# Definition:Path-Connected

## Topology

### 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:
$\map f 1 = y$

## Metric Space

Let $M = \struct {A, d}$ be a metric space.

$M$ is defined as path-connected if and only if:

$\forall m, n \in A: \exists f: \closedint 0 1 \to A: \map f 0 = m, \map f 1 = n$

where $f$ is a continuous mapping.

### Subset of Metric Space

Let $M = \struct {A, d}$ be a metric space.

Let $S \subseteq A$ be a subset of $M$.

Then $S$ is path-connected (in $M$) if and only if:

$\forall m, n \in S: \exists f: \closedint 0 1 \to S: \map f 0 = m, \map f 1 = n$

where $f$ is a continuous mapping.

## Also known as

Some sources do not hyphenate path-connected, but instead report this as path connected.

Some sources use the term arc-connected, but this normally has a more precise meaning.

## Also see

• Results about path-connected spaces can be found here.