Compactness Review

Algebraic Topology · Axiom Academy

Understanding compactness through open covers and its powerful implications 1. Open Covers and Finite Subcovers A topological space is compact if every open cover has a finite subcover. This seemingly technical definition has profound consequences. • An open cover of X is a collection U_α of open sets with X ⊆ ⋃ U_α • A subcover is a subcollection that still covers X • X is compact if every open cover has a finite subcover In other words: no matter how you cover X with open sets, you can always find finitely many that suffice. In Euclidean space ℝⁿ, we have a beautiful characterization of compact subsets. A subset K ⊆ ℝⁿ is compact if and only if K is closed and bounded. • [0, 1] is compact (closed and bounded) • (0, 1) is NOT compact (not closed) • [0, ∞) is NOT compact (not bounded) • ℝ is NOT compact (not bounded) Warning: This characterization only works in ℝⁿ! In general topological spaces, "closed and bounded" doesn't even make sense without a metric. In metric spaces (and first-countable spaces), compactness has an equivalent sequential characterization. A space X is sequentially compact if every sequence in X has a convergent subsequence. Theorem: In metric spaces, the following are equivalent: 1. X is compact (open cover definition) 3. X is complete and totally bounded Key Application: Bolzano-Weierstrass Theorem - every bounded sequence in ℝⁿ has a convergent subsequence. 4. Key Properties of Compact Spaces

This is the written version of the interactive lesson above. See the full Algebraic Topology course.