In some theorems, I see "$f: [a,b] \to \mathbb R$ on a compact interval". $[a,b]$ is actually compact, does this kind of emphasize have another meaning?
Edit for a comment: "A continuous function $f: [a,b] \to \mathbb R$ on a compact interval is Riemann integrable."
It's mostly just redundant emphasis: the phrase "on a compact interval" and writing the domain as $[a,b]$ are basically conveying the same information twice.
However, the phrase "on a compact interval" also clarifies that $[a,b]$ can be any compact interval and the claimed statement will hold. If you leave it out, it might sound like you are referring to some specific interval $[a,b]$, where you have previously defined $a$ and $b$, and that it might not hold for more general compact intervals.