# A Note on Almost Periodic Points and  Minimal  Sets inT1- and T2-Spaces

Authors: Jie-Hua Mai; Xin-He Liu
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

We show that (1) there exist almost periodic orbits inT2-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).


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## Body

## 1. Introduction

Denote  byℝ, 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=idX be the identity map of X,
and let fi=ffi−1 be the composition of f and fi−1  (i=1,2,3,…). fn is called the nth iterate of f(n∈ℤ+).Theorbit 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 {fn+i(x):i=0,1,…,N}∩U≠∅ for all n∈Z+.A subsetW 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 ofω-regular space was introduced by Mai and Sun [1].Definition 1.1 (See [1, Definition 2.1]).
A topological spaceX 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 ]).
LetX 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]).
LetX 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 ]).
LetX 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 ]).
LetX 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 aT2-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 compactT1-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 aT2-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 newT1- 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.
Let𝒯 and 𝒯′ be two topologies on a set X. If 𝒯′ is finer than 𝒯,
and (X,𝒯) is a Ti-space, i=1,2, then (X,𝒯′) is also a Ti-space.Lemma 2.2.
LetX 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 Lemma2.2, we can obtain immediately.Lemma 2.3.
LetX 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 Propositions1.5–1.7

Propositions1.5–1.7 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.LetS1={eti:t∈R} be the unit circle in the complex plane ℂ.
For any two real numbers r<s,
write A(r,s)={eti:r<t<s}.
Let 𝒯0 be the usual topology on S1,
and let(3.1)ℬ0={A(r,s):randsare real numbers, and0<s−r<π}.Then ℬ0 is a set of open arcs in S1,
which is a basis for the topology 𝒯0.Letθ∈[0,1] be a given irrational number, and let hθ:S1→S1 be the rotation defined by(3.2)hθ(e2πit)=e2πi(t+θ)for anyt∈ℝ.Then under the topology 𝒯0, hθ is a homeomorphism, S1 is the unique minimal set of hθ, and all points in S1 are almost periodic points of hθ. Let w0∈S1 be a given point, and let(3.3)W={hθn(w0):n∈Z},Y=S1−W.Then we have hθ(W)=W, and hθ(Y)=Y.Proof.
LetS1, ℬ0, 𝒯0, hθ, W, and Y be as in (3.1)–(3.3).
Let X=S1,  f=hθ, and let(3.4)𝒰1={A∩Y:A∈ℬ0},ℬ1=ℬ0∪𝒰1.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 that(3.5)z∈A⊂U,ifz∈W;z∈A∩Y⊂U,ifz∈Y.Thus we have(3.6)O(x,f)¯={X,ifx∈Y;W,ifx∈W.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.
LetS1, ℬ0 and hθ be as in (3.1)-(3.2). Take(3.7)X=S1×{0,1},ℬ20={A×{0}:A∈ℬ0}.For any A∈ℬ0 and any x∈A, write(3.8)U(A,x)=((A×{0})∪{(x,1)})−{(x,0)}.Let(3.9)ℬ21={U(A,x):A∈ℬ0,x∈A},ℬ2=ℬ20∪ℬ21.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 by(3.10)f(x,i)=(hθ(x),0),∀(x,i)∈X=S1×{0,1}.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.
LetS1, ℬ0, 𝒯0, hθ, W, and Y be as in (3.1)–(3.3). Take X=S1 and f=hθ. Let(3.11)ℬ31={A∩W:A∈ℬ0},(3.12)ℬ32={(A∩W)∪{y}:A∈ℬ0,y∈A∩Y},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 pointy∈Y is a recurrent point of f, and hence,
no point y∈Y is an almost periodic point of f.
Noting𝒯3|W=𝒯0|W,
by Lemma 2.3,
we have the following.
Claim 3.
Every pointw∈W is an almost periodic point of f.
For anyX0⊂X,
let X¯0 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 Claims3, 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.



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*Source: 262475-2008-05-06.xml*