Question about Chains complexes

Question

I have

$\mathcal{U}=\lbrace X-U, A\rbrace$ such that $\overline{U}\subset \overset{º}{A}$ and $X=\overset{º}{(X-U)}\cup \overset{º}{A}$ where $X$ is a topological space, $A$ is a subset of $X$ and $U\subset A$

Why $C_{\star}(\mathcal{U})=C_{\star}(X-U)+C_{\star}(A)$ ???

Please

Thank you

Answer 1

I am not sure of your definition of $C_*(\mathcal{U})$ but I think it means that any simplex must be in $A$ or $X-U$ due to restrictions on the diameter or because of barycentric division.