The Nature Of The Product And Soft Topological Space Topological Space Of Continuous Self-maps Of Topological Entropy

To solve problems of classical set and uncertainty set, the concept of soft set was introduced by the Russian scholar Molodtsov in1999. Then the theory of soft set has attracted attention by many mathematicians and logicians. In decade years, a great deal of new ideas and applications of soft set have been proposed. The research of these subjects has related to several area such as BCK/BCI-algebra, linear logic, the ideal theory of rings and theoretic computer science. In2011, Shabir and Na constructed a soft topological space, which contains soft sets as its objects. Based on this, we will define the concept of product soft topological space and study further fundamental properties of product soft topological spaces in this article.The topological entropy is an important concept in the theory of topological dy-namical system. In1965, Adler, Konheim and McAndrew introduced the concept of topological entropy of continuous self-mappings defined on compact spaces. In1971, Bowen generalized the concept of topological entropy of continuous self-mappings defined on metric spaces and proved that the new definition coincides with that of Adler etc within the class of compact spaces. In2007, Liu Lei, Wang Yangeng and Wei Guo proposed a new definition of topological entropy of continuous self-mappings on arbitrary topological spaces and investigated fundamental properties of the new entropy. In this paper, properties of topological entropy of a continuous self-mapping on a topological space will be studied, and the problem whether the topological entropy of a continuous self-mapping on a topological space is invariant when the space becomes smaller or its topology becomes weaker will be discussed.The arrangement of this paper is as follows:Chapter one preliminaries. In this chapter, we give the basic concepts and results of the theory of soft set, soft topology and topological entropy which will be used in the whole paper.Chapter two product soft topological spaces and their properties. Firstly, the concepts of Cartesian product of the soft set, projective order-homomorphism, basis and subbasis of a soft topological space, are introduced. Based on these con-cepts, product soft topological space is defined, and characterizations of product soft topological spaces and basic properties of product soft topological spaces are given. Finally, It is proved that product soft topological space of a family of soft topological spaces with To-separation (resp.,T1-separation, T2-separation, regular separation, connectedness) still has this kind property, and second-countability is No-multiplicative property.Chapter three the invariance of topological entropies of continuous self-mappings on topological spaces. Relevant properties of the topological entropy of a continuous self-mapping on a topological space are discussed applying the theory of open cover, invariant subset and non-wandering set. Based on this, it is proved that if non-wandering set of the continuous self-mapping f on a topological space is compact, then the topological entropy of f is the same as that of the restriction of f to its non-wandering set, and the topological entropy of a continuous self-mapping on a strong θ-compact Hausdorff space (X, J) is the same as that of the mapping on (X,θ(J)), where θ(J) is the set of all θ-open sets in (X,J).