Compactness

Compactness was formally introduced by Maurice Fréchet in 1906 in work generalizing the Bolzano–Weierstrass theorem from sets of points to spaces of functions. Later, Pavel Alexandrov and Pavel Urysohn developed the open-cover formulation that is now standard in topology. Compactness plays a central role throughout mathematics; for example, continuous real-valued functions on compact spaces attain maxima and minima, and major results such as the Arzelà–Ascoli theorem and the Peano existence theorem depend on compactness.