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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?

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up vote 2 down vote accepted

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.

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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 $U$.

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The chain proof is exactly like that, and is more general. It's a useful generalization to know for these kinds of proofs. – Henno Brandsma Aug 7 '11 at 9:51

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