Inverse limits of relative homology in Euclidean space

Question

Let $C$ be a compact set in Euclidean space and $i$ an integer. Is the inverse limit $$\underset{C\subset U}{\lim_\leftarrow}H_i(U,C)$$ over all open sets $U$ containing $C$ of relative homology groups always zero?

Answer 1

If you work with singular homology, then the answer is "no".

Let $C$ be the topologist's sine curve in $\mathbb{R}^2$. There is a cofinal inverse sequence $(U_n)$ in $\mathfrak{U}(C)$ such that each $U_n$ is homeomorphic to an open disk. Let $i_n : U_{n+1} \to U_n$ denote inclusion. Consider the long exact reduced homology sequence of the pair $(U_n,C)$. We have $H_1(U_n) = \tilde{H}_0(U_n) = 0$, hence we get an isomorphism $$\partial(n) =\partial : H_1(U_n,C) \to \tilde{H}_0(C) = \mathbb{Z} .$$ Concerning $\tilde{H}_0(C)$ recall that $C$ has two path components. By naturality we get $\partial(n) (i_n)_* = \partial(n+1)$ which shows $$\underset{C\subset U}{\lim_\leftarrow}H_1(U,C) \approx \underset{n\in \mathbb{N}}{\lim_\leftarrow}H_1(U_n,C) \approx \mathbb{Z} .$$