Explain how one could figure out 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).