This exercise is from "General Topology" by Engelking:
Give an example of a metrizable space which cannot be embedded in a locally compact metrizable space.
I don't how to start. The counterexample cannot be $\mathbb R$ since it is locally compact. Also it cannot be a discrete space since it is locally compact.
Thanks for your help.
Take any metric space $X$ that has a point $p$ such that no neighbourhood of $p$ has countable weight. E.g. the hedgehog space with $\aleph_1$ many spikes will do (as the open ball neighbourhoods of the "fusion point" are all homeomorphic to the whole space).
If we could see $X$ as a subspace of $Y$ which is locally compact, then $p$ has a compact neighbourhood $C$ in $Y$, which must be second countable (as a metrisable compact space) and must contain a neighbourhood of $p$ in $X$ as subspace, which would give $p$ a second countable neighbourhood.
So every subspace of a locally compact metrisable space is locally second countable, is what we really use. So we just need a metrisable space that doesn't have this property.