# Definition:Path-Connected/Topology/Points

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

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## Definition

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

Then $a$ and $b$ are **path-connected in $T$**.

## Also known as

Some sources do not hyphenate, but instead report this as **path connected**.

## Also see

- Path-Connectedness is Equivalence Relation
- Results about
**path-connected sets**can be found here.