Topological spaces can be infinitely varied. From topology point of view even something as deadpan as $\mathbb{R}$ has an enormous amount of structure.

Topological spaces can be infinitely varied. From topology point of view even something as deadpan as $\mathbb{R}$ has an enormous amount of structure.

What about the Zariski topology in algebraic geometry? They tend to be different than the topology you inherit from their embedding into ambient space.