How to Tell Whether a Topological Space Is Compact

Compactness is a key topological property that, intuitively, says every open cover has a finite subcover. Here are practical criteria and steps to determine whether a given space is compact, with brief explanations and where they are most useful.

1. Use the definition (open-cover criterion)

• Test: for every collection of open sets whose union equals the whole space, can you extract a finite subcollection that still covers the space?
• Best when you can reason directly about arbitrary open covers (often in proofs or for constructed examples).

2. Sequential compactness (metric and first-countable spaces)

• For metric spaces (or more generally first-countable Hausdorff spaces), compactness ⇔ every sequence has a convergent subsequence whose limit lies in the space.
• Use when you can analyze sequences; common in R^n and familiar metric spaces. (See: Heine–Borel and Bolzano–Weierstrass contexts.)

3. Heine–Borel criterion (Euclidean R^n)

• In R^n with the usual topology: compact ⇔ closed and bounded.
• Very practical for subsets of R^n. (Reference: Heine–Borel theorem.)

4. Finite intersection property (FIP)

• Equivalent characterization: every family of closed sets with the finite intersection property (every finite subfamily has nonempty intersection) has nonempty intersection overall.
• Useful when working directly with closed sets.

5. Continuous image test

• Continuous image of a compact space is compact.
• Use to show a space is compact by mapping a known compact space onto it surjectively and continuously.

6. Product spaces (Tychonoff theorem)

• Arbitrary product of compact spaces is compact in the product topology (Tychonoff; requires AC in full generality).
• Finite products: compactness preserved without AC. Use for product constructions.

7. Subspace and closed subset tests

• A closed subset of a compact space is compact.
• An open subset need not be compact. Use this to rule in compactness when you know a larger compact space.

8. Local compactness vs. compactness

• Local compactness (every point has a compact neighborhood) is weaker than compactness. Don’t confuse the two.

9. Countable compactness and limit point compactness

• In certain settings, knowing every infinite subset has an accumulation point (limit point compactness), or every countable open cover has a finite subcover (countable compactness), can help. In metric spaces these are equivalent to compactness; in general spaces they may differ.

10. Practical checklist

• If space ⊂ R^n: check closed + bounded (Heine–Borel).
• If metric space: try sequential compactness (Bolzano–Weierstrass style arguments).
• If you have a cover you can exploit: apply the open-cover definition or the FIP.
• If space is image or closed subset of known compact space: apply preservation facts.
• For products: apply finite-product compactness or Tychonoff when appropriate.

References

• Munkres, J. R., Topology, sections on compactness and product spaces.
• Willard, S., General Topology, compactness and equivalents.
• Rudin, W., Principles of Mathematical Analysis (for metric/Euclidean cases).