Path-Connected Set

A subset $S$ of a topological space is path-connected if for every pair of points $x,y\in S$, there exists a continuous path $\gamma :[0,1]\to S$ such that $\gamma (0)=x$ and $\gamma (1)=y$.

Key implications:

• Path-connected $⟹$ connected.
• The converse is false (e.g., the topologist's sine curve is connected but not path-connected).

Special case in $\mathbb{C}$:
Every open connected subset of $\mathbb{C}$ (or ${\mathbb{R}}^2$) is path-connected.