The boundaries of a chain of a compact set is in the compliment of that set?

Question

I confronted with a statement that if $z$ represents a chain in $H_n(X|A)$, where $A$ is a compact set of a given manifold $X$. Then $\partial z$ is a chain in $X-A$.

I am not sure why it’s true, but it seems can be explained by excision. Hope someone could help. Thanks!

Answer 1

If the boundary of z were not in $X-A$, then z is not in the kernel of the relative boundary operator since $\partial z$ is not zero in the relative homology, by definition.