Definition:Arc (Topology)
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Definition
Let $T$ be a topological space.
Let $\mathbb I \subset \R$ be the closed unit interval $\closedint 0 1$.
Let $a, b \in T$.
An arc from $a$ to $b$ is a path $f: \mathbb I \to T$ such that $f$ is injective.
That is, an arc from $a$ to $b$ is a continuous injection $f: \mathbb I \to T$ such that $\map f 0 = a$ and $\map f 1 = b$.
The mapping $f$ can be described as an arc (in $T$) joining $a$ and $b$.
Also known as
An arc is also known as an open curve.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arc: 1 c.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): open curve (arc)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): open curve (arc)