Let $1 \to N \to G \to H \to 1$ be a short exact sequence of topological groups. Such an exact sequence is said to be topologically split if $G$ is $N \times H$ as a
topological space.
Can someone give me an example of a topologically split short exact sequence of non-discrete connected topological groups. Of course I want an example which is not a split exact sequence.
Edit: By a split exact sequence of topological groups, I mean an exact sequence of topological groups admitting a continuous section which is also a homomorphism. In other words, $G$ is a semi-direct product of $N$ by $H$.
Best Answer
I think something like the real $ax+b$ group (a.k.a. the affine group of ${\mathbb R}$) ought to do the trick.
The following is not the full $ax+b$ group but may be easier to handle here. Take $$ G = \left\{\left( \matrix{ a & b \\ 0 & 1 } \right) \colon a >0, b \in {\mathbb R} \right\} $$
and take $N$ to be those matrices with $a=1$ (these correspond to translations if we view $G$ as acting by $x\mapsto ax+b$). Clearly $G/N \cong {\mathbb R}_{>0}$ as topological groups, and it is clear that $G\cong {\mathbb R}_{>0}\times{\mathbb R}$ as a topological space, but not as a topological group.
More generally, solvable Lie groups ought to give loads of examples, but I am very far from expert in such things.