# Connectedness is connected union-closed

From Topospaces

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 is a topological space. Suppose is a subset of and is a collection of subsets of . Suppose that:

- is a connected space in the subspace topology.
- For each , is a connected space in the subspace topology.
- is non-empty for each .

Then, the space:

is a connected space in the subspace topology from .

### Version using finite hopping

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