Finite Subcovers

Real Analysis · Axiom Academy

Cover a set with open intervals, then throw away all but finitely many. For some sets you always can — and that is what it means to be compact. An open cover, slimmed down to finitely many Take the closed interval [0,1] and surround it with open intervals — say one little open interval around every single point , an infinite pile of them. Together they cover [0,1] : every point sits inside at least one. The daring claim of compactness is that you never actually need that whole infinite pile. Watch the swarm of open intervals over [0,1] thin out: nearly all of them switch off, and just a handful are left — yet those few still cover the entire interval, endpoint to endpoint. That surviving handful is a finite subcover . From infinitely many down to five — and [0,1] is still completely covered. That reduction is the whole game. Build a finite cover of [0,1] yourself Drop a few equally spaced open intervals of the same width across [0,1] and slide their width up. With only a handful of fat-enough intervals the gaps slam shut and the whole closed interval is covered — finitely many really is enough. Five open intervals, wide enough, cover all of [0,1] . A closed, bounded interval never resists a finite cover. The open interval (0,1) refuses

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