Compactness
Central Question
Why is the finite-subcover property the right abstraction of a manageable space?
Why This Matters
Compactness extends familiar existence arguments beyond Euclidean closed-and-bounded sets and makes the role of their hypotheses visible.
Learning Prompt
Rebuild compactness from a topological perspective. Explain why finite subcovers are the appropriate abstraction and why compactness does not fundamentally mean closed and bounded. Connect compactness with sequences, continuous maps and extrema.