Definition:Path (Topology)/Also defined as

Path in context of Topology: Also defined as

The definition for path as given here is usually used in this form in complex analysis, where the details of the mapping itself tend to be important.

However, in topology it is often the case that all that is needed is merely to demonstrate the existence of such a mapping.

Thus it is usual in topology to specify $I \subset \R$ to be the closed unit interval $\closedint 0 1$, and to focus attention on the image of $I$.

From Closed Real Intervals are Homeomorphic the two definitions are seen to be equivalent.

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Path-Connectedness
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.3$: Path-connectedness: Definition $6.3.1$
• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): path: 2.