The notion of connectedness corresponds roughly to the everyday idea of an object being in one piece. One condition when we intuitively think a space to be all in one piece is if it does not have disjoint “parts”. Another condition when we would like to call a space one piece is that one can move in the space from any one point to any other point. These simple ideas have had important consequences in topology and its applications to analysis and geometry. Of course, we need to formulate the above ideas mathematically. The first idea is used in Sect. 3.1 to give a precise definition of connectedness and, using the second idea, we make a slightly different formulation, path-connectedness, discussed in Sect. 3.3. If a space X is not connected, then the knowledge of its maximal connected subspaces becomes useful for description of the structure of the topology of X. These “pieces” of the space X are considered in Sect. 3.2. As we will see later, it is important for some purposes that the space satisfy one of the two connectedness conditions locally. Localized versions of both conditions are formulated and studied in Sect. 3.4.