Compactness
Objective. Apply open-cover and sequential characterisations.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A topological space X is compact if every open cover of X has a finite subcover, meaning no matter how X is covered by open sets, finitely many of them already cover X. In metric spaces, compactness is equivalent to sequential compactness, where every sequence has a convergent subsequence with limit in X, and also equivalent to being closed and bounded by the Heine-Borel theorem in ℝⁿ. Continuous images of compact sets are compact, which is why continuous real-valued functions on compact sets attain maximum and minimum values. For example, [0,1] is compact, but (0,1) is not, since the open cover {(1/n,1):n≥2} has no finite subcover.
Worked example
Decide whether the set S=[0,1]∪[2,3] is compact as a subspace of ℝ.
- 1S is closed since it is a finite union of closed intervals
- 2S is bounded since it lies within [0,3]
- 3By Heine-Borel in ℝ, closed and bounded subsets are compact
- 4S is compact
Answer: Yes, S is compact
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Practice
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Start practisingCommon mistake. Students assume closed and bounded implies compact in every space, but this Heine-Borel equivalence is special to ℝⁿ (finite dimensions); in infinite-dimensional normed spaces the closed unit ball is bounded and closed yet not compact.
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