I've been thinking about automorphism groups of locally compact abelian groups somewhat over the last few days. It's not hard to see that in the case of the torus and integers, the (continuous) automorphisms of $(\Bbb T,\cdot)$ and $(\Bbb Z,+)$ are nothing more than the trivial automorphism and the inversion automorphism ($g\mapsto g^{-1}$). Thus the automorphism group of these is nothing more than $\Bbb Z/2\Bbb Z$. In the case of $(\Bbb R,+)$, the continuous automorphisms are nothing more than multiplication by a nonzero real number, i.e. for fixed $y$, $x\mapsto xy$ is an automorphism. In this case, the automorphism group is nothing more than $(\Bbb R^{\times},\cdot)$. In these cases, the automorphism groups are very nice objects (from a topological group standpoint).
Is there a natural (non-trivial) topology to put on the automorphism group of a locally compact abelian group? Can the topology in some way be extracted from the original group?
So as not to leave this unanswered, the compact-open topology on the space of continuous mappings between topological spaces $X$ and $Y$ has a subbase consisting of the functions that send a compact subspace $K$ of $X$ into an open subspace $U$ of $Y$.
This is a particularly nice topology in the locally compact setting as the composition map becomes continuous-i.e. the category of locally compact spaces becomes enriched over itself. Thus its natural to topologize the continuous homomorphisms between topological groups as a subspace of the space of all mappings under this topology.
I don't know whether this is always the right topology in the context of topological automorphism groups, but you can at least see quite quickly that it gives the usual topology on $\text{Aut}(\mathbb{R})$.