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I studied the questions Fundamental group of a wedge sum and Under what conditions $\pi_1(X \vee Y) \approx \pi_1(X) * \pi_1(Y)$? Both use the van Kampen theorem to prove that under certain assumptions

$$\pi_1(X \vee Y) \approx \pi_1(X) * \pi_1(Y) . \tag{+}$$ In the first question it is claimed that this is true

if we require the wedge point $p∨q$ to have a simply-connected neighborhood in $X∨Y$.

In the second question it is claimed that this is true

if there are open neighborhoods $U_p$ of $p$ in $X$ and $U_q$ of $q$ in $Y$ which are pointed contractible to $p$ and $q$.

I can see that the second assumption allows to prove (+). Define $U = X \vee U_q$ and $V = U_p \vee Y$. Then $U, V$ are open and path connected subsets of $X \vee Y$ which cover $X \vee Y$ and have contractible intersection. Also $U$ and $V$ are pointed homotopy equivalent to $X$ and $Y$. Therefore van Kampen can be used to prove (+).

But I do not see why the first assumption is sufficient to prove (+).

Can anybody explain it?

Kritiker der Elche
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    We could approach this similarly by taking the simply connected neighbourhood $U$ of $p\vee q$ and defining $U_1=U\cup X$ and $U_2=U\cup Y$. Most of the stuff remains true, except I don't know how to proove that $\pi_1(X)\simeq \pi_1(U_1)$. The obvious candidate, i.e. the homomorphism induced by the inclusion $i:X\to U_1$ is surjective, but I don't know if it is injective. In particular this would require the capability of contracting not only loops but also homotopies (e.g. $\pi_2(U)=0$?). It sounds like some pathology can happen here. Honestly, I don't think that the first claim implies (+). – freakish Jan 25 '23 at 11:24

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