Arc-Connected Space is Path-Connected
Jump to navigation
Jump to search
Theorem
Let $T = \struct {S, \tau}$ be a topological space which is arc-connected.
Then $T$ is path-connected.
Proof
Let $T = \struct {S, \tau}$ be arc-connected.
Then $\forall x, y \in S$, there exists a continuous injection $f: \closedint 0 1 \to S$, such that $\map f 0 = x$ and $\map f 1 = y$.
As $f$ is a continuous injection, it is also simply a continuous mapping.
The result follows from the definition of path-connectedness.
$\blacksquare$
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