The Functional Separation Properties Of Topological Systems,the Continuity Of Linear Order And The Kernels Of Domains

There are three parts in this paper. The first part of this paper has given the definitions of functional separation properties of topo-logical systems, which correspond to the separation properties of topological systems. We have also discussed the relations between the functional separation properties of topological systems and the separation properties of topological systems in this part (such as the proof of Urysohn Lemma under the circumstances of topological system). In the second part of this paper, we have discussed some properties of linear order on metric spaces, given the sufficient and necessary conditions of the continuity of linear order and answered the questions about the continuity of linear order on metric spaces given by Keye Martin in 2000. Then, we made some improvements to the linear order and thus made the metric spaces have even better properties under the new order. In the last part, we have discussed two questions about the kernels of Domains given by Keye Martin. We have provided several kinds of situations under which the first question will hold, given the negative answer to the second question and put forward some new questions for further studies.