R-Connected Spaces And γI-Open Sets In Ideal Topological Spaces

Connected space is an important topological space.Studying connectedness is the basic issue of general topology.Ideal topological space is a new topological space , which introduces an ideal. And ideal topological space has the similar properties to general topological space, but it also has the unique properties.We have studied the properties of R -connected space with binary relation R andγI -open sets in ideal topological space in this paper.This paper is parted into three chapters.In the first chapter, we introduced the background knowledge related to the re-search.In the second chapter, we firstly define R -open sets with the binary relation R in topological space, then concepts of R -neighborhood, R -closure, R -interior and R -separation etc. are introduced by R -open sets theory. On the basis of these con-cepts, R-connected spaces are defined and the characteristics of R-connected spaces are given. Its properties are researched at the same time. Therefore, some mainly interesting results are obtained, which can be seen in Proposition 2.4.4, Proposition 2.4.5, Theorem 2.4.9, Theorem 2.4.10, Theorem 2.4.11 and Theorem 2.4.12.In the third chapter, we defined theγI -open set in ideal topological space, studying its properties and the related properties with pre - I -open set, semi - I -open set, stronglyβ- I-open set andα- I -open set in --extremely dis-connected ideal topological spaces. Thus we obtained main results seen in Proposition 3.2.7, Proposition 3.3.3, Proposition 3.3.8.