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.