# Definition:Arc (Topology)

Jump to navigation
Jump to search

## 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 - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**open curve (arc)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**open curve (arc)**