Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set.



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> Is there a always a compact connected $L\subset X$ such that $K\subset L$?
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This is true if we assume local compactness: cover $K$ with a finite number of connected relatively compact open sets, take their closure, and then join with arcs. However, without local compactness I don't know what to do.



Is it possible to connect every compact set?
Choose a sequence $\varepsilon\_n\to 0$ and a $\varepsilon\_n$-net $N\_n$ for each $n$.
Assume $N\_0$ is a one-point set.
For each point in $x\in N\_k$ choose a closest point in $y\in N\_{k-1}$ and connect $x$ to $y$ by a curve.
Note that we can assume that diameter of the curve is at most $\delta\_k$ for a fixed sequence $\delta\_k\to 0$.


Consider the union $K'$ of all these curves with $K$; observe that $K'$ is compact and path connected.


