Hausdorff, locally connected and locally compact space exercise Let $X$ be a Hausdorff, locally connected and locally compact space. Let $U$ be a connected subset of $X$ and let $x,y \in U$. Prove there exists a compact connected subset $T$ of $U$ such that $T$ contains both $x,y$.
Well I tried using regularity and local compactness but didn't get far. Any ideas?
 A: Here is a more elementary idea: For $x\in U$ let $V_x$ be the set of all $z\in U$ such that there exists a compact connected subset of $U$ that contains $x$ and $z$. Use the hypotheses to show that $V_x$ is both open and closed in $U$, hence is all of $U$.
A: In my answer here I gave a lemma that characterizes connectedness in terms of chains in (not necessarily open) covers. 
We apply this to $U$, as we pick for each $x$ in $U$ a compact and connected neighborhood that   is a subset of $U$, which can be done by first picking a compact neighborhood inside $U$ (using Hausdorffness and local compactness) and then a connected neighborhood inside that (using local connectedness) and we take the closure (still connected) of that.
Now for every $x$ and $y$ in $U$ we have a chain (as defined in that answer) connecting $x$ to $y$ and we take the union of the chain to get a compact and connected (due to the non-empty intersections) set containing $x$ and $y$.
This exercise is a typical application of the chain-characterization of connectedness.
