That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine – Borel theorem.

The Heine – Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers.

Ultimately the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine – Borel compactness in a way that could be applied to the modern notion of a topological space.

This follows from the Heine – Borel theorem.

Again from the Heine – Borel theorem, the closed unit ball of any finite-dimensional normed vector space is compact.

# A is closed and bounded ( Heine – Borel theorem ).

This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.

* 1797 – Heinrich Heine, German poet ( d. 1856 )

by Bernt Heine, T. C. Schadeberg, Ekkehard Wolff, pp. 117 – 28 SUGIA Beiheft 1-2.

Important results include the Bolzano – Weierstrass and Heine – Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.

* December 13 – Heinrich Heine, German poet ( d. 1856 )

* February 17 – Heinrich Heine, German writer ( b. 1797 )

* Heine, R. E., 1987 " The Role of the Gospel of John in the Montanist controversy ," in Second Century v. 6, pp 1 – 18.

* The Euclidean spaces R < sup >< var > n </ var ></ sup > ( and in particular the real line R ) are locally compact as a consequence of the Heine – Borel theorem.

* Williamson, Kay & Blench, Roger ( 2000 ) ' Niger – Congo ', in Heine, Bernd & Nurse, Derek ( eds.

In the topology of metric spaces the Heine – Borel theorem, named after Eduard Heine and Émile Borel, states:

In fact, the Heine – Borel theorem for arbitrary metric spaces reads:

The history of what today is called the Heine – Borel theorem starts in the 19th century, with the search for solid foundations of real analysis.

Émile Borel in 1895 was the first to state and prove a form of what is now called the Heine – Borel theorem.

* Heinrich Heine ( 1797 – 1856 ), German poet

Indeed, the Heine – Borel definition of compactness — that every covering of a space by open sets admits a finite subcovering — is relatively recent.

Alice Heine ( February 10, 1858 – December 22, 1925 ), styled HSH The Princess of Monaco, and also The Duchess of Richelieu, was the American-born second wife of Prince Albert I of Monaco, a great-grandfather of Prince Rainier III of Monaco.

The compactness of U ( n ) follows from the Heine – Borel theorem and the fact that it is a closed and bounded subset of M < sub > n </ sub >( C ).

The Menger sponge is a closed set ; since it is also bounded, the Heine – Borel theorem implies that it is compact.

* Heine – Borel theorem,

# REDIRECT Heine – Borel theorem

# REDIRECT Heine – Borel theorem

# REDIRECT Heine – Borel theorem

An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.

In mathematics, the Heine – Cantor theorem, named after Eduard Heine and Georg Cantor, states that if M and N are metric spaces and M is compact then every continuous function

This can be seen as a generalisation of the Heine – Borel theorem from Euclidean spaces to arbitrary spaces: we must replace boundedness with total boundedness ( and also replace closedness with completeness ).

This causal relationship is unidirectional ; an observable feature of an entity does not define the underlying essence ( Dar-Nimrod & Heine, 2011 ).

There are clear echoes of the Wandering Jew in Wagner's The Flying Dutchman, whose plot line is adapted from a story by Heinrich Heine in which the Dutchman is referred to as ' the Wandering Jew of the ocean ', and his final opera Parsifal features a woman called Kundry who is in some ways a female version of the Wandering Jew.

This period in Heine's life is not very clear, but it seems that his father Samson's business deteriorated and Samson Heine effectively became the ward of his brother Salomon.

Most important was the philosopher Hegel, whose influence on Heine is hard to gauge.

It contains the ironical disillusionment which is typical of Heine:

In the French edition of " Lutetia " Heine wrote, one year before he died: " This confession, that the future belongs to the Communists, I made with an undertone of the greatest fear and sorrow and, oh !, this undertone by no means is a mask!

He is a Professor at the Heinrich Heine University of Düsseldorf.

Writing to Heinrich Heine in 1839, he offered the fatalistic view: I believe that richess is like love in the theatres and novels: no matter how often one encounters it ... it never misses its target if effectively wielded ... can grow back the foreskin of which we are robbed on the eighth day of life ; those who, on the ninth day, do not bleed from this operation shall continue to bleed an entire lifetime, even after death.

No infinite-dimensional Banach space is a Montel space, since these cannot satisfy the Heine – Borel property: the closed unit ball is closed and bounded, but not compact.

He cited Heine and Stendhal as favorites in literature.

Housman himself acknowledged the influence of the songs of William Shakespeare, the Scottish Border ballads and Heinrich Heine, but specifically denied any influence of Greek and Latin classics in his poetry.

* Heine, Bernd.

According to physicist Phil Anderson, the term was coined by himself and Volker Heine when they changed the name of their group at the Cavendish Laboratories, Cambridge from " Solid state theory " to " Theory of Condensed Matter ", as they felt it did not exclude their interests in the study of liquids, nuclear matter and so on.

In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous.

Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Dirichlet in 1854.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair.

At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis.

Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series.

A Middle High German version was described as " Perhaps the flower of religious literary creativity in the German Middle Ages " by Heinrich Heine.

Poliomyelitis was first recognized as a distinct condition by Jakob Heine in 1840.