I am working on algebraic topology.
I want to prove that for any $\Omega$, $\Lambda$ arbitrary path-connected topological spaces, $\pi_1(\Omega \vee \Lambda) = \pi_1(\Omega) \ast \pi_1(\Lambda)$, using Seifert - Van Kampen theorem and deform retract properties.
I do not understand how to construct a proper "open neighborhood" of $\Omega$ and $\Lambda$ (needed in the said theorem) that would be adapted for a deform retract.
I read Hatcher's proof in Algebraic Topology, yet I did not understand it.
How can I proceed ? Could you please provide me with a sketch of the proof ?
Thank you very much, Respectfully,
A.F