# Connected locally compact abelian groups

Is the category $\text{LCA}_c$ of connected locally compact Hausdorff abelian groups an abelian category? My feeling says no, however I can't immediately find a counterexample.

Alternatively, I'd also be happy to know whether an injective continuous map in $\text{LCA}_c$ is automatically closed.

• In general, you won't have cokernels, is that right? – Randall May 30 '18 at 14:25
• The category $\text{LCA}$ is quasi-abelian and hence has cokernels. The cokernel of $f:A\rightarrow B$ is $B/\overline{A}$, hence there is a surjection $B\rightarrow B/\overline{A}$. Moreover, the image of a connected space is connected. – Mathematician 42 May 30 '18 at 15:13

Given an irrational number $r\in\mathbb R$, the image of the continuous injective function $\mathbb R\to\mathbb R^2/\mathbb Z^2$ given by $t\mapsto(t,rt)$ is dense but not surjective.
Let me elaborate a little on Vladimir Sotirov's example for readers who may not see how it immediately answers the question. Suppose you have a continuous injective homomorphism $f:A\to B$ of locally compact abelian groups whose image is dense but not all of $B$, as in his answer. Then the kernel of $f$ is $0$, since $f$ is injective. The cokernel of $f$ is also $0$: if $g:B\to C$ is a morphism such that $gf=0$, then $g$ vanishes on the image of $f$ and hence on all of $B$ since the image of $f$ is dense and $C$ is Hausdorff. So $f$ has trivial kernel and cokernel. In an abelian category, this would imply $f$ is an isomorphism, but it is not an isomorphism since it is not surjective. Thus our category cannot be abelian.