The problem that I'm trying to proof is: Show that $\bigvee_{i=1}^kS^n$ is homotopic to $S^n/(S^n\setminus \cup_{i=1}^k e^n_i)$, where $e^n_i$ are $n$-cells disjoint attached to the sphere $S^n$.
I can visualize that it is true by making some drawings, becouse all that will be left are just the $e^n_i$ and all the rest will be identified to one point, so I glue all but the $e^n_i$ into one point, so were the $e^n_i$ is attached become just one commom point (it's like if I contract the sphere to the center, and bring the $n$-cells together).
The problem I'm having (in most exercises like this one) is how to formalize this intuition, so I would like to know how to get this idea into a formal proof.
Thanks in advance.