Definition:Arc-Connected
Definition
Points in Topological Space
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.
Subset of Topological Space
Let $T = \struct {S, \tau}$ be a topological space.
Let $U \subseteq S$ be a subset of $T$.
Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.
Then $U$ is arc-connected in $T$ if and only if every two points in $U$ are arc-connected in $T'$.
That is, $U$ is arc-connected if and only if:
- for every $x, y \in U$, there exists a continuous injection $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.
Topological Space
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
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arc-connected: 2.