Let $(X,\tau)$ be a [topological space](/page/Topological%20Space), and let $A,B \subset X$ be path-connected subspaces with the subspace topologies inherited from $X$. If $A \cap B \neq \varnothing$, then $A \cup B$ is path-connected with the [subspace topology](/page/Subspace%20Topology) inherited from $X$.