Linked from

The 13 pages that link to Heine–Borel theorem, each with the reason it gives.

• Compact spaceRelated: It translates open-cover compactness into a familiar geometric test in Euclidean space.

• Extreme value theoremRelated: It makes the theorem’s compactness hypothesis concrete for subsets of the real line and Euclidean spaces.

• Bolzano–Weierstrass theoremRelated: The sequence theorem supplies the sequential compactness behind this characterization.

• Open coverBroader topic: Its compactness conclusion guarantees finite subcovers for open covers of closed bounded sets.

• Bounded setRelated: It makes boundedness part of a precise compactness test in finite-dimensional Euclidean spaces.

• Tychonoff's theoremCompared with: It handles finite-dimensional Euclidean compactness, unlike arbitrary topological products.

• Compact operatorCompared with: Unlike finite-dimensional spaces, infinite-dimensional normed spaces have noncompact closed bounded balls.

• Banach–Alaoglu theoremCompared with: Banach–Alaoglu gives compactness from boundedness in a weaker topology, even in infinite dimensions.

• Weierstrass theoremRelated: It turns the theorem’s abstract compactness condition into a familiar test for subsets of Euclidean space.

• Heine–Cantor theoremRelated: It provides a familiar test for the compact-domain hypothesis in Euclidean settings.

• Tube lemmaCompared with: It gives a metric-space criterion for compactness, whereas the tube lemma is purely topological.

• Vitali covering lemmaCompared with: Compactness produces finite subcovers, while the Vitali lemma targets disjoint selections.

• Cantor's intersection theoremRelated: It lets the compact-space hypothesis be checked through geometry in Euclidean settings.