Definition:Path-Connected/Topology/Topological Space
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Definition
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.
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $6.3$: Path-connectedness: Definition $6.3.2$
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $4$: Connectedness
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): connected space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): connected space
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): path connected