#### Simply connected

Now that the notion of connectedness has been established, the next step is to express different kinds of connectivity. This may be done by using the notion of homotopy, which can intuitively be considered as a way to continuously warp'' or morph'' one path into another, as depicted in Figure 4.6a.

Two paths and are called homotopic (with endpoints fixed) if there exists a continuous function for which the following four conditions are met:

It is straightforward to show that homotopy defines an equivalence relation on the set of all paths from some to some . The resulting notion of equivalent paths'' appears frequently in motion planning, control theory, and many other contexts. Suppose that is path connected. If all paths fall into the same equivalence class, then is called simply connected; otherwise, is called multiply connected.