Compact-Open and Point Wise Convergence Topologies

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1. Introduction

Let $\left(X,\tau \right)$ and $\left(Y,\sigma \right)$ be two topological spaces and $C\left(X,Y\right)$ be the set of all continuous maps from X into Y. Consider all possible sets of maps of the form

$\left[K,U\right]=\left\{f\in C\left(X,Y\right):f\left(K\right)\subset U\right\},$

where K is a compact set in X and U an open set in Y. The topology ${\tau}_{3}$ generated by these sets $\left[K,U\right]$ as a subbase is called the compact-open topology on $C\left(X,Y\right)$ . Note that any open set in ${\tau}_{3}$ is called co-open set and $\left(C\left(X,Y\right),{\tau}_{3}\right)$ is called co-topological space. The compliment of co-open set is called co-closed set. We have introduced some new definitions of transitivity on $C\left(X,Y\right)$ , called the point-wise convergence transitive set, the compact-open transitive and point wise convergence topological transitive sets in C(X, Y). Relationship between these new definitions is studied. Finally, we have introduced a number of very important topological concepts and shown that every compact-open convergence transitive set implies point wise transitive set and that every compact-open-mixing system implies point wise convergence system but not conversely. Finally, we have shown that every strongly compact-open-mixing set implies strongly point wise convergence mixing set but the converse not necessarily true.

2. New Theorems of Point Wise-Convergence Topology

Definition 2.1. Consider in $C\left(X,Y\right)$ the sets

${\left\{{x}_{i},{U}_{i}\right\}}_{i1}^{k}=\left\{f\in C\left(X,Y\right):f\left({x}_{i}\right)\in {U}_{i},i=1,\cdots ,k\right\}$

where ${x}_{1},\cdots ,{x}_{k}\in X$ , ${U}_{1},\cdots ,{U}_{k}$ are open sets in Y.

The topology ${\tau}_{2}$ generated by these sets in their capacity as a subset is called the topology of point-wise convergence on $C\left(X,Y\right)$ .

Note that any open set in ${\tau}_{2}$ is called pc-open set and $\left(C\left(X,Y\right),{\tau}_{2}\right)$ is called pc-topological space. The compliment of pc-open set is called pc-closed set.

Definition 2.2. A function $F:C\left(X,Y\right)\to C\left(X,Y\right)$ is called pc-irresolute if the inverse image of each pc-open set is a pc-open set in $C\left(X,Y\right)$ .

Definition 2.3. A map $F:C\left(X,Y\right)\to C\left(X,Y\right)$ is pcr-homeomorphism if it is bijective and thus invertible and both F and ${F}^{-1}$ are pc-irresolute.

The systems $F:C\left(X,X\right)\to C\left(X,X\right)$ and $G:C\left(Y,Y\right)\to C\left(Y,Y\right)$ are topologically pcr-conjugate if there is a pcr-homeomorphism $H:C\left(X,X\right)\to C\left(Y,Y\right)$ such that $H\circ F=G\circ H$ .

Let $\left(C\left(X,Y\right),{\tau}_{2}\right)$ be a pc-topological space. The intersection of all pc-closed sets of $\left(C\left(X,Y\right),{\tau}_{2}\right)$ containing A is called the pc-closure of A and is denoted by $C{l}_{pc}\left(A\right)$ .

Definition 2.4. Let $\left(C\left(X,Y\right),{\tau}_{2}\right)$ be a point wise convergence-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a map. The map F is said to have pc-dense orbit if there exists $f\in C\left(X,Y\right)$ such that $C{l}_{pc}\left({O}_{F}\left(f\right)\right)=C\left(X,Y\right)$ .

Definition 2.5. Let $\left(C\left(X,Y\right),{\tau}_{2}\right)$ be a pc-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a pc-irresolute map, then F is said to be a point-wise-converge-transitive (shortly pc-transitive) map if for every pair of pc-open sets U and V in $\left(C\left(X,Y\right),{\tau}_{2}\right)$ there is a positive integer n such that ${F}^{n}\left(U\right)\cap V\ne \varphi $ .

Definition 2.6. Let $\left(C\left(X,Y\right),{\tau}_{2}\right)$ be a point wise convergence-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a pc-irresolute then the set $A\subseteq C\left(X,Y\right)$ is called pc-type transitive set if for every pair of non-empty pc-open sets U and V in $C\left(X,Y\right)$ with $A\cap U\ne \varphi $ and $A\cap V\ne \varphi $ there is a positive integer n such that ${F}^{n}\left(U\right)\cap V\ne \varphi $ .

Definition 2.7. 1) Let $\left(C\left(X,Y\right),{\tau}_{1}\right)$ be a point-wise convergence-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a pc-irresolute then the set $A\subseteq C\left(X,Y\right)$ is called is called topologically pc-mixing set if, given any nonempty pc-open subsets $U,V\subseteq C\left(X,Y\right)$ with $A\cap U\ne \varphi $ and $A\cap V\ne \varphi $ then $\exists N>0$ such that ${F}^{n}\left(U\right)\cap V\ne \varphi $ for all $n>N$ .

2) The set $A\subseteq C\left(X,Y\right)$ is called a weakly pc-mixing set of $\left(C\left(X,Y\right),F\right)$ if for any choice of nonempty pc-open subsets ${V}_{1},{V}_{2}$ of A and nonempty pc-open subsets ${U}_{1},{U}_{2}$ of $C\left(X,Y\right)$ with $A\cap {U}_{1}\ne \varphi $ and $A\cap {U}_{2}\ne \varphi $ there exists $n\in \text{N}$ such that ${F}^{n}\left({V}_{1}\right)\cap {U}_{1}\ne \varphi $ and ${F}^{n}\left({V}_{1}\right)\cap {U}_{2}\ne \varphi $ .

3) The set $A\subseteq C\left(X,Y\right)$ is strongly pc-mixing if for any pair of pc-open sets U and V with $U\cap A\ne \varphi $ and $V\cap A\ne \varphi $ , there exist some $n\in N$ such that ${F}^{k}\left(U\right)\cap V\ne \varphi $ for any $k\ge n$ .

4) Any element $f\in C\left(X,Y\right)$ such that its orbit ${O}_{F}\left(f\right)$ is pc-dense in X. is called hypercyclic element.

5) A system $\left(C\left(X,Y\right),F\right)$ is said to be topologically pc-mixing if, given pc-open sets U and V in $C\left(X,Y\right)$ , there exists an integer N, such that, for all $n>N$ , one has ${F}^{n}\left(U\right)\cap V\ne \varphi $ .

6) A system $\left(C\left(X,Y\right),F\right)$ is called topologically pc-mixing if for any non-empty pc-open set U, there exists $n\in \text{N}$ such that $\underset{n\ge N}{\cup}{F}^{n}\left(U\right)$ is pc-dense in $C\left(X,Y\right)$ .

3. Definitions and Theorems of Compact-Open Topology

The following definition supplies another version of a topology on the set $C\left(X,Y\right)$ .

Definition 3.1. Consider all possible sets of maps of the form [1]

$\left[K,U\right]=\left\{f\in C\left(X,Y\right):f\left(K\right)\subset U\right\}$

where K is a compact set in X and U an open set in Y. The topology ${\tau}_{3}$ generated by these sets $\left[K,U\right]$ as a subbase is called the compact-open topology on $C\left(X,Y\right)$ .

Note that any open set in ${\tau}_{3}$ is called co-open set and $\left(C\left(X,Y\right),{\tau}_{3}\right)$ is called co-topological space. The compliment of co-open set is called co-closed set.

Definition 3.2. Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a co-topological space. The map $F:C\left(X,Y\right)\to C\left(X,Y\right)$ is called co-irresolute if for every subset $A\in {\tau}_{3}$ , ${F}^{-1}\left(A\right)\in {\tau}_{3}$ . or, equivalently, F is co-irresolute if and only if for every co-closed set A, ${F}^{-1}\left(A\right)$ is co-closed set.

Definition 3.3. A map $F:C\left(X,Y\right)\to C\left(X,Y\right)$ is cor-homeomorphism if it is bijective and thus invertible and both F and ${F}^{-1}$ are co-irresolute.

The systems $F:C\left(X,X\right)\to C\left(X,X\right)$ and $G:C\left(Y,Y\right)\to C\left(Y,Y\right)$ are topologically cor-conjugate if there is a cor-homeomorphism $H:C\left(X,X\right)\to C\left(Y,Y\right)$ such that $H\circ F=G\circ H$ .

Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a co-topological space. The intersection of all co-closed sets of $\left(C\left(X,Y\right),{\tau}_{3}\right)$ containing A is called the co-closure of A and is denoted by $C{l}_{co}\left(A\right)$ .

Definition 3.4. Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a compact-open topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a map. The map F is said to have co-dense orbit if there exists $f\in C\left(X,Y\right)$ such that $C{l}_{co}\left({O}_{F}\left(f\right)\right)=C\left(X,Y\right)$ .

Definition 3.5. Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a co-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a co-irresolute map, then F is said to be a compact-open-transitive ( shortly co-transitive) map if for every pair of co-open sets U and V in $\left(C\left(X,Y\right),{\tau}_{3}\right)$ there is a positive integer n such that ${F}^{n}\left(U\right)\cap V$ is not empty.

Definition 3.6. Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a co-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a co-irresolute then the set $A\subseteq C\left(X,Y\right)$ is called co-type transitive set if for every pair of non-empty co-open sets U and V in $C\left(X,Y\right)$ with $A\cap U\ne \varphi $ and $A\cap V\ne \varphi $ there is a positive integer n such that ${F}^{n}\left(U\right)\cap V\ne \varphi $ .

Definition 3.7. 1) Let $\left(C\left(X,Y\right),{\tau}_{3}\right)$ be a co-topological space, and $F:C\left(X,Y\right)\to C\left(X,Y\right)$ be a co-irresolute then the set $A\subseteq C\left(X,Y\right)$ is called is called topologically co-mixing set if, given any nonempty co-open subsets $U,V\subseteq C\left(X,Y\right)$ with $A\cap U\ne \varphi $ and $A\cap V\ne \varphi $ then $\exists N>0$ such that ${F}^{n}\left(U\right)\cap V\ne \varphi $ for all $n>N$ .

2) The set $A\subseteq C\left(X,Y\right)$ is called a weakly co-mixing set of $\left(C\left(X,Y\right),F\right)$ if for any choice of nonempty co-open subsets ${V}_{1},{V}_{2}$ of A and nonempty co-open subsets ${U}_{1},{U}_{2}$ of $C\left(X,Y\right)$ with $A\cap {U}_{1}\ne \varphi $ and $A\cap {U}_{2}\ne \varphi $ there exists $n\in \text{N}$ such that ${F}^{n}\left({V}_{1}\right)\cap {U}_{1}\ne \varphi $ and ${F}^{n}\left({V}_{1}\right)\cap {U}_{2}\ne \varphi $ .

3) The set $A\subseteq C\left(X,Y\right)$ is strongly co-mixing if for any pair of co-open sets U and V with $U\cap A\ne \varphi $ and $V\cap A\ne \varphi $ , there exist some $n\in N$ such that ${F}^{k}\left(U\right)\cap V\ne \varphi $ for any $k\ge n$ .

4) A system $\left(C\left(X,Y\right),F\right)$ is said to be topologically co-mixing if, given co-open sets U and V in $C\left(X,Y\right)$ , there exists an integer N, such that, for all $n>N$ , one has ${F}^{n}\left(U\right)\cap V\ne \varphi $ . For related works about weakly mixing see [2] , [3] and [4] .

4. Conclusions

We have the following results:

1) Every compact-open-transitive set implies point wise convergence set but not conversely.

2) Every compact-open-mixing system implies point wise convergence system but not conversely.

3) Every strongly compact-open-mixing set implies strongly point wise convergence mixing set.

Acknowledgements

First, thanks to my family for having the patience with me for having taking yet another challenge which decreases the amount of time I can spend with them. Specially, my wife who has taken a big part of that sacrifices, also, my son Sarmad who helps me for typing my research. Thanks to all my colleagues for helping me for completing my research.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

References

[1] Borsovich, Y.U., Blizntakov, N., Izrailevich, Y.A. and Fomenko, T. (1985) Introduction to Topology. Mir Publisher, Mos-cow.

[2] Chacon, R.V. (1969) Weakly Mixing Transformations Which Are Not Strongly Mixing. Proceedings of the American Mathematical Society, 22, 559-562.

https://doi.org/10.1090/S0002-9939-1969-0247028-5

[3] Kaki, M.N.M. (2015) Chaos: Exact, Mixing and Weakly Mixing Maps. Pure and Applied Mathematics Journal, 4, 39-42.

https://doi.org/10.11648/j.pamj.20150402.11

[4] Kaki, M.N.M. (2013) Introduction to Weakly b-Transitive Maps on Topo-logical Space. Science Research, 1, 59-62.

https://doi.org/10.11648/j.sr.20130104.11