The set of reals is the most obvious example of a 1D manifold because certainly looks like (via homeomorphism) in the vicinity of every point. The range can be restricted to the unit interval to yield the manifold because they are homeomorphic (recall Example 4.5).

Another 1D manifold, which is not homeomorphic to , is a circle, . In this case , and let

If you are thinking like a topologist, it should appear that this particular circle is not important because there are numerous ways to define manifolds that are homeomorphic to . For any manifold that is homeomorphic to , we will sometimes say that the manifold

Steven M LaValle 2020-08-14