Connectedness is connected union-closed

Version using a pivoting set
Suppose $$X$$ is a topological space. Suppose $$A$$ is a subset of $$X$$ and $$B_i, i \in I$$ is a collection of subsets of $$X$$. Suppose that:


 * 1) $$A$$ is a connected space in the subspace topology.
 * 2) For each $$i \in I$$, $$B_i$$ is a connected space in the subspace topology.
 * 3) $$A \cap B_i$$ is non-empty for each $$i \in I$$.

Then, the space:

$$A \cup \bigcup_{i \in I} B_i$$

is a connected space in the subspace topology from $$X$$.