I have been looking into the history of topology. One thing I am very curious about is the history of the open cover definition of compactness. According to Raman-Sundström, this goes back to a lemma in Borels 1894 thesis. However, I have not been able to find what problem Borel was trying to solve or what kind of insight mathematicians like Heine, Borel and Lebesgue were trying to gain in studying open sets.
What motivated the open cover definition of compactness?