It is well-known that the set IR of all closed intervals of the Euclidean space R with the reverse inclusion order is a domain and its maximal point space Max(IR) endowed with the relative Scott topology is homeomorphic to the Euclidean space R. These facts inspired the fruitful study of domain (or general dcpo) models of T1 spaces. The study of the connection between domain theory and relativity theory by Martin and Panangaden also leads to the consideration of the closed interval poset IP of a general dcpo P. In the current paper, we shall carry out a systematic study on several aspects of the general closed interval posets, including their maximal point spaces, some new topologies induced on posets and certain order properties.