I think I have no or a little problem with analyzing given algebraic products. That is, I know properties of direct, semi-direct and free products. For example, the free product $G\ast H$ is a group which contains both $G$ and $H$, and $G$ and $H$ does not affect each other in this product. And the semi-direct product $H\rtimes K$ is a group in which $H$ is normal and $K$ is a complement of $H$.
However, to be honest, I think I cannot visualize them clearly. While I have a very clear image of topological products, when it comes to algebra, I lose my intuition.
I think this is maybe because I don't know motivations of those products. (Maybe without motivations, there is a way to visualize them. If you know how, please teach me.) For example, when constructing semi-direct product, I have no idea why such specific homomorphich $\phi:K\rightarrow Aut(H)$ should be used to make an object one intends.
To sum up:
How do I visualize products so that I know which product should I take to construct an object I want.
Motivations?
Thank you in advance:)
EDIT:
For example, let $X$ be a metric space and $X^I$ be a space equipped with uniform topology. I draw this for a neighborhood of $f\in X^I$.
And for a topological open sets, I always imagine it to be an region in $\mathbb{R}^n$ without a boundary.
I want to know if there is a similar way to visualize them just like these examples