Example of a subspace of the p-adic numbers which is not locally compact

Question

Let $p$ be a prime number, and let $\mathbf{Q}_p$ be the field of $p$-adic numbers together with its topology derived from the $p$-adic valuation. Can someone give me an example (together with an argument) of a subspace of $\mathbf{Q}_p$ which is not locally compact? Thank you in advance.

P.S. My motivation arises from the fact that given the field of real numbers $\mathbb{R}$ together with its euclidean topology, the subspace of rational numbers $\mathbb{Q}$ is not locally compact. Maybe the same happens if we complete $\mathbb{Q}$ with respect to the metric arising from the $p-$adic valuation?

Answer 1

A subspace $A\subseteq X$ of a locally compact Hausdorff space $X$ is locally compact if and only if it is locally closed. That is if and only if $A$ is open in its closure $\overline A$.

Since $\mathbb{Q}_p$ is Hausdorff and locally compact the statement applies. The subspace $\mathbb{Q}$ is dense in $\mathbb{Q}_p$, but not open. Hence it cannot be locally compact in the subspace topology.

Note that the same arguement shows that $\mathbb{Q}$ is not locally compact in the subspace topology inherited from $\mathbb{R}$.

Answer 2

The metric space $\Bbb{Z}$ (with the $p$-adic metric) is covered by the collection of open sets $$U_n = n+p^{|n|+3} \Bbb{Z},n\in\Bbb{Z}$$ but no finite subcollection $$\bigcup_{n=-N}^N U_n$$ covers it, because $U_n$ covers $p^{N-|n|}$ residue classes modulo $p^{N+3}$, so $\bigcup_{n=-N}^N U_n$ covers at most $\sum_{n=-N}^N p^{N-|n|}$ residue classes modulo $p^{N+3}$, ie. not them all.