Connectedness

Thus far, we have introduced the notion of topological spaces, with different types of constructions, and the mappings between topological spaces. With the concept of homeomorphism and topological invariant, we have ahead of us the task of classifying the spaces. In this chapter, we study one of the most important invariants in topology, connectedness. A space is said to be connected if there are no non-trivial partitions by open sets. Intuitively, a connected space is built up by a unique piece. The simplicity of this idea does not reduce its importance and its power appears in many other fields of mathematics. An illustrative example is Bolzano’s theorem that allows us to establish the existence of solutions of equations when the functions involving these equations are continuous. Also, connectedness is employed as a powerful tool in the classification of topological spaces. In this chapter, the concepts of connected components and path connectedness are also studied.