Can a compact topological group have the trivial topology?

Question

I want to show that the following are equivalent for a compact topological group $G$:

1. $G$ is the inverse limit of finite groups $G_i$.

2. There's a family $\left\{N_i\right\}$ of open normal subgroups that intersect trivially and $G$ is the inverse limit of the (finite) groups $G/N_i$.

3. There's a family $\left\{H_j\right\}$ of open subgroups that intersect trivially.

Now, I know how to do all the implications except 1->2 or 1->3. The reason is that if $G$ has the trivial topology, where the only open and closed sets are the empty set and the whole group, then the only open subgroup would be all of $G$, which would prevent the other two properties from holding. Is there a reason a compact topological group can't have the trivial topology?