Heine-Borel Theorem

Real Analysis · Axiom Academy

LESSON The Heine–Borel Theorem In , the abstract idea of compactness collapses into two things you can actually check: closed and bounded. 1. Compactness, Made Checkable For an arbitrary set, deciding compactness from open covers is awkward. The theorem trades that abstract test for two concrete properties — but only in with the usual distance. The animation builds K as both conditions click into place: it sits inside a big ball (bounded) and it swallows its own boundary (closed) — and exactly then it is compact. The first condition is about size . A set K is bounded when it never runs off to infinity — some single ball of finite radius holds all of it. Watch one ball grow: for a bounded set it eventually engulfs everything, but a set like keeps poking out of every ball, no matter how large. [0,1] and (0,1) both sit inside B(0,1) — bounded . The ray contains every integer n , so no finite M can cap it — not bounded . 3. Closed: It Keeps Its Limit Points The second condition is about edges . A set K is closed when every sequence inside it that converges has its limit inside K too — it never leaks a boundary point. The animation marches a sequence toward an edge: on [0,1] the limit point is captured, but on (0,1) the very same sequence escapes through the missing endpoint. In [0,1] the sequence and — the limit is kept, so it's closed . In (0,1) that same sequence still tends to 0 , but — a limit point leaks out, so it's not closed . 4. You Need Both — and a Warning

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