Which one is finer: standard topology or upper limit topology?

Question

Out of these two which one is finer over $\mathbb{R}$?

• Standard topology
• Upper limit topology

Answer 1

The upper limit topology is finer. Notice that in the upper limit topology,\begin{equation*} (a,b) =\bigcup_{n\in\mathbb{N}}\left(a,b-\frac{1}{n}\right]. \end{equation*} Hence every open interval in the standard topology is also open in the upper limit topology, but something like $(c,d]$ is not open in the standard topology.
I hope this makes sense.