Suppose \(B_\alpha\subseteq X\) are path connected for every \(\alpha\in I\) and there is a point \(p\in X\) such that \(p\in B_\alpha\) for every \(\alpha\in I\). Then \(\cup_{\alpha\in I} B_\alpha\) is path connected.
Proof
To construct a path between two points in \(\cup_{\alpha\in I} B_\alpha\) connect two paths going to the intersection point.