Compactness
Every open cover has a finite subcover.
Key idea
Every open cover has a finite subcover.
In depth
A set is compact if every open cover has a finite subcover. Compact sets behave like finite ones: continuous functions attain extremes and are uniformly continuous. In R, compact means closed and bounded.
Worked example
[0, 1] is compact; (0, 1) is not, as the cover {(1/n, 1)} has no finite subcover.
Real-world uses
- Guaranteeing that an optimisation problem on a closed bounded region has a best answer
Common mistake
Equating compact with closed in infinite-dimensional spaces.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Compactness?
Every open cover has a finite subcover. It is a topic at the expert level in Analysis & Topology.
Where is Compactness used in real life?
Real-world examples: Guaranteeing that an optimisation problem on a closed bounded region has a best answer.
What should I learn before Compactness?
Make sure you are comfortable with Interior, Closure and Boundary, the previous topic in Analysis & Topology.
What is a common mistake with Compactness?
Equating compact with closed in infinite-dimensional spaces.
How can I practise Compactness?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Compactness?
Move on to Heine–Borel Theorem next.
Related expert topics in Analysis & Topology
- Axioms of the Real Numbers – ℝ is a complete ordered field.
- Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
- Archimedean Property – For any real x there is an integer n > x.
- Density of Rationals – Between any two reals there is a rational.
- Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
- Limit Theorems for Sequences – Algebra of limits and squeeze theorem.