In the EOM entry for nowhere dense set it is stated that
In an infinite-dimensional Hilbert space, every compact subset is nowhere dense. The same holds for infinite-dimensional Banach spaces, non-locally-compact Hausdorff topological groups, and products of infinitely many non-compact Hausdorff topological spaces.
First of all, some generalizations are possible:
- For an infinite-dimensional Banach space, even relatively compact subsets are nowhere dense (see for example here, the proof of the accepted answer also works in a general infinite-dimensional Banach space)
- The statement regarding infinite products also holds without the Hausdorff assumption (see for example Theorem 16 in Kelley).
My question is about the remaining statement about topological groups, which I'm not very familiar with:
Could anybody please provide a proof? Is the Hausdorff assumption necessary here and can compactness be weakened to relative compactness?
In other words, I'm looking for the weakest condition on a topological group such that every (relatively) compact subset is nowhere dense
Edit: The proof of the accepted answer below (with a slight and straightforward modification*) shows that every relatively compact subset of a non-locally-compact Hausdorff topological group is nowhere dense.
- A relatively compact subset which is not nowhere dense has an interior point, of which the closure of the subset is a compact neighbourhood.
Best Answer
Suppose a compact set $K$ is not nowhere dense. Then it has an interior point say $g$. Note that $K$ is a compact neighborhood of $g$. For any other point $h$ apply the homeomorphism $x \to hg^{-1}x$ to see that $h$ also has compact neighborhood. Hence the space is locally compact.