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The 59 pages that link to Compact space, each with the reason it gives.

• Uniform convergenceRelated: Compactness often lets local or pointwise bounds yield global uniform control.

• TopologyRelated: Compactness is a preserved property that often enables strong classification results.

• SequenceRelated: Sequential compactness characterizes compact metric spaces through their sequences.

• Open setRelated: Compactness is defined through covers made of open sets.

• Topological spaceRelated: Compactness turns arbitrary open covers into finite data and supports powerful existence results.

• Brouwer fixed-point theoremRelated: Compactness supplies the finite-dimensional boundedness conditions behind the theorem's domain.

• Fixed-point theoremRelated: Compactness supplies existence conditions in several topological fixed-point results.

• Extreme value theoremNarrower topic: Compactness supplies the key condition that keeps a function’s values from escaping their bounds.

• HomeomorphismRelated: A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

• Closed setCompared with: Closed subsets of compact spaces are compact, but closedness alone does not imply compactness.

• Bolzano–Weierstrass theoremNarrower topic: Compactness guarantees convergent subsequences in metric spaces, beyond Euclidean boundedness alone.

• Arzelà–Ascoli theoremNarrower topic: Compactness of the domain lets local continuity control be assembled across the whole space.

• Cantor setRelated: As a closed subset of a bounded interval, the Cantor set is compact.

• Complete metric spaceCompared with: Compactness is stronger than completeness for metric spaces, requiring control beyond Cauchy limits.

• Connected spaceRelated: Compactness is independent of connectedness, though both often shape classification theorems.

• Open coverRelated: Finite subcovers are the defining test that makes open covers central to compactness.

• Uniform continuityRelated: Compactness is the domain condition that makes continuity imply uniform continuity.

• Heine–Borel theoremNarrower topic: The theorem characterizes this general topological property using closedness and boundedness in Euclidean space.

• Schwarzschild radiusRelated: Reaching the Schwarzschild radius marks the relativistic compactness threshold for horizon formation in the idealized case.

• Bounded functionRelated: Compact domains make continuous real-valued functions bounded, unlike arbitrary domains.

• Bounded setCompared with: Compact subsets of metric spaces are bounded, but bounded subsets need not be compact.

• Stone–Čech compactificationRelated: Compactness is the defining global property of the space βX.

• Sierpiński triangleRelated: As a closed subset of the initial triangle, the limiting set is compact.

• Tychonoff's theoremBroader topic: Compactness of each factor is the hypothesis preserved by the product.

• Weierstrass approximation theoremRelated: A closed bounded interval is compact, enabling uniform control of continuous functions.

• Compact operatorNarrower topic: Compact closure in the image is the defining destination property.

• Convergent sequenceRelated: Compactness yields convergent subsequences for sequences in metric spaces.

• Sequential compactnessNarrower topic: Compactness is the cover-based property often compared with sequential compactness.

• Banach–Alaoglu theoremNarrower topic: The theorem is a central compactness result for dual spaces.

• Convex bodyRelated: Compactness supplies closedness and boundedness for subsets of Euclidean space.

• Completeness of the real numbersRelated: Compactness of closed bounded real intervals reflects completeness together with boundedness.

• Felix HausdorffRelated: In Hausdorff spaces, compact subsets are closed, a central interaction between separation and compactness.

• Poincaré–Hopf theoremRelated: Compactness is a central hypothesis ensuring the global index sum is defined.

• Prokhorov's theoremNarrower topic: Compactness of the closure is the topological outcome characterized by the theorem.

• Schauder fixed-point theoremRelated: Compactness supplies the finite approximation needed in the theorem’s proof.

• Separation axiomRelated: Compactness combines with Hausdorffness to yield strong uniqueness and closedness results.

• Weierstrass theoremNarrower topic: Compactness is the domain condition that makes the theorem’s extrema attainable.

• Lefschetz fixed-point theoremRelated: Compactness is part of the common hypotheses ensuring a well-behaved Lefschetz number.

• Stone–Weierstrass theoremRelated: Compactness supplies the domain condition in the theorem’s standard form.

• Chern–Gauss–Bonnet theoremRelated: Compactness makes the manifold integral finite without boundary terms in the closed case.

• General topologyRelated: Compactness turns global covering conditions into finite ones.

• Heine–Cantor theoremNarrower topic: Finite subcovers let local continuity neighborhoods yield a global uniform bound.

• Liouville–Arnold theoremRelated: Compactness helps force a connected regular common level set to be a torus.

• Poincaré–Bendixson theoremRelated: Compactness prevents the limit set from escaping to infinity or losing accumulation points.

• Tube lemmaNarrower topic: A finite subcover lets local product neighborhoods combine into one neighborhood covering the compact factor.

• Eberlein–Šmulian theoremNarrower topic: Weak compactness is ordinary compactness applied to the weak topology.

• Gromov–Hausdorff convergenceRelated: Compactness ensures the standard Gromov–Hausdorff distance is finite and supports subsequential limit theorems.

• Krein–Milman theoremRelated: Compactness supplies the key finiteness and existence properties in the theorem.

• Carathéodory's existence theoremRelated: Compactness arguments help extract a limiting solution from approximate trajectories.

• 3-manifoldRelated: Compactness distinguishes many central classes of 3-manifolds without changing their local definition.