Definition:Path-Connected

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Definition

Topology

Points in Topological Space

Let $T = \left({S, \tau}\right)$ 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 $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

Then $a$ and $b$ are path-connected in $T$.


Set of Topological Space

Let $T = \left({S, \tau}\right)$ be a topological space.

Let $U \subseteq S$ be a subset of $S$.

Let $T' = \left({U, \tau_U}\right)$ 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 $f: \left[{0 \,.\,.\, 1}\right] \to U$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.


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 $M = \left({A, d}\right)$ be a metric space.

$M$ is defined as path-connected iff:

$\forall m, n \in A: \exists f: \left[{0 \,.\,.\, 1}\right] \to A: f \left({0}\right) = m, f \left({1}\right) = n$

where $f$ is a continuous mapping.


Subset of Metric Space

Let $M = \left({A, d}\right)$ be a metric space.

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


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

$\forall m, n \in S: \exists f: \left[{0 \,.\,.\, 1}\right] \to S: f \left({0}\right) = m, f \left({1}\right) = n$

where $f$ is a continuous mapping.


Also known as

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


Also see

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