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81-532: Grünwinkel is a borough in the southwest of Karlsruhe , Baden-Württemberg , in the southwest of Germany . There are archeological traces of a brickyard from the 1st century. Written information dates from the 15th century, when the name was Kreenwinkel , which can be read as crow corner . Today's name means green corner . Karlsruhe Karlsruhe was the capital of the Margraviate of Baden-Durlach ( Durlach : 1565–1718; Karlsruhe: 1718–1771),

162-451: A metric space is an ordered pair ( M , d ) where M is a set and d is a metric on M , i.e., a function d : M × M → R {\displaystyle d\,\colon M\times M\to \mathbb {R} } satisfying the following axioms for all points x , y , z ∈ M {\displaystyle x,y,z\in M} : If

243-751: A "structure-preserving" map is one that fully preserves the distance function: It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry . One perhaps non-obvious example of an isometry between spaces described in this article is the map f : ( R 2 , d 1 ) → ( R 2 , d ∞ ) {\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })} defined by f ( x , y ) = ( x + y , x − y ) . {\displaystyle f(x,y)=(x+y,x-y).} If there

324-399: A broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and Stefan Banach further refined and expanded the framework of metric spaces. Hausdorff introduced topological spaces as a generalization of metric spaces. Banach's work in functional analysis heavily relied on the metric structure. Over time, metric spaces became

405-399: A central part of modern mathematics . They have influenced various fields including topology , geometry , and applied mathematics . Metric spaces continue to play a crucial role in the study of abstract mathematical concepts. A distance function is enough to define notions of closeness and convergence that were first developed in real analysis . Properties that depend on the structure of

486-493: A characterization of metrizability in terms of other topological properties, without reference to metrics. Convergence of sequences in Euclidean space is defined as follows: Convergence of sequences in a topological space is defined as follows: In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in

567-412: A dream in which he dreamt of founding his new city. A variation of this story claims that he built the new palace to find peace from his wife. Charles William founded the city on June 17, 1715, after a dispute with the citizens of his previous capital, Durlach . The founding of the city is closely linked to the construction of the palace . Karlsruhe became the capital of Baden-Durlach , and, in 1771, of

648-564: A mayor was appointed from 1718. From 1812 the mayors received the title of Lord Mayor. In addition to the Lord Mayor, there are five other mayors. Mayor for: The Karlsruhe city council governs the city alongside the Mayor. The most recent city council election was held on 9 June 2024, and the results were as follows: The Verkehrsbetriebe Karlsruhe (VBK) operates the city's urban public transport network, comprising seven tram routes and

729-400: A metric space are referred to as metric properties . Every metric space is also a topological space , and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really topological properties . For any point x in a metric space M and any real number r > 0 , the open ball of radius r around x is defined to be

810-434: A metric space by measuring distances the same way we would in M . Formally, the induced metric on A is a function d A : A × A → R {\displaystyle d_{A}:A\times A\to \mathbb {R} } defined by d A ( x , y ) = d ( x , y ) . {\displaystyle d_{A}(x,y)=d(x,y).} For example, if we take

891-580: A metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane . A metric may correspond to a metaphorical , rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance , which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are

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972-459: A more effective and attractive public transport system. Metric geometry In mathematics , a metric space is a set together with a notion of distance between its elements , usually called points . The distance is measured by a function called a metric or distance function . Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry . The most familiar example of

1053-556: A network of bus routes. All city areas can be reached round the clock by tram and a night bus system. The Turmbergbahn funicular railway , to the east of the city centre, is also operated by the VBK. Similar to a premetro tramlines operating in the city centre use two tramway tunnels that were completed on 11 December 2021. The VBK is also a partner, with the Albtal-Verkehrs-Gesellschaft and Deutsche Bahn , in

1134-400: A purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis. Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges. To make this precise: a sequence ( x n ) in a metric space M is Cauchy if for every ε > 0 there

1215-500: A tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds , normed vector spaces , and graphs . In abstract algebra , the p -adic numbers arise as elements of the completion of a metric structure on the rational numbers . Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces . Many of

1296-492: A vivid and spreading startup community with well-known startups. Together, the local high tech industry is responsible for over 22,000 jobs. The current mayor of Karlsruhe is Frank Mentrup of the Social Democratic Party (SPD) since 2013. The most recent mayoral election was held on 6 December 2020, and the results were as follows: After the castle was founded in 1715, there was also a settlement in which

1377-412: A way of measuring distances between them. Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways. For example, weak solutions to differential equations typically live in a completion (a Sobolev space ) rather than the original space of nice functions for which

1458-487: Is uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for all points x and y in M 1 such that d ( x , y ) < δ {\displaystyle d(x,y)<\delta } , we have d 2 ( f ( x ) , f ( y ) ) < ε . {\displaystyle d_{2}(f(x),f(y))<\varepsilon .} The only difference between this definition and

1539-549: Is K - Lipschitz if d 2 ( f ( x ) , f ( y ) ) ≤ K d 1 ( x , y ) for all x , y ∈ M 1 . {\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.} Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of

1620-775: Is Lebesgue's number lemma , which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover. Unlike in the case of topological spaces or algebraic structures such as groups or rings , there is no single "right" type of structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that ( M 1 , d 1 ) {\displaystyle (M_{1},d_{1})} and ( M 2 , d 2 ) {\displaystyle (M_{2},d_{2})} are two metric spaces. The words "function" and "map" are used interchangeably. One interpretation of

1701-410: Is not a topological property, since R {\displaystyle \mathbb {R} } is complete but the homeomorphic space (0, 1) is not. This notion of "missing points" can be made precise. In fact, every metric space has a unique completion , which is a complete space that contains the given space as a dense subset. For example, [0, 1] is the completion of (0, 1) , and

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1782-441: Is 16.8 km (10.4 mi) and 19.3 km (12.0 mi) in the east–west direction. Karlsruhe is part of the urban area of Karlsruhe/Pforzheim, to which certain other towns in the district of Karlsruhe , such as Bruchsal , Ettlingen , Stutensee , and Rheinstetten , as well as the city of Pforzheim , belong. The city was planned with the palace tower ( Schloss ) at the center and 32 streets radiating out from it like

1863-415: Is a neighborhood of x (informally, it contains all points "close enough" to x ) if it contains an open ball of radius r around x for some r > 0 . An open set is a set which is a neighborhood of all its points. It follows that the open balls form a base for a topology on M . In other words, the open sets of M are exactly the unions of open balls. As in any topology, closed sets are

1944-833: Is a mausoleum rather than a church, and is located in the middle of the forest. The main cemetery of Karlsruhe is the oldest park-like cemetery in Germany. The crematorium was the first to be built in the style of a church. Karlsruhe is also home to a natural history museum (the State Museum of Natural History Karlsruhe ), an opera house (the Baden State Theatre ), as well as a number of independent theatres and art galleries. The State Art Gallery , built in 1846 by Heinrich Hübsch , displays paintings and sculptures from six centuries, particularly from France, Germany and Holland. Karlsruhe's newly renovated art museum

2025-559: Is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between M 1 and M 2 , they are said to be homeomorphic . Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example, R {\displaystyle \mathbb {R} } is unbounded and complete, while (0, 1) is bounded but not complete. A function f : M 1 → M 2 {\displaystyle f\,\colon M_{1}\to M_{2}}

2106-482: Is an integer N such that for all m , n > N , d ( x m , x n ) < ε . By the triangle inequality, any convergent sequence is Cauchy: if x m and x n are both less than ε away from the limit, then they are less than 2ε away from each other. If the converse is true—every Cauchy sequence in M converges—then M is complete. Euclidean spaces are complete, as is R 2 {\displaystyle \mathbb {R} ^{2}} with

2187-435: Is an isometry between the spaces M 1 and M 2 , they are said to be isometric . Metric spaces that are isometric are essentially identical . On the other end of the spectrum, one can forget entirely about the metric structure and study continuous maps , which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are: A homeomorphism

2268-496: Is bounded. To see this, start with a finite cover by r -balls for some arbitrary r . Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D . By the triangle inequality, the diameter of the whole space is at most D + 2 r . The converse does not hold: an example of a metric space that is bounded but not totally bounded is R 2 {\displaystyle \mathbb {R} ^{2}} (or any other infinite set) with

2349-496: Is dedicated to Baden's first constitution in 1818, which was one of the most liberal of its time. The Münze (mint), erected in 1826/27, was also built by Weinbrenner. The St. Stephan parish church is one of the masterpieces of neoclassical church architecture in. Weinbrenner, who built this church between 1808 and 1814, orientated it to the Pantheon, Rome . The neo-Gothic Grand Ducal Burial Chapel, built between 1889 and 1896,

2430-418: Is defined by d 1 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = | x 2 − x 1 | + | y 2 − y 1 | {\displaystyle d_{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{2}-x_{1}|+|y_{2}-y_{1}|} and can be thought of as

2511-476: Is maintained by the university. The Marktplatz has a stone pyramid marking the grave of the city's founder. Built in 1825, it is the emblem of Karlsruhe. The city is nicknamed the "fan city" ( die Fächerstadt ) because of its design layout, with straight streets radiating fan-like from the Palace. The Karlsruhe Palace ( Schloss ) is an interesting piece of architecture; the adjacent Schlossgarten includes

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2592-618: Is one of the most important art museums in Baden-Württemberg . Further cultural attractions are scattered throughout Karlsruhe's various incorporated suburbs. Established in 1924, the Scheffel Association is the largest literary society in Germany. Today the Prinz-Max-Palais , built between 1881 and 1884 in neoclassical style, houses the organisation and includes its museum. Due to population growth in

2673-445: Is the oldest part of town and lies south of the palace in the quadrant defined by nine of the radial streets. The central part of the palace runs east–west, with two wings, each at a 45° angle, directed southeast and southwest (i.e., parallel with the streets marking the boundaries of the quadrant defining the city center). The market square lies on the street running south from the palace to Ettlingen . The market square has

2754-538: The Bundesgerichtshof . The courts came to Karlsruhe after World War II, when the provinces of Baden and Württemberg were merged. Stuttgart , capital of Württemberg, became the capital of the new province ( Württemberg-Baden in 1945 and Baden-Württemberg in 1952). In compensation for the state authorities relocated to Stuttgart, Karlsruhe applied to become the seat of the high court. There are four hospitals: The Karlsruhe Municipal Hospital provides

2835-630: The Dachau concentration camp , Gurs concentration camp , Theresienstadt , and Auschwitz during the Holocaust , with 1,421 of Karlsruhe's Jews being killed. During World War II , it was the location of a forced labour camp for men, and a subcamp of the Auschwitz concentration camp, whose prisoners were mainly Poles and Russians . Much of the central area, including the palace, was reduced to rubble by Allied bombing during World War II, but

2916-538: The Heine–Cantor theorem states that if M 1 is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces. A Lipschitz map is one that stretches distances by at most a bounded factor. Formally, given a real number K > 0 , the map f : M 1 → M 2 {\displaystyle f\,\colon M_{1}\to M_{2}}

2997-552: The Karlsruher Virtueller Katalog , the first internet site that allowed researchers worldwide (for free) to search multiple library catalogues worldwide. In 2000, the regional online newspaper ka-news.de was created. As a daily newspaper, it not only provides the news, but also informs readers about upcoming events in Karlsruhe and surrounding areas. In addition to established companies, Karlsruhe has

3078-711: The Margraviate of Baden (1771–1803), the Electorate of Baden (1803–1806), the Grand Duchy of Baden (1806–1918), and the Republic of Baden (1918–1945). Its most remarkable building is Karlsruhe Palace , which was built in 1715. It contains the Baden State Museum , the large cultural, art and regional history museum of the Baden region of Baden-Württemberg. There are nine institutions of higher education in

3159-688: The Northern Black Forest . The Rhine, one of the world's most important shipping routes, forms the western limits of the city, beyond which lie the towns of Maximiliansau and Wörth am Rhein in the German state of Rhineland-Palatinate . The city centre is about 7.5 km (4.7 mi) from the river, as measured from the Marktplatz (Market Square). Two tributaries of the Rhine, the Alb and

3240-528: The Pfinz , flow through the city from the Kraichgau to eventually join the Rhine. The city lies at an altitude of between 100 and 322 m (328 and 1,056 ft), the higher figure being near the communications tower in the suburb of Grünwettersbach. Its geographical coordinates are 49°00′N 8°24′E  /  49.000°N 8.400°E  / 49.000; 8.400 ; the 49th parallel runs through

3321-529: The Rhine-Ruhr area. Summers are hot with several days registering maximum temperatures between 35 and 40 °C (95 and 104 °F). With an average of more than 2,000 sunshine hours a year, it is also one of the sunniest cities in Germany, like the Rhine-Palatinate area. Precipitation occurs mainly during the winter, while in summer it is concentrated on single evening thunderstorms. In 2008,

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3402-412: The spokes of a wheel, or the ribs of a folding fan , so that one nickname for Karlsruhe in German is the "fan city" ( Fächerstadt ). Almost all of these streets survive to this day. Because of this city layout, in metric geometry , Karlsruhe metric refers to a measure of distance that assumes travel is only possible along radial streets and along circular avenues around the centre. The city centre

3483-468: The surface of the Earth as a set of points. We can measure the distance between two such points by the length of the shortest path along the surface , " as the crow flies "; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in seismology , since it roughly corresponds to

3564-461: The town hall ( Rathaus ) to the west, the main Lutheran church ( Evangelische Stadtkirche ) to the east, and the tomb of Margrave Charles III William in a pyramid in the buildings, resulting in Karlsruhe being one of only three large cities in Germany where buildings are laid out in the neoclassical style. The area north of the palace is a park and forest. Originally the area to the east of

3645-541: The Botanical Garden with a palm, cactus and orchid house, and walking paths through the woods to the north. The so-called Kleine Kirche (Little Church), built between 1773 and 1776, is the oldest church of Karlsruhe's city centre. The architect Friedrich Weinbrenner designed many of the city's most important sights. Another sight is the Rondellplatz with its 'Constitution Building Columns' (1826). It

3726-486: The Euclidean metric and its subspace the interval (0, 1) with the induced metric are homeomorphic but have very different metric properties. Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved in many ways: in particular, they are paracompact Hausdorff spaces (hence normal ) and first-countable . The Nagata–Smirnov metrization theorem gives

3807-436: The basic notions of mathematical analysis , including balls , completeness , as well as uniform , Lipschitz , and Hölder continuity , can be defined in the setting of metric spaces. Other notions, such as continuity , compactness , and open and closed sets , can be defined for metric spaces, but also in the even more general setting of topological spaces . To see the utility of different notions of distance, consider

3888-585: The city centre, which puts it at the same latitude as much of the Canada–United States border and the cities of Vancouver (Canada), Paris (France), Regensburg (Germany), and Hulunbuir (China). Its course is marked by a stone and painted line in the Stadtgarten (municipal park). The total area of the city is 173.46 km (66.97 sq mi), hence it is the 30th largest city in Germany measured by land area. The longest north–south distance

3969-786: The city, most notably the Karlsruhe Institute of Technology . Karlsruhe/Baden-Baden Airport is the second-busiest airport in Baden-Württemberg after Stuttgart Airport , and the 17th-busiest airport in Germany. Karlsruhe lies completely to the east of the Rhine , and almost completely on the Upper Rhine Plain . It contains the Turmberg in the east, and also lies on the borders of the Kraichgau leading to

4050-495: The complements of open sets. Sets may be both open and closed as well as neither open nor closed. This topology does not carry all the information about the metric space. For example, the distances d 1 , d 2 , and d ∞ defined above all induce the same topology on R 2 {\displaystyle \mathbb {R} ^{2}} , although they behave differently in many respects. Similarly, R {\displaystyle \mathbb {R} } with

4131-407: The differential equation actually makes sense. A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. The least such r is called the diameter of M . The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r . Every totally bounded space

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4212-479: The discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points. All of these metrics make sense on R n {\displaystyle \mathbb {R} ^{n}} as well as R 2 {\displaystyle \mathbb {R} ^{2}} . Given a metric space ( M , d ) and a subset A ⊆ M {\displaystyle A\subseteq M} , we can consider A to be

4293-414: The discrete metric. Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces: One example of a compact space is the closed interval [0, 1] . Compactness is important for similar reasons to completeness: it makes it easy to find limits. Another important tool

4374-1066: The distance function d ( x , y ) = | y − x | {\displaystyle d(x,y)=|y-x|} given by the absolute difference form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in real analysis and coincide with those concepts when applied to the real line. The Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} can be equipped with many different metrics. The Euclidean distance familiar from school mathematics can be defined by d 2 ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . {\displaystyle d_{2}((x_{1},y_{1}),(x_{2},y_{2}))={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.} The taxicab or Manhattan distance

4455-774: The distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article. The maximum , L ∞ {\displaystyle L^{\infty }} , or Chebyshev distance is defined by d ∞ ( ( x 1 , y 1 ) , ( x 2 , y 2 ) ) = max { | x 2 − x 1 | , | y 2 − y 1 | } . {\displaystyle d_{\infty }((x_{1},y_{1}),(x_{2},y_{2}))=\max\{|x_{2}-x_{1}|,|y_{2}-y_{1}|\}.} This distance does not have an easy explanation in terms of paths in

4536-515: The field of non-euclidean geometry through the use of the Cayley-Klein metric . The idea of an abstract space with metric properties was addressed in 1906 by René Maurice Fréchet and the term metric space was coined by Felix Hausdorff in 1914. Fréchet's work laid the foundation for understanding convergence , continuity , and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in

4617-1168: The formula d ∞ ( p , q ) ≤ d 2 ( p , q ) ≤ d 1 ( p , q ) ≤ 2 d ∞ ( p , q ) , {\displaystyle d_{\infty }(p,q)\leq d_{2}(p,q)\leq d_{1}(p,q)\leq 2d_{\infty }(p,q),} which holds for every pair of points p , q ∈ R 2 {\displaystyle p,q\in \mathbb {R} ^{2}} . A radically different distance can be defined by setting d ( p , q ) = { 0 , if  p = q , 1 , otherwise. {\displaystyle d(p,q)={\begin{cases}0,&{\text{if }}p=q,\\1,&{\text{otherwise.}}\end{cases}}} Using Iverson brackets , d ( p , q ) = [ p ≠ q ] {\displaystyle d(p,q)=[p\neq q]} In this discrete metric , all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either. Intuitively,

4698-416: The late 1990, Karlsruhe became known as the internet capital of Germany. The DENIC , Germany's network information centre , has since moved to Frankfurt, though, where DE-CIX is located. Two major internet service providers , WEB.DE and schlund+partner / 1&1 , now both owned by United Internet  AG, are located at Karlsruhe. The library of the Karlsruhe Institute of Technology developed

4779-616: The late 19th century, Karlsruhe developed several suburban areas ( Vorstadt ) in the Gründerzeit and especially Art Nouveau styles of architecture, with many preserved examples. Karlsruhe is also home to the Majolika-Manufaktur , the only art-ceramics pottery studio in Germany. Founded in 1901, it is located in the Schlossgarten . A 'blue streak' ( Blauer Strahl ) consisting of 1,645 ceramic tiles, connects

4860-481: The length of time it takes for seismic waves to travel between those two points. The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts. Like many fundamental mathematical concepts,

4941-561: The maximum level of medical services, the St. Vincentius-Kliniken and the Diakonissen krankenhaus , connected to the Catholic and Protestant churches, respectively, offer central services, and the private Paracelsus-Klinik basic medical care, according to state hospital demand planning. Germany's largest oil refinery is located in Karlsruhe, at the western edge of the city, directly on

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5022-430: The metric d is unambiguous, one often refers by abuse of notation to "the metric space M ". By taking all axioms except the second, one can show that distance is always non-negative: 0 = d ( x , x ) ≤ d ( x , y ) + d ( y , x ) = 2 d ( x , y ) {\displaystyle 0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)} Therefore

5103-531: The metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures ) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov–Hausdorff distance between metric spaces themselves). Formally,

5184-584: The operation of the Karlsruhe Stadtbahn , the rail system that serves a larger area around the city. This system makes it possible to reach other towns in the region, like Ettlingen , Wörth am Rhein , Pforzheim , Bad Wildbad , Bretten , Bruchsal , Heilbronn , Baden-Baden , and even Freudenstadt in the Black Forest right from the city centre. The Stadtbahn is known for pioneering the concept of operating trams on train tracks, to achieve

5265-543: The other metrics described above. Two examples of spaces which are not complete are (0, 1) and the rationals, each with the metric induced from R {\displaystyle \mathbb {R} } . One can think of (0, 1) as "missing" its endpoints 0 and 1. The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in R {\displaystyle \mathbb {R} } (for example, its successive decimal approximations). These examples show that completeness

5346-411: The palace consisted of gardens and forests, some of which remain, but the Karlsruhe Institute of Technology (founded in 1825), Wildparkstadion football stadium, and residential areas have been built there. The area west of the palace is now mostly residential. Karlsruhe experiences an oceanic climate ( Köppen : Cfb ) and its winter climate is milder, compared to most other German cities, except for

5427-428: The plane, but it still satisfies the metric space axioms. It can be thought of similarly to the number of moves a king would have to make on a chess board to travel from one point to another on the given space. In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with

5508-472: The real line. Arthur Cayley , in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His distance was given by logarithm of a cross ratio . Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for elliptic geometry and hyperbolic geometry , and Felix Klein , in several publications, established

5589-407: The real numbers are the completion of the rationals. Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, in abstract algebra, the p -adic numbers are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in functional analysis . Often one has a set of nice functions and

5670-431: The river Rhine . The Technologieregion Karlsruhe is a loose confederation of the region's cities in order to promote high tech industries; today, about 20% of the region's jobs are in research and development . EnBW , one of Germany's biggest electric utility companies, with a revenue of €19.2 billion in 2012, is headquartered in the city. Due to the Karlsruhe Institute of Technology providing services until

5751-415: The second axiom can be weakened to If  x ≠ y , then  d ( x , y ) ≠ 0 {\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0} and combined with the first to make d ( x , y ) = 0 ⟺ x = y {\textstyle d(x,y)=0\iff x=y} . The real numbers with

5832-482: The set of points that are strictly less than distance r from x : B r ( x ) = { y ∈ M : d ( x , y ) < r } . {\displaystyle B_{r}(x)=\{y\in M:d(x,y)<r\}.} This is a natural way to define a set of points that are relatively close to x . Therefore, a set N ⊆ M {\displaystyle N\subseteq M}

5913-545: The studio with the Palace. It is the world's largest ceramic artwork. Another tourist attraction is the Centre for Art and Media ( Zentrum für Kunst und Medientechnologie , or ZKM), which is located in a converted ammunition factory. Karlsruhe is the seat of the German Federal Constitutional Court (Bundesverfassungsgericht) and the highest Court of Appeals in civil and criminal cases,

5994-410: The two-dimensional sphere S as a subset of R 3 {\displaystyle \mathbb {R} ^{3}} , the Euclidean metric on R 3 {\displaystyle \mathbb {R} ^{3}} induces the straight-line metric on S described above. Two more useful examples are the open interval (0, 1) and the closed interval [0, 1] thought of as subspaces of

6075-575: The united Baden until 1945. Built in 1822, the Ständehaus was the first parliament building in a German state. In the aftermath of the democratic revolution of 1848, a republican government was elected there. Karlsruhe was visited by Thomas Jefferson during his time as the American envoy to France; when Pierre Charles L'Enfant was planning the layout of Washington, D.C. , Jefferson passed to him maps of 12 European towns to consult, one of which

6156-480: The weather station in Karlsruhe, which had been in operation since 1876, was closed; it was replaced by a weather station in Rheinstetten , south of Karlsruhe. Karlsruhe is divided into 27 districts. According to legend, the name Karlsruhe , which translates as "Charles' repose" or "Charles' peace", was given to the new city after a hunting trip when Margrave Charles III William of Baden-Durlach woke from

6237-413: The ε–δ definition of continuity is the order of quantifiers: the choice of δ must depend only on ε and not on the point x . However, this subtle change makes a big difference. For example, uniformly continuous maps take Cauchy sequences in M 1 to Cauchy sequences in M 2 . In other words, uniform continuity preserves some metric properties which are not purely topological. On the other hand,

6318-619: Was a sketch he had made of Karlsruhe during his visit. In 1860, the first-ever international professional convention of chemists, the Karlsruhe Congress , was held in the city. In 1907 the town was site of the Hau Riot where large crowds caused disturbance during the trial of murderer Carl Hau . On Kristallnacht in 1938, the Adass Jeshurun synagogue was burned to the ground, and the city's Jews were later sent to

6399-419: Was founded by Charles III William , became a major city in the 19th century. In the 1950s, Karlsruhe became a significant city where the population started to grow. It gained a large student population due to the university of technology and media arts. Karlsruhe reached populations of 200,000 in 1950 and 300,000 in 2014. The Stadtgarten is a recreational area near the main railway station ( Hauptbahnhof ) and

6480-435: Was rebuilt after the war. Located in the American zone of the postwar Allied occupation , Karlsruhe was home to an American military base, established in 1945. After the war, the city was part of West Germany until 1990. In 1995, the bases closed , and their facilities were turned over to the city of Karlsruhe. Karlsruhe has a population of about 310,000 and is the 3rd largest city in Baden-Württemberg . Karlsruhe, which

6561-556: Was rebuilt for the 1967 Federal Garden Show ( Bundesgartenschau ). It is also the site of the Karlsruhe Zoo . The Durlacher Turmberg has a lookout tower (hence its name). It is a former keep dating back to the 13th century. The city has two botanical gardens: the municipal Botanischer Garten Karlsruhe , which forms part of the Palace complex, and the Botanischer Garten der Universität Karlsruhe , which

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