the following three propositions, which give negative answers to these problems.

Proposition 1.5. There exist a 𝑇₂-space 𝑋 and a continuous map 𝑓∶𝑋→𝑋 such that 𝑓 has an almost periodic orbit of which the closure is not a minimal set.Proposition 1.6. There exist a locally compact 𝑇₁-space 𝑋 and a continuous map 𝑓∶𝑋→𝑋 such that 𝑓 has a minimal set which is not compact.Proposition 1.7. There exist a 𝑇₂-space 𝑋, a continuous map 𝑓∶𝑋→𝑋, and an almost periodic point 𝑥 of 𝑓 such that not all points in the closure 𝑂(𝑥,𝑓) of the orbit 𝑂(𝑥,𝑓) are almost periodic points.

2. Finer Topologies and Subspace Topologies

In order to prove the above three propositions, we will construct several new 𝑇₁- and 𝑇₂-spaces by adding some open sets to a known topological space, or construct several new spaces with known spaces being subspaces. For this, in this section we give some lemmas on finer topological spaces and subspaces. These lemmas can be directly derived from the definitions concerned, and the proofs will be omitted.

Lemma 2.1. Let 𝒯 and 𝒯^′ be two topologies on a set 𝑋. If 𝒯^′ is finer than 𝒯, and (𝑋,𝒯) is a 𝑇_𝑖-space, 𝑖=1,2, then (𝑋,𝒯^′) is also a 𝑇_𝑖-space.Lemma 2.2. Let 𝑋 be a topological space, 𝑌 be a subspace of 𝑋, and let 𝑓∶𝑋→𝑋 be a continuous map such that 𝑓(𝑌)⊂𝑌. Then any point 𝑦∈𝑌 is an almost periodic point of 𝑓 if and only if 𝑦 is an almost periodic point of 𝑓|_𝑌.

From Lemma 2.2, we can obtain immediately.

Lemma 2.3. Let 𝑋 be a set, 𝑌 be a subset of 𝑋, and 𝑓∶𝑋→𝑋 be a map such that 𝑓(𝑌)⊂𝑌. Suppose that 𝒯 and 𝒯^′ are two topologies on 𝑋, 𝒯|_𝑌=𝒯^′|_𝑌, and 𝑓 is continuous both for 𝒯 and for 𝒯^′. Then any point 𝑦∈𝑌 is an almost periodic point of 𝑓 for topology 𝒯 if and only if 𝑦 is an almost periodic point of 𝑓 for topology 𝒯^′.

3. Proofs of Propositions 1.5–1.7

Propositions 1.5–1.7 claim that there exist continuous maps of some 𝑇₁- or 𝑇₂-spaces which have certain special properties. Hence in order to show these propositions, we need to construct maps having these special properties.

Let 𝑆¹={𝑒^𝑡𝐢∶𝑡∈𝑅} be the unit circle in the complex plane ℂ. For any two real numbers 𝑟<𝑠, write 𝐴(𝑟,𝑠)={𝑒^𝑡𝐢∶𝑟<𝑡<𝑠}. Let 𝒯₀ be the usual topology on 𝑆¹, and letℬ₀={𝐴(𝑟,𝑠)∶𝑟and𝑠arerealnumbers,and0<𝑠−𝑟<𝜋}. (3.1)Then ℬ₀ is a set of open arcs in 𝑆¹, which is a basis for the topology 𝒯₀.

Let 𝜃∈[0,1] be a given irrational number, and let ℎ_𝜃∶𝑆¹→𝑆¹ be the rotation defined byℎ_𝜃(𝑒^{2𝜋𝐢𝑡})=𝑒^{2𝜋𝐢(𝑡+𝜃)}forany𝑡∈ℝ. (3.2)Then under the topology 𝒯₀, ℎ_𝜃 is a homeomorphism, 𝑆¹ is the unique minimal set of ℎ_𝜃, and all points in 𝑆¹ are almost periodic points of ℎ_𝜃. Let 𝑤₀∈𝑆¹ be a given point, and let𝑊={ℎ^𝑛_𝜃(𝑤₀)∶𝑛∈𝑍},𝑌=𝑆¹−𝑊. (3.3)Then we have ℎ_𝜃(𝑊)=𝑊, and ℎ_𝜃(𝑌)=𝑌.

Proof. Let 𝑆¹, ℬ₀, 𝒯₀, ℎ_𝜃, 𝑊, and 𝑌 be as in (3.1)–(3.3). Let 𝑋=𝑆¹, 𝑓=ℎ_𝜃, and let𝒰₁={𝐴∩𝑌∶𝐴∈ℬ₀},ℬ₁=ℬ₀∪𝒰₁. (3.4)Then for any 𝑈, 𝑉∈ℬ₁, we have 𝑈∩𝑉∈ℬ₁∪{∅}. Thus ℬ₁ is a basis for a topology 𝒯₁ on 𝑋. It follows from ℬ₁⊃ℬ₀ that 𝒯₁⊃𝒯₀. Hence by Lemma 2.1, (𝑋,𝒯₁) is a 𝑇₂-space. Since 𝒯₁|_𝑊=𝒯₀|_𝑊 and 𝒯₁|_𝑌=𝒯₀|_𝑌, by Lemma 2.3, every point 𝑥∈𝑋 is an almost periodic point of 𝑓. Because 𝑌∈𝒯₁, that is, 𝑌 is an open set, 𝑊=𝑋−𝑌 is a closed subset of (𝑋,𝒯₁). For any 𝑥∈𝑋, let 𝑂(𝑥,𝑓) be the closure of the orbit 𝑂(𝑥,𝑓) in (𝑋,𝒯₁). For any 𝑧∈𝑋 and any 𝑈∈𝒯₁ with 𝑧∈𝑈, there exists an open arc 𝐴∈ℬ₀ such that𝑧∈𝐴⊂𝑈,if𝑧∈𝑊;𝑧∈𝐴∩𝑌⊂𝑈,if𝑧∈𝑌. (3.5)Thus we have𝑂(𝑥,𝑓)=𝑋,if𝑥∈𝑌;𝑊,if𝑥∈𝑊. (3.6)From this we see that, in the 𝑇₂-space (𝑋,𝒯₁), 𝑊 is the unique minimal set of 𝑓, and for any 𝑥∈𝑌, the closure 𝑂(𝑥,𝑓) of the almost periodic orbit 𝑂(𝑥,𝑓) is not a minimal set. Proposition 1.5 is proven.

Proof of Proposition 1.6. Let 𝑆¹, ℬ₀ and ℎ_𝜃 be as in (3.1)-(3.2). Take𝑋=𝑆¹×{0,1},ℬ₂₀={𝐴×{0}∶𝐴∈ℬ₀}. (3.7)For any 𝐴∈ℬ₀ and any 𝑥∈𝐴, write𝑈(𝐴,𝑥)=((𝐴×{0})∪{(𝑥,1)})−{(𝑥,0)}. (3.8)Letℬ₂₁={𝑈(𝐴,𝑥)∶𝐴∈ℬ₀,𝑥∈𝐴},ℬ₂=ℬ₂₀∪ℬ₂₁. (3.9)Then ℬ₂ is a basis for a topology 𝒯₂ on 𝑋, and the topological space (𝑋,𝒯₂) is a 𝑇₁-space. For any 𝐴⊂ℬ₀, let 𝐴 be the closure of 𝐴 in (𝑆¹,𝒯₀). Then 𝐴 is a closed arc, which is homeomorphic to the compact interval [0,1]. Obviously, for any 𝐴∈ℬ₀ and any 𝑥∈𝐴, ((𝐴×{0})∪{(𝑥,1)})−{(𝑥,0)} (resp., 𝐴×{0}) is a compact neighborhood of the point (𝑥,1) (resp. (𝑥,0)) in (𝑋,𝒯₂), which is homeomorphic to the subspace 𝐴 of (𝑆¹,𝒯₀). Thus (𝑋,𝒯₂) is a locally compact space. Define a map 𝑓∶𝑋→𝑋 by𝑓(𝑥,𝑖)=(ℎ_𝜃(𝑥),0),∀(𝑥,𝑖)∈𝑋=𝑆¹×{0,1}. (3.10)Then 𝑓 is continuous for the topology 𝒯₂. It is easy to see that every point (𝑥,𝑖)∈𝑋 is an almost periodic point of 𝑓, and the closure 𝑂((𝑥,𝑖),𝑓) of the orbit 𝑂((𝑥,𝑖),𝑓) in (𝑋,𝒯₂) is always the whole space 𝑋. Hence 𝑋 is the unique minimal set of 𝑓. Since ℬ₂ is an open cover of 𝑋 which has no finite subcover even has no countable subcover, the minimal set 𝑋 is not compact. Proposition 1.6 is proven.

Proof of Proposition 1.7. Let 𝑆¹, ℬ₀, 𝒯₀, ℎ_𝜃, 𝑊, and 𝑌 be as in (3.1)–(3.3). Take 𝑋=𝑆¹ and 𝑓=ℎ_𝜃. Letℬ₃₁={𝐴∩𝑊∶𝐴∈ℬ₀}, (3.11)ℬ₃₂={(𝐴∩𝑊)∪{𝑦}∶𝐴∈ℬ₀,𝑦∈𝐴∩𝑌}, (3.12)and let ℬ₃=ℬ₃₁∪ℬ₃₂. Then 𝑈∩𝑉∈ℬ₃∪{∅} for any 𝑈,𝑉∈ℬ₃, and hence, ℬ₃ is a basis for a topology 𝒯₃ on 𝑋. Clearly, under this topology 𝒯₃, we have the following.Claim 1. 𝑋 is a 𝑇₂-space, and 𝑓∶𝑋→𝑋 is continuous.Claim 2. No point 𝑦∈𝑌 is a recurrent point of 𝑓, and hence, no point 𝑦∈𝑌 is an almost periodic point of 𝑓.
Noting 𝒯₃|_𝑊=𝒯₀|_𝑊, by Lemma 2.3, we have the following.
Claim 3. Every point 𝑤∈𝑊 is an almost periodic point of 𝑓.
For any 𝑋₀⊂𝑋, let 𝑋₀ be the closure of 𝑋₀ in the space (𝑋,𝒯₃). Then we have
Claim 4. 𝑂(𝑤,𝑓)=𝑋 for any 𝑤∈𝑊, and 𝑂(𝑦,𝑓)=𝑂(𝑦,𝑓) for any 𝑦∈𝑌.
It follows from Claims 3, 4, and 2 that, for any 𝑥∈𝑊, not all points in the closure 𝑂(𝑥,𝑓) of the almost periodic orbit 𝑂(𝑥,𝑓) are almost periodic points. Proposition 1.7 is proven.

Acknowledgment

The work was supported by the Special Foundation of National Prior Basis Researches of China (Grant no. G1999075108).