Types of connectedness of the constructive real number intervals

We study different notions of connected constructive metric spaces. They differ the types of connected components and how different components relate to each other. These notions are equivalent in classical point set topology but they give differ in the constructive world. In particular the interval of constructive real number appears to be connected if we use some of the definitions of a connected space and it is not connected when we use other definitions. This study is the continuation of the previous work of the author inspired by the question of Andrej Bauer about properties of locally constant functions.