In real analysis the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states: For a subset S of Euclidean space Rn, the following two statements...

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

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...

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...

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

"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...

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...

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

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

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...

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...

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...

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...

Heckscher–Ohlin theorem (economics) Heine–Borel theorem (real analysis) Heine–Cantor theorem (metric geometry) Hellinger–Toeplitz theorem (functional analysis)...

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...

‖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...

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...

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

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)...

particular the real line R) are locally compact as a consequence of the Heine–Borel theorem. Topological manifolds share the local properties of Euclidean spaces...

in the weak* topology. If X is a normed space, a version of the Heine-Borel theorem holds. In particular, a subset of the continuous dual is weak* compact...

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

closely related to both the Bolzano–Weierstrass theorem and the Heine–Borel theorem. Every Cauchy sequence of real numbers is bounded, hence by Bolzano–Weierstrass...

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

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...

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...

a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem). Every continuous image of a compact space is compact. A compact...

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

subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, connected subsets of R n {\displaystyle \mathbb {R} ^{n}} (for n...