Definition:Path-Connected/Topology
Definition
Points in Topological Space
Let $T = \struct {S, \tau}$ be a topological space.
Let $a, b \in S$ be such that there exists a path from $a$ to $b$.
That is, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that:
- $\map f 0 = a$
and:
- $\map f 1 = b$
Then $a$ and $b$ are path-connected in $T$.
Set of Topological Space
Let $T = \struct {S, \tau}$ be a topological space.
Let $U \subseteq S$ be a subset of $S$.
Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.
Then $U$ is a path-connected set in $T$ if and only if every two points in $U$ are path-connected in $T\,'$.
That is, $U$ is a path-connected set in $T$ if and only if:
- for every $x, y \in U$, there exists a continuous mapping $f: \closedint 0 1 \to U$ 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 a path-connected space if and only if $S$ is a path-connected set of $T$.
That is, $T$ is a path-connected space if and only if:
- for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that:
- $\map f 0 = x$
- and:
- $\map f 1 = y$
Also known as
Some sources do not hyphenate path-connected, but instead report this as path connected.
Some sources use path-wise connected
Some sources use the term arc-connected or arc-wise connected, but this normally has a more precise meaning.
Also see
- Results about path-connected spaces can be found here.