Definition:Path-Connected

Definition

Points in Topological Space

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

Let $a, b \in X$ be such that there exists a path from $a$ to $b$.

That is, there exists a continuous mapping $f: \left[{0 . . 1}\right] \to X$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

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

Subsets of Topological Space

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

Let $U \subseteq X$ be a subset of $T$.

Let $T \ ' = \left({U, \vartheta_U}\right)$ be the subspace of $T$ induced by $U$.

Then $U$ is path-connected (or path connected) in $T$ iff every two points in $U$ are path-connected in $T \ '$.

That is, $U$ is path-connected if:

for every $x, y \in U, \exists$ 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 $T = \left({X, \vartheta}\right)$ be a topological space.

Then $T$ is path-connected (or path connected) iff $X$ is path-connected in $T$.

That is, $T$ is path-connected if:

for every $x, y \in X, \exists$ a continuous mapping $f: \left[{0 . . 1}\right] \to X$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.

Also see

• Results about path-connectedness can be found here.

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{III}$
• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 4$