Research Article | Open Access
Jie-Hua Mai, Xin-He Liu, "A Note on Almost Periodic Points and Minimal Sets in - and -Spaces", Discrete Dynamics in Nature and Society, vol. 2008, Article ID 262475, 5 pages, 2008. https://doi.org/10.1155/2008/262475
A Note on Almost Periodic Points and Minimal Sets in - and -Spaces
We show that (1) there exist almost periodic orbits in -spaces of which the closures are not minimal sets; (2) there exist minimal sets in locally compact -spaces which are not compact; (3) there exist almost periodic orbits in -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).
Denote by , , , and the sets of real numbers, integers, nonnegative integers, and positive integers, respectively. For any topological space , denote by the set of all continuous maps of into itself. For any , let be the identity map of , and let be the composition of and (). is called the th iterate of .
The orbit of a point under , denoted by , is the set . is called an almost periodic point of and is called an almost periodic orbit if for any neighborhood of there exists such that for all .
A subset of is said to be invariant or -invariant if , and is called a minimal set of if it is nonempty, closed, and -invariant and if no proper subset of has these three properties.
The notion of -regular space was introduced by Mai and Sun .
Definition 1.1 (See [1, Definition 2.1]). A topological space is called an -regular space if for any closed set , any point and any countable set , there exist disjoint open sets and such that and .
Mai and Sun  generalized several known results concerning almost periodic points and minimal sets of maps from regular spaces to -regular spaces, and obtained the following theorems.
Theorem A (See [1, Theorem 2.3 ]). Let be an -regular space, and . Then the closure of every almost periodic orbit of is a minimal set.
Theorem B (See [1, Theorem 3.8]). Let be a locally compact topological space which is either Hausdorff or regular, and . Then each minimal set of is compact.
Theorem C (See [1, Theorem 4.1 ]). Let be an -regular space, and . Then all points in the closure of any almost periodic orbit of 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 -spaces, respectively, and [1, Example 3.9] shows that Theorem B cannot be extended to general locally compact -spaces. However, there remain three problems which have not been solved in .
Problem 1.2 (See [1, Problem 2.5 ]). Can the condition in [1, Theorem 2.3] that is an -regular space be replaced by that is a -space? In other words, need the closure of an almost periodic orbit in a -space be a minimal set?
Problem 1.3 (See [1, Problem 3.10 ]). Let be a locally compact -space. Is each minimal set of any compact?
Problem 1.4 (See [1, Problem 4.4 ]). Can the condition in [1, Theorem 4.1] that is an -regular space be replaced by that is a -space? In other words, does the closure of any almost periodic orbit in a -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 -space and a continuous map such that has an almost periodic orbit of which the closure is not a minimal set.Proposition 1.6. There exist a locally compact -space and a continuous map such that has a minimal set which is not compact.Proposition 1.7. There exist a -space , a continuous map and an almost periodic point of such that not all points in the closure of the orbit are almost periodic points.
2. Finer Topologies and Subspace Topologies
In order to prove the above three propositions, we will construct several new - and -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 and be two topologies on a set . If is finer than , and is a -space, , then is also a -space.Lemma 2.2. Let be a topological space, be a subspace of , and let be a continuous map such that . Then any point is an almost periodic point of if and only if is an almost periodic point of .
From Lemma 2.2, we can obtain immediately.
Lemma 2.3. Let be a set, be a subset of , and be a map such that . Suppose that and are two topologies on , , and is continuous both for and for . Then any point is an almost periodic point of for topology if and only if is an almost periodic point of for topology .
Propositions 1.5–1.7 claim that there exist continuous maps of some - or -spaces which have certain special properties. Hence in order to show these propositions, we need to construct maps having these special properties.
Let be the unit circle in the complex plane . For any two real numbers , write . Let be the usual topology on , and letThen is a set of open arcs in , which is a basis for the topology .
Let be a given irrational number, and let be the rotation defined byThen under the topology , is a homeomorphism, is the unique minimal set of , and all points in are almost periodic points of . Let be a given point, and letThen we have , and .
Proof. Let , , , , , and be as in (3.1)–(3.3). Let , , and letThen for any , , we have . Thus is a basis for a topology on . It follows from that . Hence by Lemma 2.1, is a -space. Since and , by Lemma 2.3, every point is an almost periodic point of . Because , that is, is an open set, is a closed subset of . For any , let be the closure of the orbit in . For any and any with , there exists an open arc such thatThus we haveFrom this we see that, in the -space , is the unique minimal set of , and for any , the closure of the almost periodic orbit is not a minimal set. Proposition 1.5 is proven.
Proof of Proposition 1.6. Let , and be as in (3.1)-(3.2). TakeFor any and any , writeLetThen is a basis for a topology on , and the topological space is a -space. For any , let be the closure of in . Then is a closed arc, which is homeomorphic to the compact interval . Obviously, for any and any , (resp., ) is a compact neighborhood of the point (resp. ) in , which is homeomorphic to the subspace of . Thus is a locally compact space. Define a map byThen is continuous for the topology . It is easy to see that every point is an almost periodic point of , and the closure of the orbit in is always the whole space . Hence is the unique minimal set of . Since is an open cover of which has no finite subcover even has no countable subcover, the minimal set is not compact. Proposition 1.6 is proven.
Proof of Proposition 1.7. Let , , , , , and be as in (3.1)–(3.3). Take and . Letand let .
Then for any , and hence, is a basis for a topology on .
Clearly, under this topology ,
we have the following.Claim 1. is a -space, and is continuous.Claim 2. No point is a recurrent point of , and hence,
no point is an almost periodic point of .
Noting , by Lemma 2.3, we have the following.
Claim 3. Every point is an almost periodic point of .
For any , let be the closure of in the space . Then we have
Claim 4. for any , and for any .
It follows from Claims 3, 4, and 2 that, for any , not all points in the closure of the almost periodic orbit are almost periodic points. Proposition 1.7 is proven.
The work was supported by the Special Foundation of National Prior Basis Researches of China (Grant no. G1999075108).
- J.-H. Mai and W.-H. Sun, “Almost periodic points and minimal sets in -regular spaces,” Topology and Its Applications, vol. 154, no. 15, pp. 2873–2879, 2007.
Copyright © 2008 Jie-Hua Mai and Xin-He Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.