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