I am trying to understand proof of excision theorem in homology following Hatcher's algebraic topology. I came to the following simple question, which I was unable to understand completely.
For a topological space $X$, let $C_n(X)$ denotes the free abelian group on singular $n$-simplices $\sigma:\Delta^n\rightarrow X$ (continuous). Let $A,B$ be subspaces of $X$ such that their interiors cover $X$.
Q.1 Is it true that $C_n(A\cap B)=C_n(A)\cap C_n(B)$ when considered them as subgroups of $C_n(X)$?
According to Hatcher's book, let $C_n(A+B)$ denotes the sums of chains in $A$ and chains in $B$. The second simple question is
Q.2 Does it mean that $C_n(A+B)=C_n(A)+C_n(B)$ where $C_n(A),C_n(B)$ are considered as subgroups of $C_n(X)$?
I came to these questions, because I didn't understand the following statement in Hatcher:
The map $C_n(B)/C_n(A\cap B)\rightarrow C_n(A+B)/C_n(A)$ induced by inclusion is obviously an isomorphism because both quotient groups are free with basis the singular $n$-simplices in $B$ that do not lie in $A$.
(If the answer to both questionsi s yes, then quoted statement is just third isomorphism theorem; but I was wondering whether this justification is correct?)