I was trying to come up with examples of metric spaces that are not locally compact, but could only come up with totally disconnected examples ($\mathbb{Q}$ and $\mathbb{R}_l$). Are there examples that are not totally disconnected?
Wikipedia tells me something about infinite-dimensional topological vector spaces not being locally compact; I don't have much topological intuition about these spaces, so any examples not of this type would be appreciated.
Some theorems on preservation of local compactness:
If $X$ is locally compact Hausdorff then $Y \subseteq X$ is locally compact in the subspace topology iff $Y = O \cap C$, where $O$ is open in $X$ and $C$ is closed in $X$. (This holds iff $Y$ is open in $\overline{Y}$.)
E.g. $\mathbb{Q} \times \mathbb{R}$ is not totally disconnected and not locally compact. Or $(\mathbb{R} \times (0,\infty)) \cup \{(0,0)\}$. Both are not open in their planar closures.
If $X_n$, $n \in \mathbb{N}$ are metric spaces, then $\prod_n X_n$ (in the product metric) is locally compact iff all $X_n$ are locally compact and all but finitely many of the $X_n$ are compact.
So $\mathbb{R}^\mathbb{N}$ in the product metric is an example (it's also a topological vector space, granted..). But more connected examples can be made this way, as soon as you have one such example, which we have from the subspaces.