A space is connected if it does not consist of two separate pieces. This connectedness provides important consequence in topology and has let to the techniques for distinguishing between spaces.
In Ⅱ, we give the definitions and study the basic pro...
A space is connected if it does not consist of two separate pieces. This connectedness provides important consequence in topology and has let to the techniques for distinguishing between spaces.
In Ⅱ, we give the definitions and study the basic properties of connected space.
In Ⅲ, we study the basic properties of path connected space, and the relation between connected space and path connected space.
In Ⅳ, we deal with the application of connectedness. Connectedness provides a proof of the intermediate-value theorem used so often in calculus.
It also allows us to distinguish topologically between a circle and a n-sphere, and between the real line and the higher dimensional Euclidean spaces.