Volume 2008, Article ID 262475,5pages doi:10.1155/2008/262475
Research Article
A Note on Almost Periodic Points and Minimal Sets in T
1- and T
2-Spaces
Jie-Hua Mai1and Xin-He Liu2
1Institute of Mathematics, Shantou University, Shantou, Guangdong 515063, China
2Institute of Mathematics, Guangxi University, Nanning, Guangxi 530004, China
Correspondence should be addressed to Xin-He Liu,[email protected] Received 10 February 2008; Accepted 17 April 2008
Recommended by Leonid Berezansky
We show that1there exist almost periodic orbits inT2-spaces of which the closures are not minimal sets;2there exist minimal sets in locally compactT1-spaces which are not compact;3there exist almost periodic orbits inT2-spaces of which the closures contain not only almost periodic points.
These give answers to the three problems given by Mai and Sun in2007.
Copyrightq2008 J.-H. Mai and X.-H. 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.
1. Introduction
Denote byR,Z,Z, andNthe sets of real numbers, integers, nonnegative integers, and positive integers, respectively. For any topological spaceX, denote byC0Xthe set of all continuous maps ofXinto itself. For anyf∈C0X, letf0idXbe the identity map ofX, and letfiffi−1 be the composition offandfi−1i1,2,3, . . ..fnis called thenth iterate offn∈Z.
The orbit of a pointx ∈ X underf, denoted byOx, f, is the set{x, fx, f2x, . . .}.
x ∈X is called an almost periodic point off andOx, fis called an almost periodic orbit if for any neighborhoodUofxthere existsN∈Nsuch that{fnix:i0,1, . . . , N} ∩U /∅for all n∈Z.
A subsetWofX is said to be invariant orf-invariant iffW ⊂ W, andW is called a minimal set off if it is nonempty, closed, andf-invariant and if no proper subset ofW has these three properties.
The notion ofω-regular space was introduced by Mai and Sun1.
Definition 1.1see1, Definition 2.1. A topological spaceXis called anω-regular space if for any closed setW ⊂X, any pointx∈X−Wand any countable setA⊂W, there exist disjoint open setsUandV such thatx∈UandA⊂V.
Mai and Sun1generalized several known results concerning almost periodic points and minimal sets of maps from regular spaces toω-regular spaces, and obtained the following theorems.
Theorem Asee1, Theorem 2.3. LetXbe anω-regular space, andf∈C0X. Then the closure of every almost periodic orbit offis a minimal set.
Theorem B see 1, Theorem 3.8. LetX be a locally compact topological space which is either Hausdorffor regular, andf∈C0X. Then each minimal set offis compact.
Theorem Csee1, Theorem 4.1. LetXbe anω-regular space, andf∈C0X. Then all points in the closure of any almost periodic orbit offare 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, and1, Example 3.9shows that Theorem B cannot be extended to general locally compactT0-spaces. However, there remain three problems which have not been solved in1.
Problem 1.2see1, Problem 2.5. Can the condition in1, Theorem 2.3thatXis anω-regular space be replaced by thatXis aT2-space? In other words, need the closure of an almost periodic orbit in aT2-space be a minimal set?
Problem 1.3see1, Problem 3.10. LetXbe a locally compactT1-space. Is each minimal set of anyf∈C0Xcompact?
Problem 1.4see1, Problem 4.4. Can the condition in1, Theorem 4.1thatXis anω-regular space be replaced by that X is aT2-space? In other words, does the closure of any almost periodic orbit in aT2-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 aT2-spaceXand a continuous mapf :X→Xsuch thatfhas an almost periodic orbit of which the closure is not a minimal set.
Proposition 1.6. There exist a locally compactT1-spaceXand a continuous mapf :X→Xsuch that fhas a minimal set which is not compact.
Proposition 1.7. There exist aT2-spaceX, a continuous mapf:X→X,and an almost periodic point xoffsuch that not all points in the closureOx, fof the orbitOx, fare almost periodic points.
2. Finer topologies and subspace topologies
In order to prove the above three propositions, we will construct several newT1- andT2-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. LetTandTbe two topologies on a setX. IfTis finer thanT, andX,Tis aTi-space, i1,2, thenX,Tis also aTi-space.
Lemma 2.2. LetXbe a topological space,Ybe a subspace ofX, and letf:X→Xbe a continuous map such thatfY⊂Y. Then any pointy∈Yis an almost periodic point offif and only ifyis an almost periodic point off|Y.
FromLemma 2.2, we can obtain immediately.
Lemma 2.3. LetXbe a set,Ybe a subset ofX, andf:X→Xbe a map such thatfY⊂Y. Suppose thatTandTare two topologies onX,T|Y T|Y, andfis continuous both forTand forT. Then any pointy∈Y is an almost periodic point offfor topologyTif and only ifyis an almost periodic point of ffor topologyT.
3. Proofs of Propositions1.5–1.7
Propositions1.5–1.7claim that there exist continuous maps of someT1- orT2-spaces which have certain special properties. Hence in order to show these propositions, we need to construct maps having these special properties.
LetS1{eti:t∈R}be the unit circle in the complex planeC. For any two real numbers r < s, writeAr, s {eti :r < t < s}. LetT0be the usual topology onS1, and let
B0
Ar, s:r andsare real numbers, and 0< s−r < π
. 3.1
ThenB0is a set of open arcs inS1, which is a basis for the topologyT0.
Letθ∈0,1be a given irrational number, and lethθ:S1→S1be the rotation defined by
hθe2πit e2πitθ for anyt∈R. 3.2
Then under the topologyT0,hθ is a homeomorphism,S1is the unique minimal set ofhθ, and all points inS1are almost periodic points ofhθ. Letw0∈S1be a given point, and let
W
hnθ w0
:n∈Z
, Y S1−W. 3.3
Then we havehθW W, andhθY Y.
Proof ofProposition 1.5. LetS1,B0,T0,hθ,W, andYbe as in3.1–3.3. LetXS1,fhθ, and let
U1
A∩Y :A∈ B0
, B1B0∪ U1. 3.4 Then for anyU,V ∈ B1, we haveU∩V ∈ B1∪ {∅}. ThusB1 is a basis for a topology T1
onX. It follows fromB1 ⊃ B0thatT1 ⊃ T0. Hence byLemma 2.1,X,T1is aT2-space. Since T1|W T0|W andT1|Y T0|Y, byLemma 2.3, every pointx∈Xis an almost periodic point of f. BecauseY ∈ T1, that is,Y is an open set,W X−Y is a closed subset ofX,T1. For any x∈X, letOx, fbe the closure of the orbitOx, finX,T1. For anyz∈Xand anyU ∈ T1
withz∈U, there exists an open arcA∈ B0such that z∈A⊂U, if z∈W;
z∈A∩Y ⊂U, ifz∈Y. 3.5
Thus we have
Ox, f
⎧⎨
⎩
X, ifx∈Y;
W, ifx∈W. 3.6
From this we see that, in theT2-spaceX,T1,Wis the unique minimal set off, and for anyx∈ Y, the closureOx, fof the almost periodic orbitOx, fis not a minimal set.Proposition 1.5 is proven.
Proof ofProposition 1.6. LetS1,B0andhθbe as in3.1-3.2. Take XS1× {0,1}, B20
A× {0}:A∈ B0
. 3.7 For anyA∈ B0and anyx∈A, write
UA, x
A× {0}
∪ x,1
− x,0
. 3.8
Let
B21
UA, x:A∈ B0, x∈A
, B2B20∪ B21. 3.9 ThenB2is a basis for a topologyT2onX, and the topological spaceX,T2is aT1-space. For anyA⊂ B0, letAbe the closure ofAinS1,T0. ThenAis a closed arc, which is homeomorphic to the compact interval0,1. Obviously, for anyA∈ B0and anyx∈A, A×{0}∪{x,1}− {x,0}resp.,A× {0}is a compact neighborhood of the pointx,1 resp.x,0 inX,T2, which is homeomorphic to the subspaceAofS1,T0. ThusX,T2is a locally compact space.
Define a mapf :X→Xby
fx, i
hθx,0
, ∀x, i∈XS1× {0,1}. 3.10 Thenf is continuous for the topologyT2. It is easy to see that every pointx, i ∈ X is an almost periodic point of f, and the closureOx, i, fof the orbit Ox, i, f in X,T2is always the whole spaceX. HenceXis the unique minimal set off. SinceB2is an open cover ofXwhich has no finite subcover even has no countable subcover, the minimal setX is not compact.Proposition 1.6is proven.
Proof ofProposition 1.7. LetS1,B0,T0,hθ,W, andYbe as in3.1–3.3. TakeXS1andfhθ. Let
B31
A∩W:A∈ B0
, 3.11
B32
A∩W∪ {y}:A∈ B0, y∈A∩Y
, 3.12
and letB3B31∪ B32. ThenU∩V ∈ B3∪ {∅}for anyU, V ∈ B3, and hence,B3is a basis for a topologyT3onX. Clearly, under this topologyT3, we have the following.
Claim 1. Xis aT2-space, andf:X→Xis continuous.
Claim 2. No point y ∈ Y is a recurrent point of f, and hence, no point y ∈ Y is an almost periodic point off.
NotingT3|W T0|W, byLemma 2.3, we have the following.
Claim 3. Every pointw∈Wis an almost periodic point off.
For anyX0⊂X, letX0be the closure ofX0in the spaceX,T3. Then we have Claim 4. Ow, f Xfor anyw∈W, andOy, f Oy, ffor anyy∈Y.
It follows from Claims3,4, and2that, for anyx∈W, not all points in the closureOx, f of the almost periodic orbitOx, fare almost periodic points.Proposition 1.7is proven.
Acknowledgment
The work was supported by the Special Foundation of National Prior Basis Researches of China Grant no. G1999075108.
References
1J.-H. Mai and W.-H. Sun, “Almost periodic points and minimal sets inω-regular spaces,” Topology and Its Applications, vol. 154, no. 15, pp. 2873–2879, 2007.
