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68-460: Extrema may refer to: Extrema (mathematics) , maxima and minima values Extremities (disambiguation) Extrema, Minas Gerais , town in Brazil Extrema, Rondônia , town in Brazil Extrema (band) , Italian thrash/groove metal band Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

136-413: A convex polyhedron , and hence of a planar graph . The study and generalization of this formula, specifically by Cauchy (1789–1857) and L'Huilier (1750–1840), boosted the study of topology. In 1827, Carl Friedrich Gauss published General investigations of curved surfaces , which in section 3 defines the curved surface in a similar manner to the modern topological understanding: "A curved surface

204-429: A geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance . More specifically, a topological space is a set whose elements are called points , along with an additional structure called a topology , which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of

272-527: A greatest element m , then m is a maximal element of the set, also denoted as max ( S ) {\displaystyle \max(S)} . Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with (respect to order induced by T ), then m is a least upper bound of S in T . Similar results hold for least element , minimal element and greatest lower bound . The maximum and minimum function for sets are used in databases , and can be computed rapidly, since

340-415: A natural topology since it is locally Euclidean. Similarly, every simplex and every simplicial complex inherits a natural topology from . The Sierpiński space is the simplest non-discrete topological space. It has important relations to the theory of computation and semantics. Every subset of a topological space can be given the subspace topology in which the open sets are the intersections of

408-489: A set X may be defined as a collection τ {\displaystyle \tau } of subsets of X , called open sets and satisfying the following axioms: As this definition of a topology is the most commonly used, the set τ {\displaystyle \tau } of the open sets is commonly called a topology on X . {\displaystyle X.} A subset C ⊆ X {\displaystyle C\subseteq X}

476-964: A (possibly empty) set. The elements of X {\displaystyle X} are usually called points , though they can be any mathematical object. Let N {\displaystyle {\mathcal {N}}} be a function assigning to each x {\displaystyle x} (point) in X {\displaystyle X} a non-empty collection N ( x ) {\displaystyle {\mathcal {N}}(x)} of subsets of X . {\displaystyle X.} The elements of N ( x ) {\displaystyle {\mathcal {N}}(x)} will be called neighbourhoods of x {\displaystyle x} with respect to N {\displaystyle {\mathcal {N}}} (or, simply, neighbourhoods of x {\displaystyle x} ). The function N {\displaystyle {\mathcal {N}}}

544-442: A basic open set, all but finitely many of its projections are the entire space. A quotient space is defined as follows: if X {\displaystyle X} is a topological space and Y {\displaystyle Y} is a set, and if f : X → Y {\displaystyle f:X\to Y} is a surjective function , then the quotient topology on Y {\displaystyle Y}

612-399: A basis set consisting of all subsets of the union of the U i {\displaystyle U_{i}} that have non-empty intersections with each U i . {\displaystyle U_{i}.} The Fell topology on the set of all non-empty closed subsets of a locally compact Polish space X {\displaystyle X} is a variant of

680-401: A bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction ). In two and more dimensions, this argument fails. This is illustrated by the function whose only critical point is at (0,0), which is

748-427: A collection τ {\displaystyle \tau } of closed subsets of X . {\displaystyle X.} Thus the sets in the topology τ {\displaystyle \tau } are the closed sets, and their complements in X {\displaystyle X} are the open sets. There are many other equivalent ways to define a topological space: in other words

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816-426: A domain must occur at critical points (or points where the derivative equals zero). However, not all critical points are extrema. One can often distinguish whether a critical point is a local maximum, a local minimum, or neither by using the first derivative test , second derivative test , or higher-order derivative test , given sufficient differentiability. For any function that is defined piecewise , one finds

884-564: A homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In category theory , one of the fundamental categories is Top , which denotes the category of topological spaces whose objects are topological spaces and whose morphisms are continuous functions. The attempt to classify the objects of this category ( up to homeomorphism ) by invariants has motivated areas of research, such as homotopy theory , homology theory , and K-theory . A given set may have many different topologies. If

952-418: A local minimum with f (0,0) = 0. However, it cannot be a global one, because f (2,3) = −5. If the domain of a function for which an extremum is to be found consists itself of functions (i.e. if an extremum is to be found of a functional ), then the extremum is found using the calculus of variations . Maxima and minima can also be defined for sets. In general, if an ordered set S has

1020-443: A maximum (or minimum) by finding the maximum (or minimum) of each piece separately, and then seeing which one is greatest (or least). For a practical example, assume a situation where someone has 200 {\displaystyle 200} feet of fencing and is trying to maximize the square footage of a rectangular enclosure, where x {\displaystyle x} is the length, y {\displaystyle y}

1088-432: A maximum or a minimum. For example, the set of natural numbers has no maximum, though it has a minimum. If an infinite chain S is bounded, then the closure Cl ( S ) of the set occasionally has a minimum and a maximum, in which case they are called the greatest lower bound and the least upper bound of the set S , respectively. Topological space In mathematics , a topological space is, roughly speaking,

1156-428: A minimum point. An important example is a function whose domain is a closed and bounded interval of real numbers (see the graph above). Finding global maxima and minima is the goal of mathematical optimization . If a function is continuous on a closed interval, then by the extreme value theorem , global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in

1224-567: A particular sequence of functions converges to the zero function. A linear graph has a natural topology that generalizes many of the geometric aspects of graphs with vertices and edges . Outer space of a free group F n {\displaystyle F_{n}} consists of the so-called "marked metric graph structures" of volume 1 on F n . {\displaystyle F_{n}.} Topological spaces can be broadly classified, up to homeomorphism, by their topological properties . A topological property

1292-472: A real number x {\displaystyle x} if it includes an open interval containing x . {\displaystyle x.} Given such a structure, a subset U {\displaystyle U} of X {\displaystyle X} is defined to be open if U {\displaystyle U} is a neighbourhood of all points in U . {\displaystyle U.} The open sets then satisfy

1360-432: A set can have at most one minimal element and at most one maximal element. Then, due to mutual comparability, the minimal element will also be the least element, and the maximal element will also be the greatest element. Thus in a totally ordered set, we can simply use the terms minimum and maximum . If a chain is finite, then it will always have a maximum and a minimum. If a chain is infinite, then it need not have

1428-473: 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

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1496-862: A set may have many distinct topologies defined on it. If γ {\displaystyle \gamma } is an ordinal number , then the set γ = [ 0 , γ ) {\displaystyle \gamma =[0,\gamma )} may be endowed with the order topology generated by the intervals ( α , β ) , {\displaystyle (\alpha ,\beta ),} [ 0 , β ) , {\displaystyle [0,\beta ),} and ( α , γ ) {\displaystyle (\alpha ,\gamma )} where α {\displaystyle \alpha } and β {\displaystyle \beta } are elements of γ . {\displaystyle \gamma .} Every manifold has

1564-401: A surface is a topological space that is locally like a Euclidean plane . Topological spaces were first defined by Felix Hausdorff in 1914 in his seminal "Principles of Set Theory". Metric spaces had been defined earlier in 1906 by Maurice Fréchet , though it was Hausdorff who popularised the term "metric space" ( German : metrischer Raum ). The utility of the concept of a topology

1632-451: A topology native to it, and this can be extended to vector spaces over that field. The Zariski topology is defined algebraically on the spectrum of a ring or an algebraic variety . On R n {\displaystyle \mathbb {R} ^{n}} or C n , {\displaystyle \mathbb {C} ^{n},} the closed sets of the Zariski topology are

1700-404: A topology, the most commonly used of which is the definition through open sets , which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits , continuity , and connectedness . Common types of topological spaces include Euclidean spaces , metric spaces and manifolds . Although very general,

1768-467: A topology. In the usual topology on R n {\displaystyle \mathbb {R} ^{n}} the basic open sets are the open balls . Similarly, C , {\displaystyle \mathbb {C} ,} the set of complex numbers , and C n {\displaystyle \mathbb {C} ^{n}} have a standard topology in which the basic open sets are open balls. For any algebraic objects we can introduce

1836-641: Is finer than τ 1 , {\displaystyle \tau _{1},} and τ 1 {\displaystyle \tau _{1}} is coarser than τ 2 . {\displaystyle \tau _{2}.} A proof that relies only on the existence of certain open sets will also hold for any finer topology, and similarly a proof that relies only on certain sets not being open applies to any coarser topology. The terms larger and smaller are sometimes used in place of finer and coarser, respectively. The terms stronger and weaker are also used in

1904-416: Is a strict local maximum point if there exists some ε > 0 such that, for all x in X within distance ε of x with x ≠ x , we have f ( x ) > f ( x ) . Note that a point is a strict global maximum point if and only if it is the unique global maximum point, and similarly for minimum points. A continuous real-valued function with a compact domain always has a maximum point and

1972-465: Is a topological space , since the definition just given can be rephrased in terms of neighbourhoods. Mathematically, the given definition is written as follows: The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a strict extremum can be defined. For example, x is a strict global maximum point if for all x in X with x ≠ x , we have f ( x ) > f ( x ) , and x

2040-589: Is a collection of topologies on X , {\displaystyle X,} then the meet of F {\displaystyle F} is the intersection of F , {\displaystyle F,} and the join of F {\displaystyle F} is the meet of the collection of all topologies on X {\displaystyle X} that contain every member of F . {\displaystyle F.} A function f : X → Y {\displaystyle f:X\to Y} between topological spaces

2108-539: Is called continuous if for every x ∈ X {\displaystyle x\in X} and every neighbourhood N {\displaystyle N} of f ( x ) {\displaystyle f(x)} there is a neighbourhood M {\displaystyle M} of x {\displaystyle x} such that f ( M ) ⊆ N . {\displaystyle f(M)\subseteq N.} This relates easily to

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2176-399: Is called a neighbourhood topology if the axioms below are satisfied; and then X {\displaystyle X} with N {\displaystyle {\mathcal {N}}} is called a topological space . The first three axioms for neighbourhoods have a clear meaning. The fourth axiom has a very important use in the structure of the theory, that of linking together

2244-677: Is defined on the topological space X . {\displaystyle X.} The map f {\displaystyle f} is then the natural projection onto the set of equivalence classes . The Vietoris topology on the set of all non-empty subsets of a topological space X , {\displaystyle X,} named for Leopold Vietoris , is generated by the following basis: for every n {\displaystyle n} -tuple U 1 , … , U n {\displaystyle U_{1},\ldots ,U_{n}} of open sets in X , {\displaystyle X,} we construct

2312-499: Is generated by the open intervals . The set of all open intervals forms a base or basis for 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 n {\displaystyle \mathbb {R} ^{n}} can be given

2380-604: Is restricted. Since width is positive, then x > 0 {\displaystyle x>0} , and since x = 100 − y {\displaystyle x=100-y} , that implies that x < 100 {\displaystyle x<100} . Plug in critical point 50 {\displaystyle 50} , as well as endpoints 0 {\displaystyle 0} and 100 {\displaystyle 100} , into x y = x ( 100 − x ) {\displaystyle xy=x(100-x)} , and

2448-460: Is said to be closed in ( X , τ ) {\displaystyle (X,\tau )} if its complement X ∖ C {\displaystyle X\setminus C} is an open set. Using de Morgan's laws , the above axioms defining open sets become axioms defining closed sets : Using these axioms, another way to define a topological space is as a set X {\displaystyle X} together with

2516-480: Is said to possess continuous curvature at one of its points A, if the direction of all the straight lines drawn from A to points of the surface at an infinitesimal distance from A are deflected infinitesimally from one and the same plane passing through A." Yet, "until Riemann 's work in the early 1850s, surfaces were always dealt with from a local point of view (as parametric surfaces) and topological issues were never considered". " Möbius and Jordan seem to be

2584-426: Is shown by the fact that there are several equivalent definitions of this mathematical structure . Thus one chooses the axiomatization suited for the application. The most commonly used is that in terms of open sets , but perhaps more intuitive is that in terms of neighbourhoods and so this is given first. This axiomatization is due to Felix Hausdorff . Let X {\displaystyle X} be

2652-424: Is the collection of subsets of Y {\displaystyle Y} that have open inverse images under f . {\displaystyle f.} In other words, the quotient topology is the finest topology on Y {\displaystyle Y} for which f {\displaystyle f} is continuous. A common example of a quotient topology is when an equivalence relation

2720-444: Is the width, and x y {\displaystyle xy} is the area: The derivative with respect to x {\displaystyle x} is: Setting this equal to 0 {\displaystyle 0} reveals that x = 50 {\displaystyle x=50} is our only critical point . Now retrieve the endpoints by determining the interval to which x {\displaystyle x}

2788-431: 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 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. The real line can also be given

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2856-490: The lower limit topology . Here, the basic open sets are the half open intervals [ a , b ) . {\displaystyle [a,b).} This topology on R {\displaystyle \mathbb {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

2924-505: The solution sets of systems of polynomial equations. If Γ {\displaystyle \Gamma } is a filter on a set X {\displaystyle X} then { ∅ } ∪ Γ {\displaystyle \{\varnothing \}\cup \Gamma } is a topology on X . {\displaystyle X.} Many sets of linear operators in functional analysis are endowed with topologies that are defined by specifying when

2992-412: The (enlargeable) figure on the right, the necessary conditions for a local maximum are similar to those of a function with only one variable. The first partial derivatives as to z (the variable to be maximized) are zero at the maximum (the glowing dot on top in the figure). The second partial derivatives are negative. These are only necessary, not sufficient, conditions for a local maximum, because of

3060-470: The Vietoris topology, and is named after mathematician James Fell. It is generated by the following basis: for every n {\displaystyle n} -tuple U 1 , … , U n {\displaystyle U_{1},\ldots ,U_{n}} of open sets in X {\displaystyle X} and for every compact set K , {\displaystyle K,}

3128-544: The axioms given below in the next definition of a topological space. Conversely, when given the open sets of a topological space, the neighbourhoods satisfying the above axioms can be recovered by defining N {\displaystyle N} to be a neighbourhood of x {\displaystyle x} if N {\displaystyle N} includes an open set U {\displaystyle U} such that x ∈ U . {\displaystyle x\in U.} A topology on

3196-419: 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. There are many ways of defining a topology on R , {\displaystyle \mathbb {R} ,} the set of real numbers . The standard topology on R {\displaystyle \mathbb {R} }

3264-550: The concept of sequence . A topology is completely determined if for every net in X {\displaystyle X} the set of its accumulation points is specified. Many topologies can be defined on a set to form a topological space. When every open set of a topology τ 1 {\displaystyle \tau _{1}} is also open for a topology τ 2 , {\displaystyle \tau _{2},} one says that τ 2 {\displaystyle \tau _{2}}

3332-423: The concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called point-set topology or general topology . Around 1735, Leonhard Euler discovered the formula V − E + F = 2 {\displaystyle V-E+F=2} relating the number of vertices (V), edges (E) and faces (F) of

3400-457: The concepts of neighbourhood, or that of open or closed sets can be reconstructed from other starting points and satisfy the correct axioms. Another way to define a topological space is by using the Kuratowski closure axioms , which define the closed sets as the fixed points of an operator on the power set of X . {\displaystyle X.} A net is a generalisation of

3468-416: The discrete topology, under which the algebraic operations are continuous functions. For any such structure that is not finite, we often have a natural topology compatible with the algebraic operations, in the sense that the algebraic operations are still continuous. This leads to concepts such as topological groups , topological vector spaces , topological rings and local fields . Any local field has

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3536-408: The domain X is a metric space , then f is said to have a local (or relative ) maximum point at the point x , if there exists some ε > 0 such that f ( x ) ≥ f ( x ) for all x in X within distance ε of x . Similarly, the function has a local minimum point at x , if f ( x ) ≤ f ( x ) for all x in X within distance ε of x . A similar definition can be used when X

3604-427: The first to realize that the main problem about the topology of (compact) surfaces is to find invariants (preferably numerical) to decide the equivalence of surfaces, that is, to decide whether two surfaces are homeomorphic or not." The subject is clearly defined by Felix Klein in his " Erlangen Program " (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology"

3672-634: The function has a global (or absolute ) minimum point at x , if f ( x ) ≤ f ( x ) for all x in X . The value of the function at a maximum point is called the maximum value of the function, denoted max ( f ( x ) ) {\displaystyle \max(f(x))} , and the value of the function at a minimum point is called the minimum value of the function, (denoted min ( f ( x ) ) {\displaystyle \min(f(x))} for clarity). Symbolically, this can be written as follows: The definition of global minimum point also proceeds similarly. If

3740-400: The interior of the domain, or must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary, and take the greatest (or least) one.Minima For differentiable functions , Fermat's theorem states that local extrema in the interior of

3808-484: The literature, but with little agreement on the meaning, so one should always be sure of an author's convention when reading. The collection of all topologies on a given fixed set X {\displaystyle X} forms a complete lattice : if F = { τ α : α ∈ A } {\displaystyle F=\left\{\tau _{\alpha }:\alpha \in A\right\}}

3876-410: The maximum (or minimum) of a set can be computed from the maxima of a partition; formally, they are self- decomposable aggregation functions . In the case of a general partial order , the least element (i.e., one that is less than all others) should not be confused with a minimal element (nothing is lesser). Likewise, a greatest element of a partially ordered set (poset) is an upper bound of

3944-440: The maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets , such as the set of real numbers , have no minimum or maximum. In statistics , the corresponding concept is the sample maximum and minimum . A real-valued function f defined on a domain X has a global (or absolute ) maximum point at x , if f ( x ) ≥ f ( x ) for all x in X . Similarly,

4012-402: The neighbourhoods of different points of X . {\displaystyle X.} A standard example of such a system of neighbourhoods is for the real line R , {\displaystyle \mathbb {R} ,} where a subset N {\displaystyle N} of R {\displaystyle \mathbb {R} } is defined to be a neighbourhood of

4080-422: 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, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in

4148-433: The possibility of a saddle point . For use of these conditions to solve for a maximum, the function z must also be differentiable throughout. The second partial derivative test can help classify the point as a relative maximum or relative minimum. In contrast, there are substantial differences between functions of one variable and functions of more than one variable in the identification of global extrema. For example, if

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4216-462: The results are 2500 , 0 , {\displaystyle 2500,0,} and 0 {\displaystyle 0} respectively. Therefore, the greatest area attainable with a rectangle of 200 {\displaystyle 200} feet of fencing is 50 × 50 = 2500 {\displaystyle 50\times 50=2500} . For functions of more than one variable, similar conditions apply. For example, in

4284-415: The set of all subsets of X {\displaystyle X} that are disjoint from K {\displaystyle K} and have nonempty intersections with each U i {\displaystyle U_{i}} is a member of the basis. Metric spaces embody a metric , a precise notion of distance between points. Every metric space can be given a metric topology, in which

4352-477: The set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A ), then m = b . Any least element or greatest element of a poset is unique, but a poset can have several minimal or maximal elements. If a poset has more than one maximal element, then these elements will not be mutually comparable. In a totally ordered set, or chain , all elements are mutually comparable, so such

4420-458: 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. There exist numerous topologies on any given finite set . Such spaces are called finite topological spaces . Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general. Any set can be given

4488-521: The title Extrema . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Extrema&oldid=1187502617 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Extrema (mathematics) As defined in set theory ,

4556-414: The usual definition in analysis. Equivalently, f {\displaystyle f} is continuous if the inverse image of every open set is open. This is an attempt to capture the intuition that there are no "jumps" or "separations" in the function. A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called homeomorphic if there exists

4624-405: Was introduced by Johann Benedict Listing in 1847, although he had used the term in correspondence some years earlier instead of previously used "Analysis situs". The foundation of this science, for a space of any dimension, was created by Henri Poincaré . His first article on this topic appeared in 1894. In the 1930s, James Waddell Alexander II and Hassler Whitney first expressed the idea that

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