Compact Subsets of the Real Line

Topology · Axiom Academy

LESSON Compact Subsets of the Real Line Understanding the beautiful characterization: closed and bounded In topology, compactness is one of the most important and powerful properties a set can have. While the general definition involves open covers, something remarkable happens when we look at subsets of the real line . A subset of a topological space is compact if every open cover of has a finite subcover. That is, if with , then there exist finitely many indices such that . This definition, while elegant, can be difficult to work with directly. Fortunately, when we restrict our attention to the real numbers, we get a much more concrete characterization. Step 2: The Heine-Borel Theorem The fundamental result that characterizes compact subsets of the real line is the Heine-Borel Theorem. A subset of is compact if and only if is closed and bounded. Let's break down what this means: Closed: contains all its limit points. Equivalently, the complement is open, or every convergent sequence in has its limit in . Bounded: There exists such that for all . Step 3: Examples and Non-Examples Let's examine several subsets of to see which are compact: Example 1: The closed interval is compact. ✓ It is closed (contains its endpoints 0 and 1) ✓ It is bounded (all points have absolute value ≤ 1) Example 2: The open interval is NOT compact. ✗ It is not closed (missing its limit points 0 and 1) Example 3: The half-infinite interval is NOT compact. Example 4: The singleton is compact.

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