the.com/compactness
the mathematical promise that infinite searches still end in a finite number of steps.
means a space is compact if every open cover of it has a finite subcover, meaning you can always tame infinite complexity with finitely many pieces.
from formalized in the early 20th century by Fréchet and later refined via Heine-Borel, generalizing the simple fact that closed, bounded intervals on the real line behave nicely.
heine-borelclosed and bounded equals compact, only in finite dimensions
sequencesevery sequence has a convergent subsequence inside
continuous imagescompactness always survives continuous functions
infinite dimensionsclosed unit balls stop being compact there