A quick reasoning in abstract groups shows that if G/Z(G) is finite, then so is the derived subgroup G'. In this talk, we show that a similar result is true in the context of topological groups, replacing "finite" by "compact". This actually serves as an excuse to state and use a powerful structural result on central groups. https://www.uni-muenster.de/AGKramer/index.php?name=TSSS23&menu=teesemi&lang=de