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46-434: René Maurice Fréchet ( French: [ʁəne mɔʁis fʁeʃɛ, moʁ-] ; 2 September 1878 – 4 June 1973) was a French mathematician. He made major contributions to general topology and was the first to define metric spaces . He also made several important contributions to the field of statistics and probability , as well as calculus . His dissertation opened the entire field of functionals on metric spaces and introduced

92-405: A topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B . We say that the base generates the topology T . Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of

138-437: A topological space . The notation X τ may be used to denote a set X endowed with the particular topology τ . 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 (i.e., its complement is open). A subset of X may be open, closed, both ( clopen set ), or neither. The empty set and X itself are always both closed and open. A base (or basis ) B for

184-434: A base that generates them. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of the open sets of the larger space with the subset. For any indexed family of topological spaces, the product can be given the product topology , which is generated by the inverse images of open sets of the factors under the projection mappings. For example, in finite products,

230-403: A basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if X is a topological space and Y is a set, and if f  : X → Y is a surjective function , then the quotient topology on Y

276-399: A continuous function stays continuous if the topology τ Y is replaced by a coarser topology and/or τ X is replaced by a finer topology . Symmetric to the concept of a continuous map is an open map , for which images of open sets are open. In fact, if an open map f has an inverse function , that inverse is continuous, and if a continuous map g has an inverse, that inverse

322-458: A function between topological spaces is continuous in the sense above if and only if for all subsets A of X That is to say, given any element x of X that is in the closure of any subset A , f ( x ) belongs to the closure of f ( A ). This is equivalent to the requirement that for all subsets A ' of X ' Moreover, is continuous if and only if for any subset A of X . If f : X → Y and g : Y → Z are continuous, then so

368-439: A half years very near to or at the front. French egalitarian ideals caused many academics to be mobilised. They served in the trenches and many of them were lost during the war. It is remarkable that during his service in the war, he still managed to produce cutting edge mathematical papers frequently, despite having little time to devote to mathematics. After the end of the war, Fréchet was chosen to go to Strasbourg to help with

414-782: A member of the Académie des sciences until the age of 78. In 1929 he became foreign member of the Polish Academy of Arts and Sciences and in 1950 foreign member of the Royal Netherlands Academy of Arts and Sciences . Fréchet was an Esperantist , publishing some papers and articles in that constructed language. He also served as president of the Internacia Scienca Asocio Esperantista ("International Scientific Esperantist Association") from 1950 to 1953. His first major work

460-475: A set is given a different topology, it is viewed as a different topological space. Any set can be given the discrete topology , in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the trivial topology (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of

506-491: A set may have many distinct topologies defined on it. Every metric space can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any normed vector space . On a finite-dimensional vector space this topology is the same for all norms. Continuity is expressed in terms of neighborhoods : f is continuous at some point x  ∈  X if and only if for any neighborhood V of f ( x ) , there

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552-500: A similar idea can be applied to maps X → S . {\displaystyle X\rightarrow S.} Formally, a topological space X is called compact if each of its open covers has a finite subcover . Otherwise it is called non-compact . Explicitly, this means that for every arbitrary collection of open subsets of X such that there is a finite subset J of A such that Some branches of mathematics such as algebraic geometry , typically influenced by

598-473: A standard topology in which the basic open sets are open balls. The real line can also be given the lower limit topology . Here, the basic open sets are the half open intervals [ a , b ). This topology on R is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that

644-432: Is surjective , this topology is canonically identified with the quotient topology under the equivalence relation defined by f . Dually, for a function f from a set S to a topological space X , the initial topology on S has a basis of open sets given by those sets of the form f^(-1) ( U ) where U is open in X . If S has an existing topology, f is continuous with respect to this topology if and only if

690-401: Is a list of his most important works, in chronological order: Fréchet also developed ideas from the article Deux types fondamentaux de distribution statistique (1938; an English translation The Two Fundamental Types of Statistical Distribution ) of Czech geographer , demographer and statistician Jaromír Korčák . Fréchet is sometimes credited with the introduction of what is now known as

736-404: Is a neighborhood U of x such that f ( U ) ⊆  V . Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V and whose image under f contains f ( x ) . This is equivalent to the condition that the preimages of the open (closed) sets in Y are open (closed) in X . In metric spaces, this definition

782-412: Is a neighborhood U of x such that f ( U ) ⊆  V . Intuitively, continuity means no matter how "small" V becomes, there is always a U containing x that maps inside V . If X and Y are metric spaces, it is equivalent to consider the neighborhood system of open balls centered at x and f ( x ) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in

828-448: Is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which f ( A ) is open in X . If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S . Thus the final topology can be characterized as the finest topology on S that makes f continuous. If f

874-403: Is accomplished by specifying when a point is the limit of a sequence , but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set , known as nets . A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in

920-427: Is called a topology . A set with a topology is called a topological space . Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric , can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces. General topology grew out of a number of areas, most importantly

966-427: Is equivalent to the ε–δ-definition that is often used in analysis. An extreme example: if a set X is given the discrete topology , all functions to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology and the space T set is at least T 0 , then the only continuous functions are the constant functions. Conversely, any function whose range

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1012-428: Is indiscrete is continuous. Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function. Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of neighborhoods : f is continuous at some point x  ∈  X if and only if for any neighborhood V of f ( x ), there

1058-409: Is open. Given a bijective function f between two topological spaces, the inverse function f need not be continuous. A bijective continuous function with continuous inverse function is called a homeomorphism . If a continuous bijection has as its domain a compact space and its codomain is Hausdorff , then it is a homeomorphism. Given a function where X is a topological space and S

1104-477: Is sequentially continuous. If X is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces .) This motivates

1150-428: Is the collection of subsets of Y that have open inverse images under f . In other words, the quotient topology is the finest topology on Y for which f is continuous. A common example of a quotient topology is when an equivalence relation is defined on the topological space X . The map f is then the natural projection onto the set of equivalence classes . A given set may have many different topologies. If

1196-454: Is the composition g ∘ f : X → Z . If f : X → Y is continuous and The possible topologies on a fixed set X are partially ordered : a topology τ 1 is said to be coarser than another topology τ 2 (notation: τ 1 ⊆ τ 2 ) if every open subset with respect to τ 1 is also open with respect to τ 2 . Then, the identity map is continuous if and only if τ 1 ⊆ τ 2 (see also comparison of topologies ). More generally,

1242-921: The American Mathematical Society due to his contact with American mathematicians in Paris—particularly Edwin Wilson . Fréchet served at many different institutions during his academic career. From 1907 to 1908 he served as a professor of mathematics at the Lycée in Besançon , then moved in 1908 to the Lycée in Nantes to stay there for a year. After that he served at the University of Poitiers between 1910 and 1919. He married in 1908 to Suzanne Carrive (1881–1945) and had four children: Hélène, Henri, Denise and Alain. Fréchet

1288-596: The Cramér–Rao bound , but Fréchet's 1940s lecture notes on the topic appear to have been lost. In 1908 he married Suzanne Carrive. General topology The fundamental concepts in point-set topology are continuity , compactness , and connectedness : The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of open sets . If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set'

1334-625: The Sorbonne . Fréchet briefly held a position of lecturer at the Sorbonne's Rockefeller Foundation and from 1928 was a Professor (without a chair). Fréchet was promoted to tenured Chair of General Mathematics in 1933 and to Chair of Differential and Integral Calculus in 1935. In 1941 Fréchet succeeded Borel as chair in the Calculus of Probabilities and Mathematical Physics, a position Fréchet held until he retired in 1949. From 1928 to 1935 Fréchet

1380-411: The cocountable topology , in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. There are many ways to define a topology on R , the set of real numbers . The standard topology on R is generated by the open intervals . The set of all open intervals forms a base or basis for

1426-524: The French school of Bourbaki , use the term quasi-compact for the general notion, and reserve the term compact for topological spaces that are both Hausdorff and quasi-compact . A compact set is sometimes referred to as a compactum , plural compacta . Every closed interval in R of finite length is compact . More is true: In R , a set is compact if and only if it is closed and bounded. (See Heine–Borel theorem ). Every continuous image of

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1472-490: The consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions. Instead of specifying the open subsets of a topological space, the topology can also be determined by a closure operator (denoted cl), which assigns to any subset A ⊆ X its closure , or an interior operator (denoted int), which assigns to any subset A of X its interior . In these terms,

1518-467: The context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. Note, however, that if the target space is Hausdorff , it is still true that f is continuous at a if and only if the limit of f as x approaches a is f ( a ). At an isolated point , every function is continuous. In several contexts, the topology of a space is conveniently specified in terms of limit points . In many instances, this

1564-507: The existing topology is finer than the initial topology on S . Thus the initial topology can be characterized as the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S , viewed as a subset of X . A topology on a set S is uniquely determined by the class of all continuous functions S → X {\displaystyle S\rightarrow X} into all topological spaces X . Dually ,

1610-405: The following: General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of continuity , in a technically adequate form that can be applied in any area of mathematics. 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

1656-447: The latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function f : X → Y is sequentially continuous if whenever a sequence ( x n ) in X converges to a limit x , the sequence ( f ( x n )) converges to f ( x ). Thus sequentially continuous functions "preserve sequential limits". Every continuous function

1702-508: The notion of compactness . Independently of Riesz , he discovered the representation theorem in the space of Lebesgue square integrable functions . He is often referred to as the founder of the theory of abstract spaces. He was born to a Protestant family in Maligny to Jacques and Zoé Fréchet. At the time of his birth, his father was a director of a Protestant orphanage in Maligny and

1748-657: The problems caused him to live in a continual fear of not being able to solve some of them, even though he was very grateful for the special relationship with Hadamard he was privileged to enjoy. After completing high-school Fréchet was required to enroll in military service. This is the time when he was deciding whether to study mathematics or physics – he chose mathematics out of dislike of the chemistry classes he would have had to take otherwise. Thus in 1900 he enrolled to École Normale Supérieure to study mathematics. He started publishing quite early, having published four papers in 1903. He also published some of his early papers with

1794-486: The reestablishment of the university . He served as a professor of higher analysis and Director of the Mathematics Institute. Despite being burdened with administrative work, he was again able to produce a large amount of high-quality research. In 1928 Fréchet decided to move back to Paris, thanks to encouragement from Borel , who was then chair in the Calculus of Probabilities and Mathematical Physics at

1840-617: The secular system – it was not a job of a headship, however, and the family could not expect as high standards as they might have otherwise. Maurice attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard . Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the University of Bordeaux in 1894, Hadamard continuously wrote to Fréchet, setting him mathematical problems and harshly criticising his errors. Much later Fréchet admitted that

1886-411: The space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be Hausdorff spaces where limit points are unique. Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. Any set can be given

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1932-480: The topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the Euclidean spaces R can be given a topology. In the usual topology on R the basic open sets are the open balls . Similarly, C , the set of complex numbers , and C have

1978-463: Was also put in charge of lectures at the École Normale Supérieure ; in this latter capacity Fréchet was able to direct a significant number of young mathematicians toward research in probability, including Doeblin , Fortet , Loève , and Jean Ville. Despite his major achievements, Fréchet was not overly appreciated in France. As an illustration, while being nominated numerous times, he was not elected

2024-409: Was his outstanding 1906 PhD thesis Sur quelques points du calcul fonctionnel , on the calculus of functionals. Here Fréchet introduced the concept of a metric space , although the name is due to Hausdorff . Fréchet's level of abstraction is similar to that in group theory , proving theorems within a carefully chosen axiomatic system that can then be applied to a large array of particular cases. Here

2070-409: Was later in his youth appointed a head of a Protestant school. However, the newly established Third Republic was not sympathetic to religious education and so laws were enacted requiring all education to be secular. As a result, his father lost his job. To generate some income his mother set up a boarding house for foreigners in Paris. His father was able later to obtain another teaching position within

2116-590: Was planning to spend a year in the United States at the University of Illinois , but his plan was disrupted when the First World War broke out in 1914. He was mobilised on 4 August the same year. Because of his diverse language skills, gained when his mother ran the establishment for foreigners, he served as an interpreter for the British Army . However, this was not a safe job; he spent two and

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