I'm looking for an example of two locally compact subsets of the real line R, but their union isn't locally compact.
I know that generally it is not true that such union is locally compact, as we can take $\left \{ (x,y):x>0 \right \}\cup \left \{ (0,0) \right \}\ $, where the origin obviously does not have any compact neighborhood. But I don't see how it can happen in R. Thanks!
For $n\in\Bbb Z^+$ let $I_n=\left(\frac1{2n},\frac1{2n-1}\right)$, let $A=\bigcup_{n\ge 1}I_n$, and let $B=\{0\}$. Then $A$ and $B$ are locally compact, but $A\cup B$ is not locally compact at $0$. It’s really the same basic idea as your example in the plane.