Path-connected set - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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 $\gamma([0,1])\subseteq A$ .

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.