| Sober space is the most important T0space in Non-Hausdorff topology.With the development of the Domain theory,well-filtered space and d-space were becoming the other two classical important spaces.For the study of the basic properties of those three kinds of spaces,such as heredity,product,retract,and reflection,has always been a basic problem concerned by Scholars.With the development of Non-Hausdorff topology,many strongly（weakly）sober spaces were introduced.In this dissertation,three kinds of spaces:k-bounded sober spaces、k-bounded well-filtered spaces and weakly bounded well-filtered spaces,are studiedIn Chapter 1,we introduce the background and the research status at domestic and abroad about Domain theory;and we also introduce the concepts of k-bounded sober spaces、k-bounded well-filtered spaces、weakly bounded well-filtered spaces、hereditary、retracts、Smyth power spaces and so on.In Chapter 2,we mainly investigate hereditary and retracts of the k-bounded sober spaces.Moreover we also investigate the function spaces and Smyth power constructions of k-bounded sober spaces.It is proved that k-bounded sober spaces are saturated-hereditary,but not closed-hereditary,and are not closed under retracts and Smyth power constructions.At the same time,an example is constructed to show that there exists a k-bounded sober space X for which the function space[X→X]pequipped with the topology of pointwise convergence is not k-bounded sober space.In Chapter 3,we show that k-bounded well-filtered spaces are saturated-hereditary but not closed-hereditary.Furthermore,we show that if the Smyth power spaces PS（X）is a k-bounded well-filtered space,then X is also k-bounded well-filtered.but the converse does not hold.In Chapter 4,we discuss that the weakly bounded well-filteredness is closed under closed heredity,retracts,and products.And if the Smyth power spaces PS（X）is a weakly bounded well-filtered space,then X is also weakly bounded well-filtered,but the converse is not valid. |
