Let $X=\{v\in\Bbb R^2:\|v\|<1\}\cup\{(-1,0)\}$, that is, the open unit disk union a point on its boundary. (Here, $(-1,0)$ refers to a point in the plane, not an interval.)
What is $H_c^•(X)$, that is, the cohomology of $X$ with compact supports?
I'm having trouble thinking about this since $X$ is not locally compact, so it's messing with my intuition. I feel like $\chi_c(X)$ should be $2$ (i.e. $\chi_c({\rm pt})+\chi_c({\rm disk})$), but I'm having trouble making that happen.
Counter-intuitively (at least to me), I think it vanishes. For each natural $n$, let $K_n$ be the union of the closed disk of radius $1-\frac{1}{n}$ about $(0,0)$ and a line connecting $(0,0)$ to $(-1,0)$. This is an increasing sequence of compact subspaces exhausting $X$, so we can compute $H_c^{\bullet}(X)=\varprojlim H^{\bullet}(X,X\setminus K_n)$, but $X$ and $X\setminus K_n$ are contractible spaces (the latter are all homeomorphic to open squares), so $(X,X\setminus K_n)$ are acyclic pairs by their LES (and $H^0$ vanishes as well), whence $H_c^{\bullet}(X)=0$.