Let $A$, $B$ be abelian topological groups with a map $f :A \to B$. Assume also that the kernel and cokernel of this map are compact. Then we call f an isomorphism upto compactness.
Now let $A, B, C$ be abelian topological groups with maps $f :A \to B, g:B \to C$ both isomorphisms in the above sense. Then the claim is that $g \circ f$ is also such an isomorphism.
I would like a reference that proves this and ideally has some more discussion about this notion.