Examples of topologies on R

Solution 1:

I assume you're familiar with the concept of a basis for a topological space. If so, then the topology on $\mathbb{R}$ generated by $(a,b]$ is neither trivial, discrete, or the usual metric topology. Similarly, using $[a,b)$ as generating sets gives another distinct and interesting topology.

Another topology on $\mathbb{R}$, which I think was used when I first learned topology as the first example of a nontrivial non-Hausdorff topology is to let all subsets of $\mathbb{R}$ with finite complement be open. (You can check it satisfies the definition.) This is actually a pretty weird topology on $\mathbb{R}$.