In real analysis, the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S {\displaystyle S} of Euclidean space R n {\displaystyle...

lemma Borel's law of large numbers Borel measure Borel–Kolmogorov paradox Borel–Cantelli lemma Borel–Carathéodory theorem Heine–Borel theorem Borel determinacy...

as the intermediate value theorem, the Bolzano–Weierstrass theorem, the extreme value theorem, and the Heine–Borel theorem. It is usually taken as an...

Andréief–Heine identity Heine–Borel theorem Heine–Cantor theorem Heine definition of continuity Heine's Reciprocal Square Root Identity Heine–Stieltjes...

closed and bounded subsets. This form of the theorem makes especially clear the analogy to the Heine–Borel theorem, which asserts that a subset of R n {\displaystyle...

covers, the Heine-Borel property can be inferred. For every natural number n, the n-sphere is compact. Again from the Heine–Borel theorem, the closed...

version follows from the general topological statement in light of the Heine–Borel theorem, which states that sets of real numbers are compact if and only if...

Andréief–Heine identity Heine–Borel theorem Heine–Cantor theorem Heine–Stieltjes polynomials Heine definition of continuity Heine functions Heine's identity...

forms an open cover of I. Since I is closed and bounded, by the Heine–Borel theorem I is compact, implying that this covering admits a finite subcover...

theorem Goodstein's theorem Green's theorem (to do) Green's theorem when D is a simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's...

"every open cover of K {\displaystyle K} has a finite subcover". The Heine–Borel theorem asserts that a subset of the real line is compact if and only if...

interval is also an interval Heine–Borel theorem – sometimes used as the defining property of compactness Bolzano–Weierstrass theorem – states that each bounded...

Ax–Grothendieck theorem (model theory) Barwise compactness theorem (mathematical logic) Borel determinacy theorem (set theory) Büchi-Elgot-Trakhtenbrot theorem (mathematical...

review. Heine–Borel theorem. This theorem was proved in 1872 by Émile Borel, not by Eduard Heine. Borel used techniques similar to those that Heine used...

if and only if it is closed and bounded. This is also called the Heine-Borel theorem. In topological vector spaces, a different definition for bounded...

‖x‖, so it is closed; Sn is also bounded, so it is compact by the Heine–Borel theorem. More generally, in a metric space (E,d), the sphere of center x...

Fyodorov–Schoenflies–Bieberbach theorem Jordan–Schoenflies theorem Schoenflies notation Schoenflies displacement Heine–Borel theorem Arthur Moritz Schoenflies...

equivalent to weak Kőnig's lemma and thus to WKL0 over RCA0: The Heine–Borel theorem for the closed unit real interval, in the following sense: every...

point in A. Compactness (or Heine-Borel compactness): Every open cover of A admits a finite subcover. The Eberlein–Šmulian theorem states that the three are...

is complete and totally bounded. This is a generalization of the Heine–Borel theorem, which states that any closed and bounded subspace S {\displaystyle...

Bornological space – Space where bounded operators are continuous Heine–Borel theorem – Subset of Euclidean space is compact if and only if it is closed...

student of Henri Poincaré, in 1895, and it extends the original Heine–Borel theorem on compactness for arbitrary covers of compact subsets of R n {\displaystyle...

The Menger sponge is a closed set; since it is also bounded, the Heine–Borel theorem implies that it is compact. It has Lebesgue measure 0. Because it...

the completion of the space is compact). This is analogous to the Heine-Borel theorem. In contrast, no infinite-dimensional normed space has this property...

non-compactness are useless for subsets of Euclidean space Rn: by the Heine–Borel theorem, every bounded closed set is compact there, which means that γ(X)...

hypercubes get divided by every midplane), as in the proof of the Heine–Borel theorem. A finite subdivision rule R {\displaystyle R} consists of the following...

category theorem Nowhere dense Baire space Banach–Mazur game Meagre set Comeagre set Compact space Relatively compact subspace Heine–Borel theorem Tychonoff's...

paradox, Borel–Cantelli lemma, Borel–Carathéodory theorem, Heine–Borel theorem, Borel summation, Borel distribution Alexander Borodin, Russian composer and...

another curve. This space is closed and bounded and so compact by the Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too...

Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, Taylor's theorem, the fundamental theorem of calculus, the...