connected sum of two surfaces

Question

I was reading Massey's textbook on Algebraic topology and the author claims that if $S_2$ is a 2-sphere then $S_1 \# S_2$ is homeomorphic to $S_1$. I don't know why that is true and since I'm very much of a beginner in this field, I am looking more for an answer that provides intuition rather than a construction of the homeomorphism.

Answer 1

Think about what the connected sum is. You cut out a 2-disk from $S_1$. You cut out a 2-disk from $S_2$, and since $S_2$ is a sphere you're just left with a 2-disk!. Finally you glue the 2-disk you obtain in place of the 2-disk you cut out from $S_1$. In other words, you haven't done anything.

Answer 2

Try to think about the question again with the help of the picture below. In the case of the picture, $S_1$ is a torus.