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:

- (
**Start with first path**) for all . - (
**End with second path**) for all . - (
**Hold starting point fixed**) for all . - (
**Hold ending point fixed**) for all .

During the warping process, the path image cannot make a discontinuous jump. In , this prevents it from moving over holes, such as the one shown in Figure 4.6b. The key to preventing homotopy from jumping over some holes is that must be continuous. In higher dimensions, however, there are many different kinds of holes. For the case of , for example, suppose the space is like a block of Swiss cheese that contains air bubbles. Homotopy can go around the air bubbles, but it cannot pass through a hole that is drilled through the entire block of cheese. Air bubbles and other kinds of holes that appear in higher dimensions can be characterized by generalizing homotopy to the warping of higher dimensional surfaces, as opposed to paths [439].

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*.

Steven M LaValle 2020-08-14