Connectedness is connected union-closed

This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) satisfying a topological space metaproperty (i.e., connected union-closed property of topological spaces)
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Statement

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 $X$.

Version using finite hopping

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