Topological Nearly Entropy on Nearly Compact Spaces

TL;DR

This paper explores properties of topological nearly entropy on nearly compact spaces, introduces a new entropy notion for nearly compact spaces, and examines their relationships and properties in product spaces.

Contribution

It introduces a new topological nearly entropy for nearly compact spaces and studies its properties and relationships with existing notions.

Findings

Topological nearly entropy is studied on nearly compact spaces.

A new entropy notion, Ent_n(f), is introduced for nearly compact spaces.

Fundamental properties of entropy in product spaces are established.

Abstract

In our previous paper [9], we have introduced topological nearly entropy, Ent_N (f) by restricting X into a class of nearly compact spaces. In the present paper, some additional properties of this notion are studied. Furthermore, we introduce another new notion of topological nearly entropy of f denoted by Ent_n (f) when the whole space X itself is nearly compact. We show the relationship between these two notions for the class of nearly compact subspaces. We also propose new space, namely, R-space in studying the topological nearly entropy on nearly compact and Hausdorff space. As a consequence, the topological nearly entropy of f and it restriction f|K coincides. Finally, some fundamental properties of topological nearly entropy for product space are obtained.

Equations

$$\left\{K\in P\left(X\right)\setminus \left\{\varnothing \right\}:K\text{ is nearly compact relative to }X\text{ such that }f(K)\subseteq K\right\}.$$

$$En{t}_N\left(f,\mathcal{U},K\right)=\underset{n\to \infty }{\lim }\frac{1}{n}{M}_K\left(\underset{i=0}{\overset{n-1}{\bigvee }}{f}^{-i}\left(\mathcal{U}\right)\right)$$

$$En{t}_N\left(f,K\right)=\underset{\mathcal{U}}{\sup }\left\{En{t}_N\left(f,\mathcal{U},K\right):\mathcal{U}\text{is a regular open cover of}X\right\}$$

$$En{t}_N\left(f\right)=\underset{K}{\sup }\left\{En{t}_N\left(f,K\right):K\in H\left(X,f\right)\right\}$$

$$En{t}_N\left(f^{-1},\mathcal{U},K\right)$$

$$N_n\left(\mathcal{U}\right)=\min \left\{\mathrm{card}\left(\mathcal{V}\right):\mathcal{V}\text{ is a subcover of }\mathcal{U},X=\bigcup _{V\in \mathcal{V}}V\right\}.$$

$$\underset{m\to \infty }{\lim }\frac{1}{m}{M}_n\left(\underset{i=0}{\overset{m-1}{\bigvee }}{f}^{-i}\left(\mathcal{U}\right)\right)$$

$$En{t}_n\left(f,\mathcal{U}\right)=\underset{m\to \infty }{\lim }\frac{1}{m}{M}_n\left(\underset{i=0}{\overset{m-1}{\bigvee }}{f}^{-i}\left(\mathcal{U}\right)\right)$$

$$En{t}_n\left(f\right)=\underset{\mathcal{U}}{\sup }\left\{En{t}_n\left(f,\mathcal{U}\right):\mathcal{U}\text{is a regular open of}X\right\}$$

$$K=\left(\bigcup _{l=1}^s{U}_l\right)\cap K=\bigcup _{l=1}^s\left(U_l\cap K\right)\subseteq \bigcup _{l=1}^s\left(U_l^{{}^{\prime }}\cap K\right)\subseteq \bigcup _{l=1}^s{U}_l^{{}^{\prime }}.$$

$$N_K\left(\underset{i=0}{\overset{m-1}{\bigvee }}{f}^{-i}\left(\mathcal{U}\right)\right)\le s=N_n\left(\underset{i=0}{\overset{m-1}{\bigvee }}{\left(f{|}_K\right)}^{-i}\left(\mathcal{U}{|}_K\right)\right)$$

$$En{t}_N\left(f,\mathcal{U},K\right)$$

$$\left\{V_j^{{}^{\prime }}:V_j^{{}^{\prime }}=\bigcap _{i=0}^{m-1}{\left(f{|}_K\right)}^{-i}\left(V_j^i\cap K\right)\right\}$$

$$V_j\cap K=\bigcap _{i=0}^{m-1}\left(f^{-i}\left(V_j^i\right)\cap K\right)\subseteq \bigcap _{i=0}^{m-1}\left(f^{-i}\left(V_j^i\right)\cap f^{-i}\left(K\right)\right)=\bigcap _{i=0}^{m-1}{f}^{-i}\left(V_j^i\cap K\right).$$

$$V_j\cap K=\left(\bigcap _{i=0}^{m-1}{f}^{-i}\left(V_j^i\cap K\right)\right)\cap K=\bigcap _{i=0}^{m-1}{\left(f{|}_K\right)}^{-i}\left(V_j^i\cap K\right)=V_j^{{}^{\prime }}$$

$$K=\bigcup _{j=1}^q\left(V_j\cap K\right)=\bigcup _{j=1}^q{V}_j^{{}^{\prime }}.$$

$$M_n\left(\underset{i=0}{\overset{m-1}{\bigvee }}{\left(f{|}_K\right)}^{-i}\left(\mathcal{U}{|}_K\right)\right)\le M_K\left(\underset{i=0}{\overset{m-1}{\bigvee }}{f}^{-i}\left(\mathcal{U}\right)\right).$$

$$En{t}_n\left(f{|}_K,\mathcal{U}{|}_K\right)$$

$$En{t}_N\left(f\right)=\underset{K}{\sup }\left\{En{t}_N\left(f,K\right):K\in H\left(X,f\right)\right\}=En{t}_N\left(f,X\right).$$

$$En{t}_N\left(f,X\right)$$

$$G\cap H$$

$$En{t}_n\left(f{|}_K,K\right)=\underset{{\mathcal{U}}_K}{\sup }\left\{En{t}_n\left(f{|}_K,{\mathcal{U}}_K,K\right)\right\}=\underset{\mathcal{U}}{\sup }\left\{En{t}_n\left(f{|}_K,\mathcal{U}{|}_K,K\right)\right\}.$$

$$En{t}_N\left(f,K\right)=\underset{\beta }{\sup }\left\{En{t}_N\left(f,\beta ,K\right):\beta \text{ runs over all regular open covers of }X\right\}.$$

$$\underset{\mathcal{U}}{\sup }\left\{En{t}_N\left(f,\mathcal{U},K\right)\right\}\le En{t}_N\left(f,K\right)$$

$$En{t}_n\left(f{|}_K,K\right)\le En{t}_N\left(f,K\right).$$

$$En{t}_N\left(f,K\right)=\underset{\mathcal{U}}{\sup }\left\{En{t}_N\left(f,\mathcal{U},K\right)\right\}=\underset{\mathcal{U}}{\sup }\left\{En{t}_n\left(f{|}_K,\mathcal{U}{|}_K,K\right)\right\}.$$

$$\underset{\mathcal{U}}{\sup }\left\{En{t}_n\left(f{|}_K,\mathcal{U}{|}_K,K\right)\right\}\le En{t}_n\left(f{|}_K,K\right).$$

$$En{t}_N\left(f,K\right)$$

$$En{t}_N\left(f\right)=\underset{K}{\sup }\left\{En{t}_N\left(f,K\right):K\in H\left(X,f\right)\right\}.$$

$$En{t}_N\left(f,K\right)=\underset{\mathcal{U}}{\sup }\left\{En{t}_N\left(f,\mathcal{U},K\right):\mathcal{U}\text{is a regular open cover of}X\right\}.$$

Full text

1. Introduction

Compactness is a crucial property of topological space which play an important role for the development of topological entropy of a dynamical system. Adler et al. [1] introduced the concept of topological entropy of continuous mapping for a compact space of dynamical system in 1965. Subsequently, Bowen [3] generalized the concept of topological entropy due to Adler et al. [1] which is metric-dependent and hold for noncompact spaces, see also [11] for the detail explanation of these two notions. Next, Liu et al. [10] generalized the notion of Adler et al. where separation axioms not necessarily required which intiate the concept of topological entropy for arbitrary topological spaces. In 2015, Afsan [2] defined topological $H$-entropy for topological $H$-dynamical system based on $H$-set and $H$-closed topological space.

In a recent paper, Gulamsarwar and Salleh [9] presented a new generalized notion of topological entropy on nearly compact space by using $R$-map [5], namely topological nearly entropy. Recall that every compact space is nearly compact but not the converse. Thus, the concept of topological nearly entropy for topological $R$-dynamical system was initiated in accordance with several fundamental properties of Adler et al. [1] and Liu et al. [10]. Until now, many researchers are interested with the concept of topological entropy as we can see more of their works in [4] and [8], etc.

This paper explore more fundamental properties of topological nearly entropy. Moreover this paper presenting the notion of topological nearly entropy for $R$-map on nearly compact space and some its fundamental properties are studied. These fundamental properties are further extension of Gulamsarwar and Salleh [9] topological nearly entropy. Furthermore, we introduce $R$-space in order to study the effect of nearly compactness on Hausdorff space to the topological nearly entropy. Finally, we show that $R$-map preserves nearly compact property and obtain several properties of topological nearly entropy for product space.

In Section 3, we explore more properties and show the results obtained for topological nearly entropy of the funtion $f$ and it inverse $f^{-1}$ if $f$ is bijective. The definition of topological nearly entropy on nearly compact space and its fundamental properties are discussed in Section 4. We also study topological nearly entropy on nearly compact and Hausdorff space which has triggered the idea to introduce $R$-space. Consequently, we obtain several properties of topological nearly entropy for product space which are discusses in the Section 5.

2. Preliminaries

Throughout this paper $X$ is always mean arbitrary topological space unless explicitly stated. We denote the closure and interior of $A\subseteq X$ by $\mathop{\mathrm{c}l}\left(A\right)$ and $\mathop{\mathrm{i}nt}\left(A\right)$, respectively. A subset $A$ of $X$ is said to be regular open if and only if $A=\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(A\right)\right)$. By regular open cover of $X$, we mean a cover of $X$ by regular open sets of $X$. The join of two covers $\mathcal{U}$ and $\mathcal{V}$ are described as $\mathcal{U}\vee\mathcal{V}=\left\{A\cap B:A\in\mathcal{U},B\in\mathcal{V}\right\}$ and a cover $\mathcal{V}$ is said to be refinement of a cover $\mathcal{U}$, denoted as $\mathcal{U}\prec\mathcal{V}$, if for every element $V\in\mathcal{V}$ there exists an element $U\in\mathcal{U}$ such that $V\subseteq U$. The complement of a set $A$ is denoted as $A^{c}$.

A subset $Y$ of $X$ is said to be nearly compact relative to $X$ if every cover $\mathcal{U}=\{U_{\alpha}:\alpha\in\Delta\}$ of $X$ by open sets of $X$ contains a finite subfamily $\mathcal{U}_{0}=\{U_{1},U_{2},\ldots,U_{n}\}$ such that $Y\subseteq\bigcup_{i=0}^{n}\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(U_{i}\right)\right)$, or equivalently, $Y$ is covered by finitely regular open sets of $X$. If $Y=X$ is nearly compact relative to $X$, the space $X$ is called nearly compact. In other words, the space $X$ is nearly compact if and only if every regular open cover of $X$ contains a finite subcover, see [12].

A function $f:X\rightarrow Y$ is called $R$-map [5] if the inverse image of every regular open set in $Y$ is regular open in $X$. If $f:X\rightarrow X$ is an $R$-map, then the pair $\left(X,f\right)$ is called topological $R$-dynamical system, and if $X$ is nearly compact, $\left(X,f\right)$ is called nearly compact $R$-dynamical system, see [9]. Now, we recall some definitions of topological nearly entropy and its fundamental properties from [9] as follows.

Definition 2.1**.**

[9]** Let $\left(X,f\right)$ be a topological $R$-dynamical system, $\mathcal{U}$ be a regular open cover of $X$ and $K$ be a nonempty nearly compact relative to the space $X$ such that $f\left(K\right)\subseteq K$. Let $N_{K}\left(\mathcal{U}\right)=\min\left\{\mathop{\mathrm{c}ard}\left(\mathcal{V}\right):\mathcal{V}\text{ is a subcover of }\mathcal{U},K\subseteq\mathop{\displaystyle\bigcup}\limits_{V\in\mathcal{V}}V\right\}$. Since $K$ is a nearly compact relative to $X$, $N_{K}\left(\mathcal{U}\right)$ is a positive integer. Let $M_{K}\left(\mathcal{U}\right)=\log N_{K}\left(\mathcal{U}\right)$.

Theorem 2.1**.**

[9]** Let $\left(X,f\right)$ be a topological $R$-dynamical system. Let $\mathcal{U}$ and $\mathcal{V}$ be a regular open covers of $X$, and $K$ be a nearly compact relative to $X$ such that $f\left(K\right)\subseteq K$. Then the following statements hold:

$\left(a\right)$* $M_{K}\left(\mathcal{U}\right)\geq 0$*

$\left(b\right)$* $\mathcal{U}\prec\mathcal{V}$ implies $M_{K}\left(\mathcal{U}\right)\leq M_{K}\left(\mathcal{V}\right)$*

$\left(c\right)$* $M_{K}\left(\mathcal{U}\vee\mathcal{V}\right)\leq M_{K}\left(\mathcal{U}\right)+M_{K}\left(\mathcal{V}\right)$*

$\left(d\right)$* $M_{K}\left(f^{-1}\left(\mathcal{U}\right)\right)\leq M_{K}\left(\mathcal{U}\right)$. When $f\left(K\right)=K$, the equality holds.*

Let $f:X\rightarrow X$ be an $R$-map and let $\left(X,f\right)$ be a topological $R$-dynamical system. Denote by $H\left(X,f\right)$ the set of all $f$-invariant nonempty nearly compact relative to $X$, i.e.,

[TABLE]

If $X$ is nearly compact, it follows from $f\left(X\right)\subseteq X$ that $H\left(X,f\right)\neq\emptyset$. However, when $X$ is not nearly compact, $H\left(X,f\right)$ could be empty.

Definition 2.2**.**

[9]** Let $\left(X,f\right)$ be a topological $R$-dynamical system. $H\left(X,f\right)$ be the set $\left\{K\in P\left(X\right)\setminus\left\{\emptyset\right\}:K\text{ is nearly compact relative to }X\text{ such that }f\left(K\right)\subseteq K\right\}$ and $\mathcal{U}$ be a regular open cover of $X$. Then for $K\in H\left(X,f\right)$

[TABLE]

is called the topological nearly entropy of $f$ on $K$ relative to $\mathcal{U}$ and

[TABLE]

is called topological nearly entropy of $f$ on $K$.

Theorem 2.2**.**

[9]** Let $\left(X,f\right)$ be a topological $R$-dynamical system, $K_{1},K_{2}\in H\left(X,f\right)$ with $K_{1}\subseteq K_{2}$ and $\mathcal{U}$ be a regular open cover of $X$. Then

$\left(a\right)$* $Ent_{N}\left(f,\mathcal{U},K_{1}\right)\leq Ent_{N}\left(f,\mathcal{U},K_{2}\right)$ and*

$\left(b\right)$* $Ent_{N}\left(f,K_{1}\right)\leq Ent_{N}\left(f,K_{2}\right)$.*

Definition 2.3**.**

[9]** Let $\left(X,f\right)$ be a topological $R$-dynamical system. Then

[TABLE]

is called topological nearly entropy of $f$. For the special case $H\left(X,f\right)=\emptyset,Ent_{N}\left(f\right)=0$ is to be taken.

3. Further Properties of Topological Nearly Entropy

Several fundamental properties of topological nearly entropy has been proved in our previous paper (see [9]). Now we shall prove another further result about topological nearly entropy. The topological nearly entropy of a map $f$ is coincide with topological nearly entropy of it inverse $f^{-1}$ whenever $f,f^{-1}$ are bijections and $R$-maps as follows.

Theorem 3.1**.**

Let $\left(X,f\right)$ be a topological $R$-dynamical system and the function $f:X\rightarrow X$ is bijective. Assume that $f^{-1}:X\rightarrow X$ is $R$-map, then if $K\in H\left(X,f\right)$ and $f,f^{-1}:K\rightarrow K$ are bijections and $R$-maps, then $Ent_{N}\left(f\right)=Ent_{N}\left(f^{-1}\right)$.

Proof.

If $H\left(X,f\right)=\emptyset$, then $Ent_{N}\left(f\right)=0$ and $Ent_{N}\left(f\right)\leq Ent_{N}\left(f^{-1}\right)$. So, let $H\left(X,f\right)\neq\emptyset$. Now consider $K\in H\left(X,f\right)$, from the asumption $f,f^{-1}:K\rightarrow K$ are bijections and $R$-maps, we have $K\in H\left(X,f^{-1}\right)$ so that $H\left(X,f\right)\subseteq H\left(X,f^{-1}\right)$. Let $\mathcal{U}$ be any regular open cover of $X$ and $K\in H\left(X,f\right)$. Since $f:K\rightarrow K$ is bijective and $R$-map, then $f\left(K\right)=K$. By Theorem 2.1(d) in [9], $M_{K}\left(f^{-1}\left(\mathcal{U}\right)\right)=M_{K}\left(\mathcal{U}\right)$. So,

[TABLE]

Then, we have $Ent_{N}\left(f\right)\leq Ent_{N}\left(f^{-1}\right)$ by Definitions 2.2 and 2.3.

By interchange the role of $f$ and $f^{-1}$ in the preceding argument, we will have $Ent_{N}\left(f^{-1}\right)\leq Ent_{N}\left(\left(f^{-1}\right)^{-1}\right)=Ent_{N}\left(f\right)$. Thus, $Ent_{N}\left(f\right)=Ent_{N}\left(f^{-1}\right)$.

4. Topological Nearly Entropy on Nearly Compact Spaces

In this section, we define topological nearly entropy on nearly compact space by using $R$-map. Moreover, we show that this definition is equivalent to the previous definition in, see Definition 2.3 whenever $X$ is nearly compact.

Definition 4.1**.**

Let $\left(X,f\right)$ be a nearly compact $R$-dynamical system and $\mathcal{U}$ be a regular open cover of $X$. Let

[TABLE]

Since $X$ is nearly compact, $N_{n}\left(\mathcal{U}\right)$ is a positive integer. Let $M_{n}\left(\mathcal{U}\right)=\log N_{n}\left(\mathcal{U}\right).$

Theorem 4.1**.**

Let $\left(X,f\right)$ be a nearly compact $R$-dynamical system and $\mathcal{U}$ and $\mathcal{V}$ be a regular open covers of $X$. Then the following statements hold:

1. $\left(a\right)$

$M_{n}\left(\mathcal{U}\right)\geq 0$. 2. $\left(b\right)$

$\mathcal{U}\prec\mathcal{V}$* implies $M_{n}\left(\mathcal{U}\right)\leq M_{n}\left(\mathcal{V}\right)$.* 3. $\left(c\right)$

$M_{n}\left(\mathcal{U}\vee\mathcal{V}\right)\leq M_{n}\left(\mathcal{U}\right)+M_{n}\left(\mathcal{V}\right)$. 4. $\left(d\right)$

$M_{n}\left(f^{-1}\left(\mathcal{U}\right)\right)\leq M_{n}\left(\mathcal{U}\right)$. The equality holds when $f$ is onto.

Proof.

The proofs are almost analogous to the proof of Theorem 2.1 (see [9]) by replacing the role of $K$ as $X$ and slight modification of the proof, one can easily establish this theorem.

Theorem 4.2**.**

Let $\left(X,f\right)$ be a nearly compact $R$-dynamical system and $\mathcal{U}$ be a regular open cover of $X$. Then

[TABLE]

exists.

Proof.

The proof are almost analogous to the proof af Theorem 2.3 in [9] by replacing the role of $K$ as $X$ and slight modification of the proof, one can easily establish this theorem.

Definition 4.2**.**

Let $\left(X,f\right)$ be a nearly compact $R$-dynamical system and $\mathcal{U}$ be a regular open cover of $X$. Then

[TABLE]

is called topological nearly entropy of $f$ relative to $\mathcal{U}$ and

[TABLE]

is called the topological nearly entropy of $f$.

In the next theorem, we show the relationship between this new notion with the previous Definition 2.2.

Theorem 4.3**.**

Let $\left(X,f\right)$ be a topological $R$-dynamical system, $K\in H\left(X,f\right)$ and $\mathcal{U}$ be any regular open cover of $X$. Then for the subspace $K$, $Ent_{n}\left(f|_{K},\mathcal{U}|_{K}\right)=Ent_{N}\left(f,\mathcal{U},K\right)$, where $\mathcal{U}|_{K}=\left\{U\cap K:U\in\mathcal{U}\right\}$ and mapping $f|_{K}:K\rightarrow K$ is defined by $f|_{K}\left(k\right)=f\left(k\right)$ for each $k\in K$.

Proof.

Let $N_{n}\left(\bigvee_{i=0}^{m-1}\left(f|_{K}\right)^{-i}\left(\mathcal{U}|_{K}\right)\right)=s$. Then there exist a finite subfamily $\left\{U_{1},U_{2},\ldots,U_{s}\right\}$ of $\bigvee_{i=0}^{m-1}$ $\left(f|_{K}\right)^{-i}\left(\mathcal{U}|_{K}\right)$ such that $K\subseteq\bigcup_{l=1}^{s}U_{l}$. Then, for each $l\in\left\{1,2,\ldots,s\right\}$, there exist $\left\{U_{l}^{0},U_{l}^{1},U_{l}^{2},\ldots,U_{l}^{m-1}\right\}\in\mathcal{U}$ such that $U_{l}=\bigcap_{i=0}^{m-1}\left(f|_{K}\right)^{-i}\left(U_{l}^{i}\cap K\right)$. Clearly, $\left\{U_{l}^{{}^{\prime}}:U_{l}^{{}^{\prime}}=\bigcap_{i=0}^{m-1}f^{-i}\left(U_{l}^{i}\right)\right\}$ is a subfamily of $\bigvee_{i=0}^{m-1}f^{-i}\left(\mathcal{U}\right)$ with $U_{l}\subset U_{l}^{{}^{\prime}}$. Moreover,

[TABLE]

Thus,

[TABLE]

and so $M_{K}\left(\bigvee_{i=0}^{m-1}f^{-i}\left(\mathcal{U}\right)\right)\leq M_{n}\left(\bigvee_{i=0}^{m-1}\left(f|_{K}\right)^{-i}\left(\mathcal{U}|_{K}\right)\right)$. Hence,

[TABLE]

To establish the reverse inequality, consider $N_{K}\left(\bigvee_{i=0}^{m-1}f^{-i}\left(\mathcal{U}\right)\right)=q$. Then there exists a finite subfamily $\left\{V_{1},V_{2},\ldots,V_{q}\right\}$ of $\bigvee_{i=0}^{m-1}f^{-i}\left(\mathcal{U}\right)$ such that $K\subseteq\bigcup_{j=1}^{q}V_{j}$. Then for each $j\in\left\{1,2,\ldots,q\right\}$, there exist $\left\{V_{j}^{0},V_{j}^{1},V_{j}^{2},\ldots,V_{j}^{m-1}\right\}$ $\in\mathcal{U}$ such that $V_{j}=\bigcap_{i=0}^{m-1}f^{-i}\left(V_{j}^{i}\right)$. Consider the subfamily

[TABLE]

of $\bigvee_{i=0}^{m-1}\left(f|_{K}\right)^{-i}\left(\mathcal{U}|_{K}\right)$. Since $K\subseteq\bigcup_{j=1}^{q}V_{j}$, then $\bigcup_{j=1}^{q}\left(V_{j}\cap K\right)=K$. Again, $f\left(K\right)\subseteq K$ ensure that $K\subseteq f^{-i}\left(K\right)$ for each $i=0,1,2,\ldots$, and

[TABLE]

Thus,

[TABLE]

and so

[TABLE]

Hence, $N_{n}\left(\bigvee_{i=0}^{m-1}\left(f|_{K}\right)^{-i}\left(\mathcal{U}|_{K}\right)\right)\leq q=N_{K}\left(\bigvee_{i=0}^{m-1}f^{-i}\left(\mathcal{U}\right)\right)$ and so,

[TABLE]

Therefore,

[TABLE]

Thus, $Ent_{n}\left(f|_{K},\mathcal{U}|_{K}\right)=Ent_{N}\left(f,\mathcal{U},K\right).$

The following theorem indicates that the concept of topological nearly entropy $Ent_{N}\left(f\right)$ is coincides with $Ent_{n}\left(f\right)$ when $X$ itself is nearly compact.

Theorem 4.4**.**

Let $\left(X,f\right)$ be a nearly compact $R$-dynamical system. Then $Ent_{n}\left(f\right)=Ent_{N}\left(f\right)$.

Proof.

It is clear that $X\in H\left(X,f\right)$ and by Definition 2.3,

[TABLE]

Again for any regular open cover $\mathcal{U}$ of $X$, Theorem 4.3 ensures that $Ent_{n}\left(f,\mathcal{U}\right)=Ent_{N}\left(f,\mathcal{U},X\right)$ and so

[TABLE]

Thus, $Ent_{N}\left(f\right)=Ent_{N}\left(f,X\right)=Ent_{n}\left(f\right)$.

Next , we study some properties of of topological nearly entropy on nearly compact and Hausdorff space by introducing $R$-space. So, the result demonstrate that topological nearly entropy of a map $f$ is coincide with topological nearly entropy of it restriction $f|_{K}$ whenever $K\in H\left(X,f\right)$.

Recall that in general, the union of two regular open sets is not regular open. For example, let $\left(1,2\right)$ and $\left(2,3\right)$ be two regular open sets in $\mathbb{R}$. Hence, the union of both intervals are not regular open since $\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(\left(1,2\right)\cup\left(2,3\right)\right)\right)=\left(1,3\right)$ but $\left(1,3\right)\neq\left(1,2\right)\cup\left(2,3\right)$. As an upshot, we introduce $R$-space as follows.

Definition 4.3**.**

A topological space $X$ is said to be $R$-space if the union of each regular open subset of $X$ is also regular open.

Now, by using $R$-space we study some basic properties of nearly compact in Hausdorff space.

Lemma 4.1**.**

Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly compact relative to $X$ and $p\notin A$, then there exist regular open sets $G,H$ of $X$ such that $p\in G,A\subset H$ and $G\cap H=\emptyset.$

Proof.

Let $a\in A$. Since $p\notin A$, $p\neq a$. By hypothesis, $X$ is Hausdorff, hence there exist open sets $G_{a},H_{a}$ of $X$ such that $p\in G_{a},a\in H_{a}$ and $G_{a}\cap H_{a}=\emptyset.$ Since $A\subset\bigcup\left\{H_{a}:a\in A\right\}$, then $\left\{H_{a}:a\in A\right\}$ forms a cover of $A$ by open subsets of $X$. As $A$ is nearly compact relative to $X$, so there exist a finite subfamily $H_{a_{1}},\ldots,H_{a_{m}}$ of $\left\{H_{a}\right\}$ such that $A\subset\bigcup_{i=1}^{m}\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(H_{a_{i}}\right)\right)$.

Let $H=\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(H_{a_{1}}\right)\right)\cup\cdots\cup\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(H_{a_{m}}\right)\right)$ and $G=\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(G_{a_{1}}\right)\right)\cap\cdots\cap\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(G_{a_{m}}\right)\right)$. Since $X$ is $R$-space, then $H$ and $G$ are regular open. Furthermore, $A\subset H$ and $p\in G$ since $p$ belongs to each $G_{a_{i}}$ individually. Lastly we claim that $G\cap H=\emptyset$. Note first that, $\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(G_{a_{i}}\right)\right)$ $\cap$ $\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(H_{a_{i}}\right)\right)=\emptyset$ implies $G\cap\mathop{\mathrm{i}nt}\left(\mathop{\mathrm{c}l}\left(H_{a_{i}}\right)\right)=\emptyset$. Thus, by the distributive law,

[TABLE]

This completes the proof.

Lemma 4.2**.**

Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly compact relative to $X$ and $p\notin A$, then there is a regular open set $G$ such that $p\in G\subset A^{c}$.

Proof.

By Lemma 4.1, there exist regular open sets $G$ and $H$ in $X$ such that $p\in G,A\subset H$ and $G\cap H=\emptyset$. Hence $G\cap A=\emptyset$, and thus $p\in G\subset A^{c}.$

Lemma 4.3**.**

Let $X$ be a Hausdorff and $R$-space. If $A$ is a nearly compact relative to $X$, then $A$ is a regular closed subset of $X$.

Proof.

We prove, equivalently, that $A^{c}$ is a regular open subset of $X$. Let $p\in A^{c}$, i.e., $p\notin A$. Then by Lemma 4.2 there exists a regular open set $G_{p}$ of $X$ such that $p\in G_{p}\subset A^{c}$. Hence $A^{c}=\bigcup_{p\in A^{c}}G_{p}$. Thus $A^{c}$ is regular open as it is the union of regular open sets in $R$-space $X$. Therefore $A$ is a regular closed subset of $X$.

The next theorem shows that the conditions on which the topological nearly entropy of $f$ and it restriction $f|_{K}$ coincides.

Theorem 4.5**.**

Let $\left(X,f\right)$ be a topological $R$-dynamical system. Assume that $X$ is Hausdorff and $R$-space. If $K\in H\left(X,f\right)$ then

$\left(a\right)$* $Ent_{N}\left(f,K\right)=Ent_{n}\left(f|_{K},K\right)$*

$\left(b\right)$* $Ent_{N}\left(f\right)=Ent_{n}\left(f|_{K}\right),$*

where $f|_{K}$ is defined as in the Theorem 4.3.

Proof.

$\left(a\right)$ Let $\mathcal{U}_{K}$ be any regular open cover of $K$ (by regular open subsets of $K$). Since $X$ is Hausdorff and $K$ is nearly compact relative to $X$, then by Lemma 4.3, $K$ is a regular closed subset of $X$. For every $A\in\mathcal{U}_{K}$, there exist a regular open subset $U_{A}$ of $X$ satisfying $A=U_{A}\cap K$. Denote $\mathcal{U}^{\prime}=\left\{U_{A}:A\in\mathcal{U}_{K}\right\}$. Clearly, $\mathcal{U}=\mathcal{U}^{\prime}\cup\left\{X\setminus K\right\}$ is a regular open cover of $X$ satisfying $\mathcal{U}|_{K}=\mathcal{U}_{K}\cup\left\{\emptyset\right\}$. Hence, every regular open cover $\mathcal{U}_{K}$ of $K$ is a restriction $\mathcal{U}|_{K}$ of some special regular open cover $\mathcal{U}$ of $X$ that includes regular open subset $X\setminus K$ of $X$. From Definition 4.2, we have

[TABLE]

By Definition 2.2, we have

[TABLE]

It follows from Theorem 4.3 that, $Ent_{N}\left(f,\mathcal{U},K\right)=Ent_{n}\left(f|_{K},\mathcal{U}|_{K},K\right)$, thus $Ent_{n}\left(f|_{K},K\right)=\sup_{\mathcal{U}}\left\{Ent_{n}\left(f|_{K},\mathcal{U}|_{K},K\right)\right\}=\sup_{\mathcal{U}}$ $\left\{Ent_{N}\left(f,\mathcal{U},K\right)\right\}$. Recall that the regular open covers $\mathcal{U}$ are some special regular open covers of $X$. Hence,

[TABLE]

implying

[TABLE]

For reverse inequality, i.e., $Ent_{n}\left(f|K,K\right)\geq Ent_{N}\left(f,K\right)$, let $\mathcal{U}$ be any regular open cover of $X$. From Theorem 4.3, we have $Ent_{N}\left(f,\mathcal{U},K\right)=Ent_{n}\left(f|_{K},\mathcal{U}|_{K},K\right)$ and thus from Definition 2.2,

[TABLE]

As $\mathcal{U}|_{K}$ are some special regular open covers of $K$, we have

[TABLE]

Thus,

[TABLE]

$\left(b\right)$ By Definition 2.3,

[TABLE]

Now let $\mathcal{U}$ be any regular open cover of $X$. From Definition 2.2,

[TABLE]

Thus,

[TABLE]

By Theorem 4.3,

[TABLE]

Hence,

[TABLE]

The second equality follows from equation $\left(\ref{2}\right)$. This completes the proof.

5. Topological Nearly Entropy of Product Space

Let $\left(X,f\right)$ and $\left(Y,h\right)$ be two topological $R$-dynamical systems. For the product space $X\times Y$, define function $f\times h:X\times Y\rightarrow X\times Y$ by $\left(f\times h\right)\left(x,y\right)=\left(f\left(x\right),h\left(y\right)\right)$. This function $f\times h$ is $R$-map and $\left(X\times Y,f\times h\right)$ forms a topological $R$-dynamical system. If $\mathcal{U}$ and $\mathcal{V}$ are regular open covers of $X$ and $Y$, respectively, then $\mathcal{U}\times\mathcal{V}$ is a regular open cover of $X\times Y$.

As a consequence, we obtain the properties of topological nearly entropy for product space. But, firstly we show that $R$-map preserve nearly compactness relative to the space.

Lemma 5.1**.**

Let $f:X\rightarrow Y$ be an $R$-map. If $A$ is nearly compact relative to $X$, then $f\left(A\right)$ is nearly compact relative to $Y$.

Proof.

Let $\left\{V_{\alpha}:\alpha\in\Delta\right\}$ be a cover of $f\left(A\right)$ by regular open subsets of $Y$. For each $x\in A$, take $\alpha_{x}\in\Delta$ such that $f\left(x\right)\in V_{\alpha_{x}}$. Since $f$ is $R$-map, $f^{-1}\left(V_{\alpha_{x}}\right)$ is a regular open subset of $X$ containing $x$. Now $\left\{f^{-1}\left(V_{\alpha_{x}}\right):x\in A\right\}$ forms a regular open cover of $A$ by regular open subsets of $X$. Since $A$ is nearly compact relative to $X$, there exists a finite subset of points $x_{1},\ldots,x_{n}$ of $A$ such that $A\subseteq\bigcup_{k=1}^{n}f^{-1}\left(V_{\alpha_{x_{k}}}\right)$. Now,

[TABLE]

Therefore $f\left(A\right)$ is nearly compact relative to $Y$.

Lemma 5.2**.**

Let $X$ and $Y$ be two nearly compact $R$-dynamical system. Suppose that $\mathcal{U}$ and $\mathcal{V}$ are regular open covers of $X$ and $Y$, respectively. Then for a regular open cover $\mathcal{U}\times\mathcal{V}$ of $X\times Y$, we have

[TABLE]

Proof.

Let $\left\{U_{1},\ldots,U_{k}\right\}$ and $\left\{V_{1},\ldots,V_{p}\right\}$ be subcovers of $\mathcal{U}$ and $\mathcal{V}$ of minimal cardinality, respectively. Then $\left\{U_{i}\times V_{j}:i=1,\ldots,k,j=1,\ldots,p\right\}$ is a subcover of $\mathcal{U}\times\mathcal{V}$ with cardinality $kp$. It follows that

[TABLE]

Thus,

[TABLE]

Lemma 5.3**.**

Let $\left(X,f\right)$ and $\left(Y,h\right)$ be two topological $R$-dynamical systems. Suppose $T_{x}:X\times Y\rightarrow X$ and $T_{y}:X\times Y\rightarrow Y$ are the $R$-map projections on $X$ and $Y$, respectively. If $K\in H\left(X\times Y,f\times h\right)$, then $T_{x}\left(K\right)\in H\left(X,f\right)$, $T_{y}\left(K\right)\in H\left(Y,h\right)$ and $K\subseteq T_{x}\left(K\right)\times T_{y}\left(K\right)$.

Proof.

Since $T_{x}$ and $T_{y}$ are $R$-maps and $K\in H\left(X\times Y,f\times h\right)$, $T_{x}\left(K\right)$ is nearly compact relative to $X$ and $T_{y}\left(K\right)$ is nearly compact relative to $Y$ by Lemma 5.1. It suffices to show that $T_{x}\left(K\right)$ and $T_{y}\left(K\right)$ are invariant nearly compact relative to $X$ and $Y$ by $f$ and $h$, respectively. For any $x\in T_{x}\left(K\right)$, there exists a $y\in Y$ which satisfying $\left(x,y\right)\in K$. Since $K$ is an invariant nearly compact relative to $X\times Y$ by $f\times h$, we have

[TABLE]

Thus

[TABLE]

Hence $f\left(T_{x}\left(K\right)\right)\subseteq T_{x}\left(K\right)$ and therefore $T_{x}\left(K\right)\in H\left(X,f\right)$.

There also exists an $x\in X$ for any $y\in T_{y}\left(K\right)$ satisfying $\left(x,y\right)\in K$. Since $K$ is an invariant nearly compact relative to $X\times Y$ by $f\times h$, we have

[TABLE]

Thus

[TABLE]

Hence $h\left(T_{y}\left(K\right)\right)\subseteq T_{y}\left(K\right)$ and therefore $T_{y}\left(K\right)\in H\left(Y,h\right).$

Finally, if $\left(x,y\right)\in K$, then by the definitions of $T_{x}\left(K\right)$ and $T_{y}\left(K\right)$, we will have $x\in T_{x}\left(K\right)$ and $y\in T_{y}\left(K\right)$ and thus $\left(x,y\right)\in$ $T_{x}\left(K\right)\times T_{y}\left(K\right)$.

Lemma 5.4**.**

[12]** The product of two nearly compact spaces is nearly compact.

Lemma 5.5**.**

Let $\left(X,f\right)$ and $\left(Y,h\right)$ be two nearly compact $R$-dynamical systems. If $\mathcal{W}$ is a regular open cover of $X\times Y$, then there exist regular open cover $\mathcal{U}$ of $X$ and regular open cover $\mathcal{V}$ of $Y$ satisfying $\mathcal{W}\prec\mathcal{U}\times\mathcal{V}$.

Proof.

Since $X$ and $Y$ are nearly compact, by Lemma 5.4 the product space of $X\times Y$ is also nearly compact and for each $x\in X$, $\left\{x\right\}\times Y$ is a nearly compact subset of $X\times Y$. Since $\mathcal{W}$ is a regular open cover of $X\times Y$, then there exist $A_{y}\in\mathcal{W}$ for any $y\in Y$ which satisfying $\left(x,y\right)\in A_{y}$. Let $U\left(x,y\right)$ be a regular open neighborhood of $x$ in $X$ and $V\left(x,y\right)$ be a regular open neigborhood of $y$ in $Y$ with

[TABLE]

Since $Y$ is nearly compact, the regular open cover $\left\{V\left(x,y\right):y\in Y\right\}$ of $Y$ has a finite subcover, say $\left\{V\left(x,y_{1}\right),\ldots,V\left(x,y_{m\left(x\right)}\right)\right\}$. Hence, for each $U\left(x,y_{j}\right)\times V\left(x,y_{j}\right),j\in\left\{1,2,\ldots,m\left(x\right)\right\}$, there exist an $A_{j}\in\mathcal{W}$ such that $U\left(x,y_{j}\right)\times V\left(x,y_{j}\right)\subseteq A_{j}$. Denote $U\left(x\right)=\mathop{\displaystyle\bigcap}\limits_{j=1}^{m\left(x\right)}U\left(x,y_{j}\right)$, then

[TABLE]

On the other hand, as $X$ is nearly compact, the regular open cover $\left\{U\left(x\right):x\in X\right\}$ of $X$ has a finite subcover, say $\left\{U\left(x_{1}\right),\ldots,U\left(x_{n}\right)\right\}$. Observe that each $U\left(x_{i}\right)\times V\left(x_{i},y_{j}\right),i\in\left\{1,2,\ldots,n\right\},j\in\left\{1,2,\ldots,m\left(x_{i}\right)\right\}$ is contained in an element of $\mathcal{W}$. For any $y\in Y$, let $V\left(y\right)$ be the intersection of those elements of the collection $\left\{V\left(x_{i},y_{j}\right):i=1,\ldots,n;j=1,\ldots,m\left(x_{i}\right)\right\}$ that contain $y$. Since this collection has finitely many elements and each of them is regular open subset of $Y$, then $V\left(y\right)$ is also regular open subset of $Y$. Notice that $\left\{V\left(y\right):y\in Y\right\}$ is a regular open cover of $Y$ and hence it has a finite subcover as $Y$ is nearly compact. Denote this subcover by $\left\{V\left(y_{1}\right),\ldots,V\left(y_{m}\right)\right\}$. Now, $\left\{U\left(x_{i}\right)\times V\left(y_{j}\right):i=1,\ldots,n;j=1,\ldots,m\right\}$ is a regular open cover of $X\times Y$ and each $U\left(x_{i}\right)\times V\left(y_{j}\right)$ is contained in an element of $\mathcal{W}$. Let $\mathcal{U}=\left\{U\left(x_{i}\right):i=1,\ldots,n\right\}$ and $\mathcal{V}=\left\{V\left(y_{j}\right):j=1,\ldots,m\right\}$. Then $\mathcal{U}\times\mathcal{V}$ is a regular open cover of $X\times Y$ and $\mathcal{W}\prec\mathcal{U}\times\mathcal{V}$.

Theorem 5.1**.**

Let $\left(X,f\right)$ and $\left(Y,h\right)$ be two nearly compact $R$-dynamical systems. Then

[TABLE]

where $f\times h:X\times Y\rightarrow X\times Y$ is $R$-map defined by $\left(f\times h\right)\left(x,y\right)=\left(f\left(x\right),h\left(y\right)\right)$.

Proof.

Let $\mathcal{C}$ be an arbitrary regular open cover of $X\times Y$. By Lemma 5.5, there exist regular open cover $\mathcal{U}$ of $X$ and regular open cover $\mathcal{V}$ of $Y$ such that $\mathcal{U}=\{U_{i}:i=1,\ldots,n\}$ where $U_{i}$ is a regular open subset of $X$ and $\mathcal{V}=\{V_{j}:j=1,\ldots,m\}$ where $V_{j}$ is a regular open subset of $Y$ satisfying $\mathcal{C}\prec\mathcal{U}\times\mathcal{V}$. By Theorem 2.1 and Lemma 5.2

[TABLE]

So,

[TABLE]

Theorem 5.2**.**

Let $\left(X,f\right)$ and $\left(Y,h\right)$ be two topological $R$-dynamical systems with $X\times Y$ are $R$-space. If $X$ and $Y$ are Hausdorff, then $Ent_{N}\left(f\times h\right)\leq Ent_{N}\left(f\right)+Ent_{N}\left(h\right)$, where $f\times h:X\times Y\rightarrow X\times Y$ is $R$-map defined by $\left(f\times h\right)\left(x,y\right)=\left(f\left(x\right),h\left(y\right)\right)$.

Proof.

If $H\left(X\times Y,f\times h\right)=\emptyset$, then the result follows since $Ent_{N}\left(f\times h\right)=0$ by Definition 2.3.

If $H\left(X\times Y,f\times h\right)\neq\emptyset$, let $K\in H\left(X\times Y,f\times h\right)$ and the projections $T_{x}:X\times Y\rightarrow X$, $T_{y}:X\times Y\rightarrow Y$. For any regular open cover $\mathcal{W}$ of $X\times Y$, $T_{x}\left(K\right)\in H\left(X,f\right),T_{y}\left(K\right)\in H\left(X,f\right)$ and $K\subseteq T_{x}\left(K\right)\times T_{y}\left(K\right)$ by Lemma 5.3. Since $X$ and $Y$ are Hausdorff and $R$-spaces, then by Lemmas 5.1 and 4.3 $T_{x}\left(K\right)$ and $T_{y}\left(K\right)$ are regular closed subset of $X$ and $Y$, respectively.

Denote $T_{x}\left(K\right)$ and $T_{y}\left(K\right)$ as $K_{x}$ and $K_{y}$, respectively. Thus $K\subseteq K_{x}\times K_{y}$ and $K_{x}\times K_{y}\in H\left(X\times Y,f\times h\right)$ by Lemma 5.4. By Theorem 2.2,

[TABLE]

From Theorem 4.3, we have

[TABLE]

where $\mathcal{W}|_{K_{x}\times K_{y}}$ is a regular open cover of $K_{x}\times K_{y}$. By Lemma 5.5 , there exist a regular open cover of $\mathcal{U^{\prime}}$ of $K_{x}$ and a regular open cover $\mathcal{V^{\prime}}$ of $K_{y}$ such that $\mathcal{U^{\prime}}=\left\{U_{i}:i=1,\ldots,n\right\}$ where $U_{i}$ is a regular open subset of $K_{x}$ and $\mathcal{V^{\prime}}=\left\{V_{j}:j=1,\ldots,n\right\}$ where $V_{j}$ is a regular open subset of $K_{y}$ satisfying $\mathcal{W}|_{K_{x}\times K_{y}}\prec\mathcal{U^{\prime}}\times\mathcal{V^{\prime}}$. For every $U_{i}$, there exist a regular open subset $A_{i}$ of $X$ such that $U_{i}=A_{i}\cap K_{x}$. Denote $\mathcal{U}=\left\{A_{1},\ldots,A_{n},X\setminus K_{x}\right\}$. Hence $\mathcal{U}$ is regular open cover of $X$ and $\mathcal{U}|_{K_{x}}=\mathcal{U^{\prime}}\cup\left\{\emptyset\right\}$. Similarly, there exist a regular open cover $\mathcal{V}$ of $Y$ such that $\mathcal{V}|_{K_{y}}=\mathcal{V^{\prime}}\cup\left\{\emptyset\right\}$. Hence, from $\mathcal{W}|_{K_{x}\times K_{y}}\prec\mathcal{U^{\prime}}\times\mathcal{V^{\prime}}$, we have

[TABLE]

Therefore, we have $\mathcal{W}|_{K_{x}\times K_{y}}\prec\mathcal{U}|_{K_{x}}\times\mathcal{V}|_{K_{y}}$. Moreover, we also have

[TABLE]

By Theorem 4.1, equation (5.2) and Theorem 5.1, we have

[TABLE]

By Theorem 4.3,

[TABLE]

Hence, $Ent_{N}\left(f,\mathcal{U},K_{x}\right)=Ent_{n}\left(f|_{K_{x}},\mathcal{U}|_{K_{x}}\right)$ and $Ent_{N}\left(h,\mathcal{V},K_{y}\right)=Ent_{n}\left(h|_{K_{y}},\mathcal{V}|_{K_{y}}\right)$. Thus from equation $\left(\ref{(1)}\right)$,

[TABLE]

By Theorem 2.2 and equation $\left(\ref{(1)}\right)$,

[TABLE]

Finally by Definitions 2.2 and 2.3, we have

[TABLE]

In the next example, we shows $Ent_{N}\left(f\right)=0$ for $f\left(x\right)=kx$ on $\mathbb{R}$ with usual topology.

Example 5.1**.**

Let a map $f:\mathbb{R}\rightarrow\mathbb{R}$ define by $f\left(x\right)=kx$, where $k\in\left\{2,3,4,\ldots\right\}$. The only invariant compact subset of $\mathbb{R}$ and hence nearly compact subset of $\mathbb{R}$ is $\left\{0\right\}$, so $H\left(\mathbb{R},f\right)=\left\{\left\{0\right\}\right\}$. Denote $K=\left\{0\right\}$, thus for any regular open cover $\mathcal{U}$ of $\mathbb{R}$, any subcover $\mathcal{V}$ of $\mathcal{U}$ such that $K\subseteq\mathop{\displaystyle\bigcup}\limits_{V\in\mathcal{V}}V$ with the smallest cardinality contains a single element of $\mathcal{U}$, i.e., $\left\{0\right\}$. Hence, $N_{K}\left(\mathcal{U}\right)=1$, so that $N_{K}\left(\bigvee_{i=0}^{n-1}f^{-i}\left(\mathcal{U}\right)\right)=1$ and $M_{K}\left(\bigvee_{i=0}^{n-1}f^{-i}\left(\mathcal{U}\right)\right)=0$ which implies

[TABLE]

Therefore, from Definition 2.3, $Ent_{N}\left(f\right)=0$. If we replaced $\mathbb{R}$ by $\mathbb{R}^{+}$, then $H\left(\mathbb{R}^{+},f\right)=\emptyset$. Again from Definition 2.3, $Ent_{N}\left(f\right)=0$. Recall that the role of topological entropy is to describe the complexity of a topological dynamical system and positivity of entropy implies the complexity of dynamical behaviour of the system. Thus, we could say that topological nearly entropy is describe the complexity of topological $R$-dynamical system and it positive value shows that more complicated $R$-dynamical properties of the system. Hence, we conclude that $Ent_{N}\left(f\right)=0$ is a characterization for $R$-dynamical property of the map $f\left(x\right)=kx$.

6. Conclusion

We present a notion of topological nearly entropy on nearly compact spaces and investigate some its properties which is an extension from our previous paper [9]. Both of the notions are coincide in the special case of nearly compact spaces. Yet, they retain various fundamental properties of Adler et al. [1]. A new space, namely, $R$-space was introduced for nearly compact on Hausdorff space. Consequently, the properties of topological nearly entropy for product spaces are obtained.