The Heine-Borel Theorem

Topology · Axiom Academy

LESSON The Heine-Borel Theorem Unit 5: Compactness - The fundamental characterization for ℝⁿ In topology and analysis, compactness is one of the most important properties a set can have. Compact sets behave "nicely" in many ways—continuous functions achieve their maximum and minimum on them, sequences have convergent subsequences, and many theorems require compactness as a hypothesis. A subset of a metric space is compact if every open cover of has a finite subcover. That is, if is a collection of open sets with , then there exist finitely many such that . This definition is abstract and not always easy to check directly. For subsets of Euclidean space ℝⁿ, the Heine-Borel Theorem gives us a much simpler characterization. Step 2: The Heine-Borel Theorem A subset of is compact if and only if is closed and bounded . This is remarkable! The abstract topological property of compactness reduces to two simple geometric conditions in Euclidean space. What does "closed and bounded" mean? Intuitively: The set fits inside some large ball. Intuitively: The set contains its boundary—no "holes" or missing edge points. Step 3: Examples and Counterexamples Let's test our understanding with concrete examples in ℝ and ℝ². Compact Sets (Closed AND Bounded): NOT Compact (Missing Closed or Bounded): — Bounded but not closed (open interval) — Closed but not bounded (infinite ray) Step 4: Proof Idea — Compact ⇒ Closed and Bounded

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