Compactness makes infinite spaces behave like finite sets
In topology, compactness is the infinite analogue of a finite list: you can always thin a sprawling open cover down to finitely many sets and still blanket the space. Maurice Fréchet named the idea in 1906; Alexandrov and Urysohn later standardized today’s cover language.
Nineteenth-century roots include Bolzano’s 1817 insight that bounded sequences have convergent subsequences, proved by nested bisection decades before Weierstrass popularized it. By the 1880s Ascoli and Arzelà treated functions as points in abstract spaces, extending Bolzano–Weierstrass beyond numbers.
Fréchet generalized those theorems from point sets to function spaces. Compactness now underwrites analysis: continuous functions on compact domains attain maxima, and many existence proofs reduce to extracting convergent subsequences.
A second strand came from studying the real line. In 1870 Eduard Heine proved that a continuous function on a closed, bounded interval is uniformly continuous, using a lemma that any countable covering of the interval by open pieces can be cut to finitely many. Émile Borel saw its importance in 1895, and Pierre Cousin and Henri Lebesgue extended it to arbitrary collections, giving the Heine–Borel theorem: in Euclidean space, compact means closed and bounded. Lebesgue used this passage from local to global facts in building his integral, and Alexandrov and Urysohn's 1929 work carried it into general topological spaces.
Examples make the boundary matter. The closed interval from 0 to 1 is compact, while the open interval and the whole real line are not, because a sequence can drift toward a missing endpoint or off to infinity. A closed disc and a sphere are compact, but an open disc or a sphere with one point removed fails the test. Hilbert and Erhard Schmidt's work on integral equations also led to compact operators.
Source: Compact space
