While working on some problem (not relevant here), it turned out to be convenient to be able to enclose arbitrary compact subsets in "nicer" compact subsets, hence the question:



> 
> if $(M,g)$ is a Riemannian manifold and $K \subseteq M$ is compact, does there exist a connected open subset $U \subseteq M$ such that $K \subseteq U$ and $\overline U$ is compact?
> 
> 
> 


If $M$ is complete, then enclosing $K$ in a large enough ball will do, using the Hopf-Rinow theorem. If $M$ is not complete, then I have been able to construct either a relatively compact but not connected $U$, or a connected but not relatively compact one. Is it possible to get both properties?



Riemannian manifolds: every compact subset is contained in a connected relatively compact open subset
You will need to assume $K$ lies in a connected component of $M$.


Then for every point $x\in K$ there is a small open balls in $M$ whose closure is compact (you can just do this in charts). Since $K$ is compact it is covered by finitely many of such balls. Now choose a path that goes through these (finitely many) open balls. 


Take an open subset $V$ containing this path whose closure is compact. Then define $U$ to be the union of $V$ with the finitely many open balls constructed before. 


