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

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.

The Heine – Borel theorem implies that a Euclidean n-sphere is compact.

* 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

As a result, various literary figures visited the place, including Heinrich Heine ( Die Harzreise ) and Theodor Fontane and especially the Bode Gorge.

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.

Heine now had to search for a job.

A monument to this can now be found in the center of the square, consisting of a glass panel opening onto an underground white room with empty shelf space for 20, 000 volumes and a plaque, bearing an epigraph from an 1820 work by Heinrich Heine: " Das war ein Vorspiel nur, dort wo man Bücher verbrennt, verbrennt man am Ende auch Menschen " (" That was only a prelude ; where they burn books, they ultimately burn people ").

In June 1995 Eddie Bauer Germany was announced as part of a joint venture between Eddie Bauer Inc. and two members of the Otto-Versand Group ( now Otto GmbH & Co KG ): Heinrich Heine GmbH ( Heine Group ) and Sport Scheck.