[Math] How to compare open sets from $\mathbb{R}$ with the finite complement topology and the standard topology

Question

I am being asked about which of the two topologies are finer than the other and I am having troubles once I am not able to say what are a basis for $\mathbb{R}$ with the finite complement topology, so I don't know how to compare such topologies.

Any comments will be very helpful.

Answer 1

Observe that finite sets are closed if $\mathbb R$ is equipped with the standard topology, so that cofinite sets belong to that topology. That means that the standard topology is finer than the finite complement topology.

Also there are sets belonging to the standard topology that are not cofinite. This allows us to conclude that the standard topology is properly finer than the finite complement topology.