A Study On Discretely Generated Spaces,Baire Properties And Topologicasl Dynamics

In this thesis, we mainly study problems on properties of discretely gener-ated spaces and topological transitive property in topological dynamics system. In the first part of this thesis, the properties in producs of (weakly) discretely generated spaces are investigated. By these properties, we prove some conditions by which a Baire spaec is equivalent to a Volterra space. In the last part of this thesis, topological transitive properties in functional spaces are discussed, and the characterzations of Li-Yorke chaos in Co-semigroups are considered.Firstly, the properties in producs of (weakly) discretely generated spaces are investigated. We show that the product of a regular nested space with a discretely generated space is discretely generated. We also show that the product of a gen-eralized ordered topological space with a discretely generated space is discretely generated. Some other properties in producs of weakly discretely generated spaces are discussed. We get the following conclusion. Let X be a locally compact gener-alized ordered topological space and let Y be a weakly discretely generated space. If there exists a closed discrete subset F of X such that for any point x ∈ X\F, there is a neighborhood Vx of x in X such that Vx x Y is weakly discretely gen-erated, then X x Y is weakly discretely generated. We finally prove that if X is a locally compact generalized ordered topological space, IX is a linearly ordered compactification of X and Y is a weakly discretely generated space such that X x Y is weakly discretely generated and lX\X is scattered, then IX x Y is weakly discretely generated.Secondly, some conditions by which a Baire space is equivalent to a Volterra space are discussed. An example is given to show that in a topological space X, the sequential closure of a set may not be a sequential subspace of X. Then we point out that there is a gap in a conclusion in a paper of Gruenhage and Lutzer. We finally show that the regular property which appears in some results of Gruenhage and Lutzer’s article can be replaced by Hausdorff. By the properties of discretely generated spaces, we show that if X is a monotonically normal T1-space with countable pseudocharacter and has a σ-discrete dense subspace D of X, then X is a Baire space if and only if X is Volterra. We also get that if X be a metacompact normal sequential T1-space and has a σ-closed discrete dense subspace, then X is a Baire space if and only if X is Volterra.Finally, topologically transitive properties of composition operators in CP(X) and in Ck(X) are investigated. We get the following theorem. Let(X, d) be a countable metric space. Suppose that G is a semigroup of continuous self maps of X with the following properties:(1) Every element of G is one-to-one on X. (2) The action of G is strongly runs away on X. Let G be the semigroup of composition operators induced by the elements of G, then the action of G on CP(X) is topologically transitive and hypercyclic. We also show that the action of G on Cκ(R\Z) is hypercyclic, where G is the set of one-to-one and continuous self maps of R\Z.Some topologically dynamical properties in Co-semigroups are discussed. We prove that if a Co-semigroup is hereditarily hypercyclic with respect to a syndetic sequence then it is mixing. We give the definitions of irregular vectors and Li-Yorke Chaos Criterion for Co-semigroups. We finally get characterizations of Li-Yorke chaos in Co-semigroups.