In my computation book by Sipser, he says that since every language that can be decided in time $o(n \log n)$ is regular, then that can be used to show $TIME(n \log (\log n))\setminus TIME(n)$ must be the empty set. Can anyone show me why this is?
both $TIME(n\log(\log n))$ and $TIME(n)$ are regular. I think that only means we can subtract the two sets and the result will still be regular. I just dont understand how its possible to subtract the collection of $O(n\log(\log n))$ time TM decidable languages from the collection of $O(n)$ time TM decidable languages and get the empty set. These two collections are not equal so I feel like there will be something left over