In this paper, we have investigated and introduced some new definitions of transitivity in topological space. To study the dynamics of a self-map f : X → X means to study the qualitative behavior of the sequences { f n ( x ) } as n goes to infinity when x varies in X, where f n denotes the composition of f with itself n times:

By a topological system I mean a pair ( X , f ) , where X is a locally compact Hausdorff topological space (the phase space), and f : X → X is a continuous function. The dynamics of the system is given by x n + 1 = f ( x n ) , x 0 ∈ X , n ∈ N and the solution passing through x is the sequence { f ( x n ) } where n ∈ N .

Let x ∈ X , then the set { x , f ( x ) , f 2 ( x ) , ⋯ } is called an orbit of x under f and is denoted by O f ( x ) , so O f ( x ) is the set of points which occur on the orbit of x at some positive time, and the sequence x , f ( x ) , f 2 ( x ) , ⋯ is called the trajectory of x. Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive.

Topological dynamics is concerned with the behavior of iterations of a continuous map f from a space X into itself. Suppose for some x ∈ X , sequence x , f ( x ) , f 2 ( x ) , ⋯ converges to some point say x 0 ∈ X , then we must have f ( x 0 ) = x 0 , because f is continuous. Such points we call as fixed points. We say that the point x is attracted by the fixed point x 0 . The set of all points in X attracted by x 0 is called the stable set or the basin of attraction of the fixed point x 0 and is denoted by W f ( x 0 ) . A fixed point x 0 is said to be attracting if its stable set is a neighborhood of it.

A point x ∈ X is said to be periodic if there exists a positive integer n ∈ N such that f n ( x ) = x . The set of all periodic points of the map f is denoted by per(f).

A point x ∈ X is called a θ-adherent point of A [1] , if A ∩ C l ( U ) ≠ ϕ for every open set U containing x. The set of all θ-adherent points of a subset A of X is called the θ-closure of A and is denoted by C l θ ( A ) . A subset A of X is called θ-closed if A = C l   θ ( A ) . Dontchev and Maki [2] have shown that if A and B are subsets of a space X, then C l θ ( A ∪ B ) = C l θ ( A ) ∪ C l θ ( B ) and that C l θ ( A ∩ B ) = C l θ ( A ) ∩ C l θ ( B ) . Recall that a space (X, τ) is Hausdorff if and only if every compact set is θ-closed. The complement of a θ-closed set is called a θ-open set. The family of all θ-open sets forms a topology on X and is denoted by τ θ . This topology is coarser than τ and that a space (X, τ) is regular if and only if τ = τ θ [3] .

Definition 2.1 [4] By a topological system I mean a pair ( X , f ) , where X is a locally compact Hausdorff topological space (the phase space), and f : X → X is a continuous function. The dynamics of the system is given by x n + 1 = f ( x n ) , x 0 ∈ X , n ∈ N and the solution passing through x 0 is the sequence { f ( x n ) } where n ∈ N .

Definition 2.2. 1) Let x ∈ X , then the set { x , f ( x ) , f 2 ( x ) , ⋯ } is called an orbit of x under f and is denoted by O f ( x ) , so O f ( x ) is the set of points which occur on the orbit of x at some positive time, and the sequence x , f ( x ) , f 2 ( x ) , ⋯ is called the trajectory of x.

2) Let X be a topological space, f : X → X , { f n ( x 0 ) } n = 0 ∞ be a sequence in X, and let x ∈ X . Then { f n ( x 0 ) } converges to x if for all open sets U containing x, there exists an integer N such that f n ( x 0 ) ∈ U for all n >N, Note that if this sequence is convergence then it converges to a fixed point, say y, i.e. f ( y ) = y .

Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive.

Definition 2.3. 1) (Transitivity) Let X be a topological space with no isolated point. Then the function f : X → X is said to be transitive if for any two open sets U and V in X, there is a point x ∈ U and an n > 0 such that f n ( x ) ∈ V . It is easily to show that if f is transitive then for every pair U, V of non-empty open sets, there exist a positive integer n such that f n ( U ) ∩ V ≠ ϕ .

2) Let X be a topological space, the function f : X → X , is said to be topologically mixing if for every pair U, V of non-empty open sets, if there exist N such that f n ( U ) ∩ V ≠ ϕ for all n > N .

Definition 2.4. (topological weak mixing) Let X has no isolated point. g is topologically weakly mixing, if the product of two functions g × g is topologically transitive.

Proposition 2.5. Every topological mixing function implies topological weak mixing. But the converse is no necessarily true.

Proof: It is easily to prove the foregoing theorem.

Definition 2.6. A map f is said to be transitive (resp., θ-transitive [5] ) if for any non-empty open (resp., θ-open) sets U and V in X, there exists n ∈ N such that f n ( U ) ∩ V ≠ ϕ .

Theorem 2.7 [5] . Let X be a non-empty locally θ-compact Hausdorff space. Then the intersection of a countable collection of θ-open θ-dense subsets of X is θ-dense in X.

Corollary 2.8. A subset A of a space ( X , τ ) is θ-dense if and only if A ∩ U ≠ ϕ for all U ∈ τ α other than U = ϕ .

Two topological spaces ( X , τ ) and ( Y , τ 1 ) are called homeomorphic if there exists a one-to-one onto function f : ( X , τ ) → ( Y , τ 1 ) such that f and f − 1 are both continuous.

Note that any homeomorphic spaces have the same dynamics, if we have any notion about first space then we have the same notion about the other one.

A map h : X → Y is a homeomorphism if it is continuous, bijective and has a continuous inverse.

A function f : X → X is called θ-irresolute [6] if the inverse image of each θ-open set is a θ-open set in X.

A map h : X → Y is θr-homeomorphism if it is bijective and thus invertible and both h and h − 1 are θ-irresolute.

Theorem 2.9. Let ( X , f ) be a topological system where X is a non-empty θ-compact topological space and f : X → X is θ-irresolute map and that X is separable. Suppose that f is topologically θ-transitive. Then there is an element x ∈ X such that the orbit O f ( x ) = { x , f ( x ) , f 2 ( x ) , ⋯ , f n ( x ) , ⋯ } is θ-dense in X.

Proof: Let B = { U i } , i = 1 , 2 , 3 , ⋯ be a countable basis for the θ-topology of X. For each i, let O i = { x ∈ X : f n ( x ) ∈ U i   forsome   n ≥ 0 }

Then, clearly O i is θ-open and θ-dense. It is θ-open since f is θ-irresolute, so, O i = ∪ i = 1 ∞ f − 1 ( U i ) is θ-open and θ-dense since f is topological θ-transitive map. Further, for every θ-open set V, there is a positive integer n such that f n ( V ) ∩ U i ≠ ϕ , since f is θ transitive.

Now, apply theorem 2.7 to the countable θ-dense set { O i } to say that ∩ i = 0 ∞ O i is θ-dense and so non-empty. Let y ∈ ∩ i = 0 ∞ O i . This means that, for each i, there is a positive integer n such that f n ( y ) ∈ U i for every i. By Corollary 2.8 this implies that O f ( x ) is θ-dense in X.

Definition 2.10. The function f : X → X , is strongly transitive [7] if for any nonempty open set U ⊂ X , X = ∪ k = 0 s f k ( U ) for some s > 0. It is easily seen that X = ∪ k = 0 ∞ f k ( U ) for any nonempty open set U ⊂ X if and only if ∪ k = 0 ∞ f − k ( x ) is dense in X for any x ∈ X .

We may consider that, the last statement of the foregoing definition as lemma, because we can use this statement to prove the following corollary.

Lemma 2.11. X = ∪ k = 0 ∞ f k ( U ) for any nonempty open set U ⊂ X if and only if ∪ k = 0 ∞ f − k ( x ) is dense in X for any x ∈ X .

According to the definition 2.10 and lemma 2.11, we have the following important corollary.

Corollary 2.12. If ∪ k = 0 ∞ f − k ( x ) is dense in X for any x ∈ X , then the function f : X → X , is strongly transitive.

There are the following results:

Proposition 3.1. Every topological mixing function implies topological weak mixing. But the converse is no necessarily true.

Theorem 3.2. Let ( X , f ) be a topological system where X is a non-empty θ-compact topological space and f : X → X is θ-irresolute map and that X is separable. Suppose that f is topologically θ-transitive. Then there is an element x ∈ X such that the orbit O f ( x ) = { x , f ( x ) , f 2 ( x ) , ⋯ , f n ( x ) , ⋯ } is θ-dense in X.

The author declares no conflicts of interest.

Mohammed, D.M. (2022) On Theta Transitivity in a Topological Space with Countable Base. Open Access Library Journal, 9: e4998. https://doi.org/10.4236/oalib.1104998