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I'm trying to endow affine group $\mathrm{GA}(X)$, $X$ the given affine space, with canonical topology based on the semidirect product of two of its subgroups $T(X)$ and $\mathrm{GA}_a(X)$: $\mathrm{GA}(X)=T(X)\mathrm{GA}_a(X)$, where $T(X)$ is the set of all translations of $X$ and $\mathrm{GA}_a(X)$ is the stabilizer of $a\in X$ in $\mathrm{GA}(X)$ (note that $X$ is a homogeneous space for $\mathrm{GA}(X)$). I have had the canonical topology on subgroups $T(X)$ and $\mathrm{GA}_a(X)$, respectively, based on group isomorphisms $T(X)\cong\vec X$ and $\mathrm{GA}_a(X)\cong\mathrm{GL}(\vec X)$. But my math background does not support me to construct the topology based on semidirect product.

Abstracting from the above concrete motivation, the problem is as follows:

Given topological groups $(H,\tau)$ and $(K,\sigma)$, group $G$ is the semidirect product of its subspaces $H$ and $K$, assuming, without loss of generality, $H$ is normal. How to, if at all, construct the induced topology $\nu$ on $G$ to make it a topological group and $\tau,\sigma$ happen to be the subspace topology of $\nu$?

The condition that $\tau,\sigma$ are subspace topologies of $\nu$ is added by me because I think that's a natural requirement. Please correct me if not. As I said, I'm not able to perform this construction, so I ask here in hope to get some help. It is also ok if you refer me to some textbook or online tutorial (I searched but having no luck) that addresses construction of topological group out of semidirect product. Thank you.

PS, for those who doubt feasibility/motivation/context of such a construction, the question comes from the following paragraph in a geometry text: enter image description here Note: the $X$ on the 3rd line should be $\mathrm{GA}(X)$ in my understanding.

user5280911
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Upon request, here we go: if $H$ and $K$ are topological groups equipped with an action $$\alpha: K \to \text{Aut}(H),$$ which is continuous in the sense that $$ (k,h)\in K\times H \mapsto \alpha_k(h)\in H $$ is a continuous map, then we define the semidirect product group $ H\rtimes_{\alpha}K $ to be the Cartesian product $H\times K$, equipped with the multiplication operation $$ (h_1,k_1)(h_2,k_2) = (h_1\alpha_{k_1}(h_2),k_1k_2). $$

If we equip $H\rtimes_{\alpha} K$ with the product topology, then it becomes a topological group.

Ruy
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