Definition:Arc-Connected/Points
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Definition
Let $T = \struct {S, \tau}$ be a topological space.
Let $a, b \in S$ be such that there exists an arc from $a$ to $b$.
That is, there exists a continuous injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.
Then $a$ and $b$ are arc-connected.
It is also declared that any point $a$ is arc-connected to itself.
Also known as
The term arc-connected can also be seen unhyphenated: arc connected.
Some sources also refer to this condition as:
but the extra syllable does not appear to add to the understanding.
Also see
- Results about arc-connected spaces can be found here.
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