# Definition:Path-Connected/Topology/Topological Space

< Definition:Path-Connected | Topology(Redirected from Definition:Path-Connected Space)

## Contents

## Definition

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

## 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.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{I}: \ \S 4$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.3$: Path-connectedness: Definition $6.3.2$