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91-465: Topology (from the Greek words τόπος , 'place, location', and λόγος , 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations , such as stretching , twisting , crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space

182-577: A Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the world's oldest recorded living language . Among the Indo-European languages, its date of earliest written attestation is matched only by the now-extinct Anatolian languages . The Greek language is conventionally divided into the following periods: In the modern era, the Greek language entered

273-598: A group , called the homeomorphism group of X , often denoted Homeo ( X ) . {\textstyle {\text{Homeo}}(X).} This group can be given a topology, such as the compact-open topology , which under certain assumptions makes it a topological group . In some contexts, there are homeomorphic objects that cannot be continuously deformed from one to the other. Homotopy and isotopy are equivalence relations that have been introduced for dealing with such situations. Similarly, as usual in category theory, given two spaces that are homeomorphic,

364-411: A robot can be described by a manifold called configuration space . In the area of motion planning , one finds paths between two points in configuration space. These paths represent a motion of the robot's joints and other parts into the desired pose. Disentanglement puzzles are based on topological aspects of the puzzle's shapes and components. In order to create a continuous join of pieces in

455-497: A smooth structure on a manifold to be defined. Smooth manifolds are "softer" than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold – that is, one can smoothly "flatten out" certain manifolds, but it might require distorting

546-577: A basis for coinages: anthropology , photography , telephony , isomer , biomechanics , cinematography , etc. Together with Latin words , they form the foundation of international scientific and technical vocabulary ; for example, all words ending in -logy ('discourse'). There are many English words of Greek origin . Greek is an independent branch of the Indo-European language family. The ancient language most closely related to it may be ancient Macedonian , which, by most accounts,

637-417: A certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a line segment to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts. In this case, for example, the line segment possesses infinitely many points, and therefore cannot be put into a bijection with a set containing only

728-411: A convenient proof that any subgroup of a free group is again a free group. Differential topology is the field dealing with differentiable functions on differentiable manifolds . It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require only

819-508: A faster, more convenient cursive writing style with the use of ink and quill . The Greek alphabet consists of 24 letters, each with an uppercase ( majuscule ) and lowercase ( minuscule ) form. The letter sigma has an additional lowercase form (ς) used in the final position of a word: In addition to the letters, the Greek alphabet features a number of diacritical signs : three different accent marks ( acute , grave , and circumflex ), originally denoting different shapes of pitch accent on

910-495: A finite number of points, including a single point. This characterization of a homeomorphism often leads to a confusion with the concept of homotopy , which is actually defined as a continuous deformation, but from one function to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space X correspond to which points on Y —one just follows them as X deforms. In

1001-470: A function exists, X {\displaystyle X} and Y {\displaystyle Y} are homeomorphic . A self-homeomorphism is a homeomorphism from a topological space onto itself. Being "homeomorphic" is an equivalence relation on topological spaces. Its equivalence classes are called homeomorphism classes . The third requirement, that f − 1 {\textstyle f^{-1}} be continuous ,

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1092-426: A given space. Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric . In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x

1183-420: A homeomorphism and the domain of the function is said to be homeomorphic to the range. Another way of saying this is that the function has a natural extension to the topology. If two spaces are homeomorphic, they have identical topological properties, and are considered topologically the same. The cube and the sphere are homeomorphic, as are the coffee cup and the doughnut. However, the sphere is not homeomorphic to

1274-590: A lesser extent, in the Western Mediterranean in and around colonies such as Massalia , Monoikos , and Mainake . It was also used as the official language of government and religion in the Christian Nubian kingdoms , for most of their history. Greek, in its modern form, is the official language of Greece, where it is spoken by almost the entire population. It is also the official language of Cyprus (nominally alongside Turkish ) and

1365-400: A limited but productive system of compounding and a rich inflectional system. Although its morphological categories have been fairly stable over time, morphological changes are present throughout, particularly in the nominal and verbal systems. The major change in the nominal morphology since the classical stage was the disuse of the dative case (its functions being largely taken over by

1456-459: A mixed syllable structure, permitting complex syllabic onsets but very restricted codas. It has only oral vowels and a fairly stable set of consonantal contrasts . The main phonological changes occurred during the Hellenistic and Roman period (see Koine Greek phonology for details): In all its stages, the morphology of Greek shows an extensive set of productive derivational affixes ,

1547-527: A modular construction, it is necessary to create an unbroken path in an order which surrounds each piece and traverses each edge only once. This process is an application of the Eulerian path . Greek language Greek ( Modern Greek : Ελληνικά , romanized :  Elliniká , [eliniˈka] ; Ancient Greek : Ἑλληνική , romanized :  Hellēnikḗ ) is an Indo-European language, constituting an independent Hellenic branch within

1638-533: A set (for instance, determining if a cloud of points is spherical or toroidal ). The main method used by topological data analysis is to: Several branches of programming language semantics , such as domain theory , are formalized using topology. In this context, Steve Vickers , building on work by Samson Abramsky and Michael B. Smyth , characterizes topological spaces as Boolean or Heyting algebras over open sets, which are characterized as semidecidable (equivalently, finitely observable) properties. Topology

1729-409: A state of diglossia : the coexistence of vernacular and archaizing written forms of the language. What came to be known as the Greek language question was a polarization between two competing varieties of Modern Greek: Dimotiki , the vernacular form of Modern Greek proper, and Katharevousa , meaning 'purified', a compromise between Dimotiki and Ancient Greek developed in the early 19th century that

1820-466: A topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not result into homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as

1911-457: Is a π -system . The members of τ are called open sets in X . A subset of X is said to be closed if its complement is in τ (that is, its complement is open). A subset of X may be open, closed, both (a clopen set ), or neither. The empty set and X itself are always both closed and open. An open subset of X which contains a point x is called an open neighborhood of x . A function or map from one topological space to another

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2002-409: Is a set endowed with a structure, called a topology , which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity . Euclidean spaces , and, more generally, metric spaces are examples of topological spaces, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies . A property that

2093-660: Is a quantum field theory that computes topological invariants . Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory , the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson , Jones , Witten , and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory , as different manifolds can sustain different kinds of strings. In cosmology, topology can be used to describe

2184-689: Is bijective and continuous, but not a homeomorphism ( S 1 {\textstyle S^{1}} is compact but [ 0 , 2 π ) {\textstyle [0,2\pi )} is not). The function f − 1 {\textstyle f^{-1}} is not continuous at the point ( 1 , 0 ) , {\textstyle (1,0),} because although f − 1 {\textstyle f^{-1}} maps ( 1 , 0 ) {\textstyle (1,0)} to 0 , {\textstyle 0,} any neighbourhood of this point also includes points that

2275-406: Is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus . If a continuous function is one-to-one and onto , and if the inverse of the function is also continuous, then the function is called

2366-523: Is essential. Consider for instance the function f : [ 0 , 2 π ) → S 1 {\textstyle f:[0,2\pi )\to S^{1}} (the unit circle in ⁠ R 2 {\displaystyle \mathbb {R} ^{2}} ⁠ ) defined by f ( φ ) = ( cos ⁡ φ , sin ⁡ φ ) . {\textstyle f(\varphi )=(\cos \varphi ,\sin \varphi ).} This function

2457-444: Is invariant under such deformations is a topological property . The following are basic examples of topological properties: the dimension , which allows distinguishing between a line and a surface ; compactness , which allows distinguishing between a line and a circle; connectedness , which allows distinguishing a circle from two non-intersecting circles. The ideas underlying topology go back to Gottfried Wilhelm Leibniz , who in

2548-776: Is point-set topology. The basic object of study is topological spaces , which are sets equipped with a topology , that is, a family of subsets , called open sets , which is closed under finite intersections and (finite or infinite) unions . The fundamental concepts of topology, such as continuity , compactness , and connectedness , can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words nearby , arbitrarily small , and far apart can all be made precise by using open sets. Several topologies can be defined on

2639-451: Is relevant to physics in areas such as condensed matter physics , quantum field theory and physical cosmology . The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science . Electrical and mechanical properties depend on the arrangement and network structures of molecules and elementary units in materials. The compressive strength of crumpled topologies

2730-843: Is sometimes called aljamiado , as when Romance languages are written in the Arabic alphabet. Article 1 of the Universal Declaration of Human Rights in Greek: Transcription of the example text into Latin alphabet : Article 1 of the Universal Declaration of Human Rights in English: Proto-Greek Mycenaean Ancient Koine Medieval Modern Homeomorphism Very roughly speaking,

2821-763: Is still used internationally for the writing of Ancient Greek . In Greek, the question mark is written as the English semicolon, while the functions of the colon and semicolon are performed by a raised point (•), known as the ano teleia ( άνω τελεία ). In Greek the comma also functions as a silent letter in a handful of Greek words, principally distinguishing ό,τι ( ó,ti , 'whatever') from ότι ( óti , 'that'). Ancient Greek texts often used scriptio continua ('continuous writing'), which means that ancient authors and scribes would write word after word with no spaces or punctuation between words to differentiate or mark boundaries. Boustrophedon , or bi-directional text,

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2912-463: Is studied in attempts to understand the high strength to weight of such structures that are mostly empty space. Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT)

3003-437: Is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it

3094-427: Is the set of all points whose distance to x is less than r . Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line , the complex plane , real and complex vector spaces and Euclidean spaces . Having a metric simplifies many proofs. Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal

3185-632: Is to modern spoken English ". Greek is spoken today by at least 13 million people, principally in Greece and Cyprus along with a sizable Greek-speaking minority in Albania near the Greek-Albanian border. A significant percentage of Albania's population has knowledge of the Greek language due in part to the Albanian wave of immigration to Greece in the 1980s and '90s and the Greek community in

3276-437: Is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups , homology, and cohomology . Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for

3367-562: The British Overseas Territory of Akrotiri and Dhekelia (alongside English ). Because of the membership of Greece and Cyprus in the European Union, Greek is one of the organization's 24 official languages . Greek is recognized as a minority language in Albania, and used co-officially in some of its municipalities, in the districts of Gjirokastër and Sarandë . It is also an official minority language in

3458-657: The European canon . Greek is also the language in which many of the foundational texts in science and philosophy were originally composed. The New Testament of the Christian Bible was also originally written in Greek. Together with the Latin texts and traditions of the Roman world , the Greek texts and Greek societies of antiquity constitute the objects of study of the discipline of Classics . During antiquity , Greek

3549-516: The Indo-Iranian languages (see Graeco-Aryan ), but little definitive evidence has been found. In addition, Albanian has also been considered somewhat related to Greek and Armenian, and it has been proposed that they all form a higher-order subgroup along with other extinct languages of the ancient Balkans; this higher-order subgroup is usually termed Palaeo-Balkan , and Greek has a central position in it. Linear B , attested as early as

3640-780: The United Kingdom , and throughout the European Union , especially in Germany . Historically, significant Greek-speaking communities and regions were found throughout the Eastern Mediterranean , in what are today Southern Italy , Turkey , Cyprus , Syria , Lebanon , Israel , Palestine , Egypt , and Libya ; in the area of the Black Sea , in what are today Turkey, Bulgaria , Romania , Ukraine , Russia , Georgia , Armenia , and Azerbaijan ; and, to

3731-468: The genitive ). The verbal system has lost the infinitive , the synthetically -formed future, and perfect tenses and the optative mood . Many have been replaced by periphrastic ( analytical ) forms. Pronouns show distinctions in person (1st, 2nd, and 3rd), number (singular, dual , and plural in the ancient language; singular and plural alone in later stages), and gender (masculine, feminine, and neuter), and decline for case (from six cases in

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3822-429: The geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable ( Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from

3913-465: The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick ." This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg , the result does not depend on

4004-597: The plane , the sphere, and the torus, which can all be realized without self-intersection in three dimensions, and the Klein bottle and real projective plane , which cannot (that is, all their realizations are surfaces that are not manifolds). General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology

4095-505: The real line , the complex plane , and the Cantor set can be thought of as the same set with different topologies. Formally, let X be a set and let τ be a family of subsets of X . Then τ is called a topology on X if: If τ is a topology on X , then the pair ( X , τ ) is called a topological space. The notation X τ may be used to denote a set X endowed with the particular topology τ . By definition, every topology

4186-417: The syntax of Greek have remained constant: verbs agree with their subject only, the use of the surviving cases is largely intact (nominative for subjects and predicates, accusative for objects of most verbs and many prepositions, genitive for possessors), articles precede nouns, adpositions are largely prepositional, relative clauses follow the noun they modify and relative pronouns are clause-initial. However,

4277-443: The 17th century envisioned the geometria situs and analysis situs . Leonhard Euler 's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although, it was not until the first decades of the 20th century that the idea of a topological space was developed. The motivating insight behind topology

4368-566: The 20th century on), especially from French and English, are typically not inflected; other modern borrowings are derived from Albanian , South Slavic ( Macedonian / Bulgarian ) and Eastern Romance languages ( Aromanian and Megleno-Romanian ). Greek words have been widely borrowed into other languages, including English. Example words include: mathematics , physics , astronomy , democracy , philosophy , athletics , theatre, rhetoric , baptism , evangelist , etc. Moreover, Greek words and word elements continue to be productive as

4459-544: The Greek alphabet since approximately the 9th century BC. It was created by modifying the Phoenician alphabet , with the innovation of adopting certain letters to represent the vowels. The variant of the alphabet in use today is essentially the late Ionic variant, introduced for writing classical Attic in 403 BC. In classical Greek, as in classical Latin, only upper-case letters existed. The lower-case Greek letters were developed much later by medieval scribes to permit

4550-549: The Greek language was the Cypriot syllabary (also a descendant of Linear A via the intermediate Cypro-Minoan syllabary ), which is closely related to Linear B but uses somewhat different syllabic conventions to represent phoneme sequences. The Cypriot syllabary is attested in Cyprus from the 11th century BC until its gradual abandonment in the late Classical period, in favor of the standard Greek alphabet. Greek has been written in

4641-593: The Indo-European language family. It is native to Greece , Cyprus , Italy (in Calabria and Salento ), southern Albania , and other regions of the Balkans , Caucasus , the Black Sea coast, Asia Minor , and the Eastern Mediterranean . It has the longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet , which has been used for approximately 2,800 years; previously, Greek

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4732-404: The acute during the late 20th century, and it has only been retained in typography . After the writing reform of 1982, most diacritics are no longer used. Since then, Greek has been written mostly in the simplified monotonic orthography (or monotonic system), which employs only the acute accent and the diaeresis. The traditional system, now called the polytonic orthography (or polytonic system),

4823-479: The arrival of Proto-Greeks, some documented in Mycenaean texts ; they include a large number of Greek toponyms . The form and meaning of many words have changed. Loanwords (words of foreign origin) have entered the language, mainly from Latin, Venetian , and Turkish . During the older periods of Greek, loanwords into Greek acquired Greek inflections, thus leaving only a foreign root word. Modern borrowings (from

4914-419: The case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: homotopy equivalence . There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an isotopy between

5005-507: The concepts now known as homotopy and homology , which are now considered part of algebraic topology . Unifying the work on function spaces of Georg Cantor , Vito Volterra , Cesare Arzelà , Jacques Hadamard , Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space potentially giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined

5096-796: The country. Prior to the Greco-Turkish War and the resulting population exchange in 1923 a very large population of Greek-speakers also existed in Turkey , though very few remain today. A small Greek-speaking community is also found in Bulgaria near the Greek-Bulgarian border. Greek is also spoken worldwide by the sizable Greek diaspora which has notable communities in the United States , Australia , Canada , South Africa , Chile , Brazil , Argentina , Russia , Ukraine ,

5187-441: The dative led to a rise of prepositional indirect objects (and the use of the genitive to directly mark these as well). Ancient Greek tended to be verb-final, but neutral word order in the modern language is VSO or SVO. Modern Greek inherits most of its vocabulary from Ancient Greek, which in turn is an Indo-European language, but also includes a number of borrowings from the languages of the populations that inhabited Greece before

5278-457: The definition of sheaves on those categories, and with that the definition of general cohomology theories. Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecular chains based on

5369-643: The doughnut. While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n -dimensional manifold has a neighborhood that is homeomorphic to the Euclidean space of dimension n . Lines and circles , but not figure eights , are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces , although not all surfaces are manifolds. Examples include

5460-528: The earliest forms attested to four in the modern language). Nouns, articles, and adjectives show all the distinctions except for a person. Both attributive and predicative adjectives agree with the noun. The inflectional categories of the Greek verb have likewise remained largely the same over the course of the language's history but with significant changes in the number of distinctions within each category and their morphological expression. Greek verbs have synthetic inflectional forms for: Many aspects of

5551-412: The extent that one can speak of a new language emerging. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language. It is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, " Homeric Greek is probably closer to Demotic than 12-century Middle English

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5642-434: The function maps close to 2 π , {\textstyle 2\pi ,} but the points it maps to numbers in between lie outside the neighbourhood. Homeomorphisms are the isomorphisms in the category of topological spaces . As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms X → X {\textstyle X\to X} forms

5733-407: The hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking

5824-401: The hole into a handle. Homeomorphism can be considered the most basic topological equivalence . Another is homotopy equivalence . This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object. Topology, as a well-defined mathematical discipline, originates in the early part of

5915-418: The homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous deformation . A function f : X → Y {\displaystyle f:X\to Y} between two topological spaces is a homeomorphism if it has the following properties: A homeomorphism is sometimes called a bicontinuous function. If such

6006-449: The language show both conservative and innovative tendencies across the entire attestation of the language from the ancient to the modern period. The division into conventional periods is, as with all such periodizations, relatively arbitrary, especially because, in all periods, Ancient Greek has enjoyed high prestige, and the literate borrowed heavily from it. Across its history, the syllabic structure of Greek has varied little: Greek shows

6097-467: The late 15th century BC, was the first script used to write Greek. It is basically a syllabary , which was finally deciphered by Michael Ventris and John Chadwick in the 1950s (its precursor, Linear A , has not been deciphered and most likely encodes a non-Greek language). The language of the Linear B texts, Mycenaean Greek , is the earliest known form of Greek. Another similar system used to write

6188-452: The many other countries of the Greek diaspora . Greek roots have been widely used for centuries and continue to be widely used to coin new words in other languages; Greek and Latin are the predominant sources of international scientific vocabulary . Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, or possibly earlier. The earliest written evidence is

6279-465: The morphological changes also have their counterparts in the syntax, and there are also significant differences between the syntax of the ancient and that of the modern form of the language . Ancient Greek made great use of participial constructions and of constructions involving the infinitive, and the modern variety lacks the infinitive entirely (employing a raft of new periphrastic constructions instead) and uses participles more restrictively. The loss of

6370-480: The number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theorem, signaling the birth of topology. Further contributions were made by Augustin-Louis Cauchy , Ludwig Schläfli , Johann Benedict Listing , Bernhard Riemann and Enrico Betti . Listing introduced the term "Topologie" in Vorstudien zur Topologie , written in his native German, in 1847, having used

6461-473: The overall shape of the universe . This area of research is commonly known as spacetime topology . In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect , and then generalized in other areas of physics, for instance in photonics by F.D.M Haldane . The possible positions of

6552-464: The pairwise arrangement of their intra-chain contacts and chain crossings. Knot theory , a branch of topology, is used in biology to study the effects of certain enzymes on DNA. These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis . Topological data analysis uses techniques from algebraic topology to determine the large scale structure of

6643-481: The planar and higher-dimensional Schönflies theorem . In high-dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature /spherical, zero curvature/flat, and negative curvature/hyperbolic – and

6734-425: The point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure. Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow

6825-643: The regions of Apulia and Calabria in Italy. In the framework of the European Charter for Regional or Minority Languages , Greek is protected and promoted officially as a regional and minority language in Armenia, Hungary , Romania, and Ukraine. It is recognized as a minority language and protected in Turkey by the 1923 Treaty of Lausanne . The phonology , morphology , syntax , and vocabulary of

6916-490: The shape of the sphere; it applies to any kind of smooth blob, as long as it has no holes. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism . The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and

7007-407: The space and affecting the curvature or volume. Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability , handle decompositions , local flatness , crumpling and

7098-625: The space of homeomorphisms between them, Homeo ( X , Y ) , {\textstyle {\text{Homeo}}(X,Y),} is a torsor for the homeomorphism groups Homeo ( X ) {\textstyle {\text{Homeo}}(X)} and Homeo ( Y ) , {\textstyle {\text{Homeo}}(Y),} and, given a specific homeomorphism between X {\displaystyle X} and Y , {\displaystyle Y,} all three sets are identified. The intuitive criterion of stretching, bending, cutting and gluing back together takes

7189-437: The stressed vowel; the so-called breathing marks ( rough and smooth breathing ), originally used to signal presence or absence of word-initial /h/; and the diaeresis , used to mark the full syllabic value of a vowel that would otherwise be read as part of a diphthong. These marks were introduced during the course of the Hellenistic period. Actual usage of the grave in handwriting saw a rapid decline in favor of uniform usage of

7280-649: The term "topological space" and gave the definition for what is now called a Hausdorff space . Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski . Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series . For further developments, see point-set topology and algebraic topology. The 2022 Abel Prize

7371-570: The twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler . His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron . This led to his polyhedron formula , V − E + F = 2 (where V , E , and F respectively indicate

7462-515: The word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". Their work was corrected, consolidated and greatly extended by Henri Poincaré . In 1895, he published his ground-breaking paper on Analysis Situs , which introduced

7553-574: Was a distinct dialect of Greek itself. Aside from the Macedonian question, current consensus regards Phrygian as the closest relative of Greek, since they share a number of phonological, morphological and lexical isoglosses , with some being exclusive between them. Scholars have proposed a Graeco-Phrygian subgroup out of which Greek and Phrygian originated. Among living languages, some Indo-Europeanists suggest that Greek may be most closely related to Armenian (see Graeco-Armenian ) or

7644-557: Was also used in Ancient Greek. Greek has occasionally been written in the Latin script , especially in areas under Venetian rule or by Greek Catholics . The term Frankolevantinika / Φραγκολεβαντίνικα applies when the Latin script is used to write Greek in the cultural ambit of Catholicism (because Frankos / Φράγκος is an older Greek term for West-European dating to when most of (Roman Catholic Christian) West Europe

7735-435: Was awarded to Dennis Sullivan "for his groundbreaking contributions to topology in its broadest sense, and in particular its algebraic, geometric and dynamical aspects". The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance,

7826-556: Was by far the most widely spoken lingua franca in the Mediterranean world . It eventually became the official language of the Byzantine Empire and developed into Medieval Greek . In its modern form , Greek is the official language of Greece and Cyprus and one of the 24 official languages of the European Union . It is spoken by at least 13.5 million people today in Greece, Cyprus, Italy, Albania, Turkey , and

7917-439: Was impossible to find a route through the town of Königsberg (now Kaliningrad ) that would cross each of its seven bridges exactly once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory . Similarly,

8008-485: Was recorded in writing systems such as Linear B and the Cypriot syllabary . The alphabet arose from the Phoenician script and was in turn the basis of the Latin , Cyrillic , Coptic , Gothic , and many other writing systems. The Greek language holds a very important place in the history of the Western world. Beginning with the epics of Homer , ancient Greek literature includes many works of lasting importance in

8099-580: Was under the control of the Frankish Empire ). Frankochiotika / Φραγκοχιώτικα (meaning 'Catholic Chiot') alludes to the significant presence of Catholic missionaries based on the island of Chios . Additionally, the term Greeklish is often used when the Greek language is written in a Latin script in online communications. The Latin script is nowadays used by the Greek-speaking communities of Southern Italy . The Yevanic dialect

8190-637: Was used for literary and official purposes in the newly formed Greek state. In 1976, Dimotiki was declared the official language of Greece, after having incorporated features of Katharevousa and thus giving birth to Standard Modern Greek , used today for all official purposes and in education . The historical unity and continuing identity between the various stages of the Greek language are often emphasized. Although Greek has undergone morphological and phonological changes comparable to those seen in other languages, never since classical antiquity has its cultural, literary, and orthographic tradition been interrupted to

8281-673: Was written by Romaniote and Constantinopolitan Karaite Jews using the Hebrew Alphabet . In a tradition, that in modern time, has come to be known as Greek Aljamiado , some Greek Muslims from Crete wrote their Cretan Greek in the Arabic alphabet . The same happened among Epirote Muslims in Ioannina . This also happened among Arabic-speaking Byzantine rite Christians in the Levant ( Lebanon , Palestine, and Syria ). This usage

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