I began reading this paper (also provided below), and I would really appreciate if someone could help me with the premises behind the first sentence. I have not been able to find the given long exact sequence in my books, and I don't want to wrongly start generalizing it.
This is the sequence when $U$ is an open subspace of X:
$$\cdots \to H^n_c(U) \to H^n_c(X) \to H^n_c(X-U) \to H^{n+1}_c(U) \to \cdots$$
First Attempt of an Example
The first example I tried was $X=\mathbb{R}^2$ with $(X-U)$ a closed disk in the plane. I then made the compact exhaustion of $U$ to be 2-dimensional rings around $U$ that grow in size to cover $U$. The 2-rings get larger and larger and also closer and closer to the closed disk. Then each 2-ring quotiented by the rest of $U$ is a torus. That is a 2-ring quotiented by the inner and outer ring, which is a 1-sphere. So $H_c^n(U)$ is equivalent to the cohomology of a 2-sphere, which I think is R, 0, R for n=0, 1, 2 resp.
Second Attempt of an Example
Next I wanted to try an example where neither $U$ nor $X-U$ are compact. So I let $X$ be an infinite space that is made up of infinitely many copies of $\mathbb{R}^2$ branching off each other. I took $U$ to be the infinite open subset that is made up of infinitely many copies of $\mathbb{R}^2 - S$ where $S$ is a 2-simplex that is a square. Then $X-U$ is a connected space that is made up of infinitely many copies of a square. There are infinitely many intersecting planes in this example, but $U$ is open, although infinite.
Long story short, the exact sequence given in the paper says that $H^n_c(X) = H^n_c(X-U)$ if $H^n_c(U)$ is trivial for all $n$, but I don't want to work out more examples before I know all the conditions on $X$ and $U$. Which leads to:
Question:
Where is the error in my reasoning in the above two examples?
Does this long exact sequence hold when $U$ and its complement are unbounded?
Is the definition of an "open subspace" of a CW complex really more strict than I am making it? I am using the definiton given here.
(I will keep looking for the answers to these questions myself and update when I find something, but help from a mathematician would be awesome.)
Update: The answer written by @Hanspeter Kraft here reiterates that the long exact sequence holds for CW complexes $X$, but there is no discussion about the open and closed subspaces.
