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I have a question on the wording of the Van Kampen Theorem in Hatcher's Algebraic Topology. Here's the theorem as written:

If $X$ is the union of path-connected open sets $A_\alpha$ each containing the basepoint $x_0 \in X$ and if each intersection $A_\alpha \cap A_\beta$ is path-connected, then the homomorphism $\Phi : *_\alpha \pi_1(A_\alpha) \to \pi_1(X)$ [induced by the homomorphisms induced by the inclusions $A_\alpha \hookrightarrow X$] is surjective. If in addition each intersection $A_\alpha \cap A_\beta \cap A_\gamma$ is path-connected, then the kernel of $\Phi$ is the normal subgroup $N$ generated by all elements of the form $i_{\alpha\beta}(\omega)i_{\beta\alpha}(\omega)^{-1}$ [where $i_{\alpha\beta}$ and $i_{\beta\alpha}$ are the homomorphisms induced by the inclusions $A_\alpha \cap A_\beta \hookrightarrow A_\alpha$ and $A_\alpha \cap A_\beta \hookrightarrow A_\beta$ respectively, and $\omega \in \pi_1(A_\alpha \cap A_\beta)$.]

When Hatcher says "the normal subgroup generated by elements of the form..." does he mean the smallest normal subgroup containing these elements, or does he mean the smallest ordinary subgroup containing these elements, which he claims is normal? I think it's the first, because I'm having trouble showing that the subgroup generated by those elements is normal, but I'm not totally sure.

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He means the smallest normal subgroup containing the elements. Typically, the subgroup they generate will not be normal. More generally, "the normal subgroup generated by..." usually means the smallest normal subgroup containing the elements, unless context makes it clear otherwise.

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