Concepts of Modern Mathematics by Ian Stewart

Author:Ian Stewart [Stewart, Ian]
Language: eng
Format: epub, pdf
ISBN: 9780486134956
Publisher: Dover Publications
Published: 2012-06-05T16:00:00+00:00

Figure 133

This is important: if we want to use paths to detect holes, the way we go round the path affects the way the path wraps around the hole. In Figure 134 one path can be ‘pulled away from’ the hole, and the other one can’t.

Figure 134

The easiest way to specify how to travel along a path is to imagine a point moving along it. At time t this point will be in position p(t). It starts at some time to and ends at some time t1. Since the path has no breaks, this means that p specifies a continuous function whose domain is the set of real numbers x in the interval t0 ≤ x ≤ t1, and whose target is the given topological space. Each such function defines a path, and each path defines such a function.

If one path ends where another begins, we can compose them by travelling along the first and then along the second as in Figure 135.

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