Definition:Arc-Connected/Topological Space
Jump to navigation
Jump to search
Definition
Let $T = \struct {S, \tau}$ be a topological space.
Then $T$ is arc-connected if and only if every two points in $T$ are arc-connected in $T$.
That is, $T$ is arc-connected if and only if:
- $\forall x, y \in S: \exists$ a continuous injection $f: \closedint 0 1 \to X$ such that $\map f 0 = x$ and $\map f 1 = y$.
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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): arc connected