A topological space is said to be connected if it cannot be represented as the union of two disjoint, nonempty, open sets. While this definition is rather elegant and general, if is connected, it does not imply that a path exists between any pair of points in thanks to crazy examples like the topologist's sine curve:
How can we avoid such problems? The standard way to fix this is to use the path definition directly in the definition of connectedness. A topological space is said to be path connected if for all , there exists a path such that and . It can be shown that if is path connected, then it is also connected in the sense defined previously.
Another way to fix it is to make restrictions on the kinds of topological spaces that will be considered. This approach will be taken here by assuming that all topological spaces are manifolds. In this case, no strange things like (4.8) can happen,4.7 and the definitions of connected and path connected coincide . Therefore, we will just say a space is connected. However, it is important to remember that this definition of connected is sometimes inadequate, and one should really say that is path connected.
Steven M LaValle 2020-08-14