Path-connected set

A path-connected set is a subset $A\subseteq X$ of a topological space $X$ such that for any $x,y\in A$ there exists a path $\gamma\colon [0,1]\to X$ with $\gamma(0)=x$, $\gamma(1)=y$, and $\gamma([0,1])\subseteq A$.

Path-connectedness is stronger than connectedness (every path-connected set is connected), and it leads to a decomposition of spaces into “path components,” analogous to connected components .

Examples:

• Any interval in $\mathbb{R}$ is path-connected, and more generally any convex subset of $\mathbb{R}^n$ is path-connected.
• The set $(-1,0)\cup(0,1)\subseteq\mathbb{R}$ is not path-connected.