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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