I would like to use the fact that if two path-connected pointed topological spaces $(X,p)$ and $(Y,q)$ admit two contractible open neighbourhoods of $p$ and $q$, then $$ \pi_1(X\vee Y) = \pi_1(X)*\pi_1(Y), $$ in order to prove that $\pi_1(S^1 \vee S^1)=\mathbb Z*\mathbb Z$. Unfortunately, I cannot find any contractible neighbourhoods of the base points $1\in S^1$. The thing is that any homotopy that deformation retracts to $1$ would work also for the whole $S^1$ which, I know, is not contractible.
Any help find such homotopy? Thank you.
Take $U = S^1 \vee S^1 \setminus \{p_1,p_2\}$, where $p_i$ is a point in the $i$-th circle which is distinct from the basepoint. Then $ U \approx (0,1) \vee (0,1)$ with basepoint $1/2 \in (0,1)$.