Abstract: By means of the deep researches on basic concepts of point set topology and relationships among them, and the thought and method of dual category the concept of internal derived set was introduced in point set topology. Based on the concept of internal derived set, the concept of correlative subset was proposed, and its fundamental properties and relationships between it and other concepts were studied, and some good results were obtained. At last, as the application of internal derived set and correlative subset, several equivalent characterizations of disconnected space were given by using the properties of correlativetype disconnected space and disconnected space and relationships between them.

Key words: topology, derived set, internal derived set, correlativetype disconnected space, disconnected space