Definition:Path-Connected/Topology/Points
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Definition
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 $T$.
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 sets can be found here.
Sources
- 2003: John M. Lee: Introduction to Smooth Manifolds: $\S 1.1$: Smooth Manifolds. Topological Manifolds