Union of Simply Connected Sets with Path-Connected Intersection is Simply Connected

Theorem

Let $\struct {T, \tau}$ be a topological space.

Consider all subsets of $T$ as subspaces, equipped with the subspace topology induced by $\tau$.

Let $U$ and $V$ be open subsets of $T$ that are simply connected.

Let $U \cap V$ be non-empty and path-connected.

Let $U \cup V = T$.

Then $\struct {T, \tau}$ is simply connected.

Proof

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Sources

• 2000: James R. Munkres: Topology (2nd ed.): $9$: The Fundamental Group: $\S 59$: The Fundamental Group of $S^n$