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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.

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

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

• 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.

• 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.

• 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.

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

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

• 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.

• 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.

• 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.

• 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.

• Pavel AlexandrovRelated: Compactness is the property his best-known compactification construction supplies.

• Cantor's intersection theoremRelated: Compactness ensures nested nonempty closed sets cannot lose all their common points.

• Fundamental lemma of the calculus of variationsRelated: Compactly supported tests use compact regions to keep localization and integrability controlled.

• Hopf conjecture (Euler characteristic)Related: Compactness is an explicit hypothesis, excluding noncompact examples.

• Riesz's lemmaRelated: Separated sequences constructed using the lemma obstruct compactness of closed unit balls.

• Soul theoremRelated: The soul is compact even though the ambient manifold is noncompact.

• Stone's representation theorem for Boolean algebrasRelated: Compactness is one of the defining properties of the representing space.