Definition of Compactness

Real Analysis · Axiom Academy

LESSON Definition of Compactness A set is compact when every open cover has a finite subcover — the topological heart of "closed and bounded." An open cover of a set K is a collection of open sets whose union contains K — every point of K lands in at least one of the sets. A finite subcover is a finite subcollection that still covers K . The animation starts with an open cover of K made of many overlapping sets, then thins it down to just a handful — a finite subcover that already does the whole job. 2. The Closed Interval [0,1] Is Compact Take any open cover of [0,1] . Because the endpoints 0 and 1 are included , each must lie inside some open set of the cover. Starting at 0 , follow overlapping sets to the right; since there are no gaps to fall through, a finite chain of them reaches all the way to 1 . The animation shows a deliberately wasteful cover of [0,1] (seven overlapping sets), then keeps only four — U_1,U_2,U_3,U_4 — that already cover every point from 0 to 1 . [0,1] is both closed (it contains 0 and 1 ) and bounded . The closed endpoints are exactly what stop an "almost-cover" from leaking out the ends — there is no point arbitrarily close to the boundary that escapes every finite collection. 3. The Open Interval (0,1) Is NOT Compact Now the endpoint 0 is missing — and that single gap lets us build an open cover with no finite subcover. Consider the sets

This is the written version of the interactive lesson above. See the full Real Analysis course.