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Homeomorphic
and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. In the mathematics|mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words ''ὅμοιος'' (''homoios'') = similar or same and ''μορφή'' (''morphē'') = shape, form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the ...
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Mug And Torus Morph
A mug is a type of cup typically used for drinking hot drinks, such as coffee, hot chocolate, or tea. Mugs usually have handles and hold a larger amount of fluid than other types of cup. Typically, a mug holds approximately of liquid. A mug is a less formal style of drink container and is not usually used in formal place settings, where a teacup or coffee cup is preferred. Shaving mugs are used to assist in wet shaving. Ancient mugs were usually carved in wood or bone, ceramic or shaped of clay, while most modern ones are made of ceramic materials such as bone china, earthenware, porcelain, or stoneware. Some are made from strengthened glass, such as Pyrex. Other materials, including enameled metal, plastic, or steel are preferred, when reduced weight or resistance to breakage is at a premium, such as for camping. A travel mug is insulated and has a cover with a small sipping opening to prevent spills. Techniques such as silk screen printing or decals are used to apply decorations ...
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Sphere
of a sphere A sphere (from Greek language|Greek —, "globe, ball") is a geometrical object in three-dimensional space that is the surface of a ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk"). Like a circle in a two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance from a given point in a three-dimensional space.. This distance is the radius of the ball, which is made up from all points with a distance less than (or, for a closed ball, less than ''or equal to'') from the given point, which is the center of the mathematical ball. These are also referred to as the radius and center of the sphere, respectively. The longest straight line segment through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of both the sphere and its ball. While outside mathematics the terms "sphere" and ...
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Interval (mathematics)
In mathematics, an (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other examples of intervals are the set of numbers such that , the set of all real numbers \R, the set of nonnegative real numbers, the set of positive real numbers, the empty set, and any singleton (set of one element). Real intervals play an important role in the theory of integration, because they are the simplest sets whose "size" (or "measure" or "length") is easy to define. The concept of measure can then be extended to more complicated sets of real numbers, leading to the Borel measure and eventually to the Lebesgue measure. Intervals are central to interval arithmetic, a general numerical computing technique that automatically provides guaranteed enclosures for arbitrary formulas, even in the presence of uncertainties, mathematical appr ...
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Blue Trefoil Knot
Blue is one of the three primary colours of pigments in painting and traditional colour theory, as well as in the RGB colour model. It lies between violet and green on the spectrum of visible light. The eye perceives blue when observing light with a dominant wavelength between approximately 450 and 495 nanometres. Most blues contain a slight mixture of other colours; azure contains some green, while ultramarine contains some violet. The clear daytime sky and the deep sea appear blue because of an optical effect known as Rayleigh scattering. An optical effect called Tyndall effect explains blue eyes. Distant objects appear more blue because of another optical effect called aerial perspective. Blue has been an important colour in art and decoration since ancient times. The semi-precious stone lapis lazuli was used in ancient Egypt for jewellery and ornament and later, in the Renaissance, to make the pigment ultramarine, the most expensive of all pigments. In the eighth century Ch ...
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Equivalence Class
In mathematics, when the elements of some set have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set into equivalence classes. These equivalence classes are constructed so that elements and belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set and an equivalence relation on , the ''equivalence class'' of an element in , denoted by /math>, is the set :\ of elements which are equivalent to . It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of . This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of by , and is denoted by . When the set has some structure (such as a group operation or a topology) and the equivalence relation is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exampl ...
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Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements and of a set are equivalent with respect to an equivalence relation ; the most common are "" and "", which are used when is implicit, and variations of "", "", or "" to specify explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation ~ on a set ''X'' is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all ''a'', ''b'' and ''c'' in ''X'': *''a'' ~ ''a''. (Reflexivity) *''a'' ~ ''b'' if and only i ...
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Open Mapping
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa. Open and closed maps are not necessarily continuous. Further, continuity is independent of openness and closedness in the general case and a continuous function may have one, both, or neither property; this fact remains true even if one restricts oneself to metric spaces. Although their definitions seem more natural, open and closed maps are much less important than continuous maps. Recall that, by definition, a function f : X \to Y is continuous if the preimage of every open set of Y is open in X. (Equivalently, if the preimage of every closed set of Y is closed in X). ...
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Continuity (topology)
In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be ''discontinuous''. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it. Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers ...
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Onto
In mathematics, a function ''f'' from a set ''X'' to a set ''Y'' is surjective (also known as onto, or a surjection), if for every element ''y'' in the codomain ''Y'' of ''f'', there is at least one element ''x'' in the domain ''X'' of ''f'' such that ''f''(''x'') = ''y''. It is not required that ''x'' be unique; the function ''f'' may map one or more elements of ''X'' to the same element of ''Y''. The term ''surjective'' and the related terms ''injective'' and ''bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word ''sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a ...
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Injective Function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements of its codomain. In other words, every element of the function's codomain is the image of ''at most'' one element of its domain. The term ''one-to-one function'' must not be confused with ''one-to-one correspondence'' that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an ''injective homomorphism'' is also called a ''monomorphism''. However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures ...
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Bijection
In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. A bijective funct ...
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Function (mathematics)
In mathematics, a functionThe words map, mapping, transformation, correspondence, and operator are often used synonymously. . is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept. A function is a process or a relation that associates each element of a set ...
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Mathematical Joke
A mathematical joke is a form of humor which relies on aspects of mathematics or a stereotype of mathematicians. The humor may come from a pun, or from a double meaning of a mathematical term, or from a lay person's misunderstanding of a mathematical concept. Mathematician and author John Allen Paulos in his book ''Mathematics and Humor'' described several ways that mathematics, generally considered a dry, formal activity, overlaps with humor, a loose, irreverent activity: both are forms of "intellectual play"; both have "logic, pattern, rules, structure"; and both are "economical and explicit". Some performers combine mathematics and jokes to entertain and/or teach math. Humor of mathematicians may be classified into the esoteric and exoteric categories. Esoteric jokes rely on the intrinsic knowledge of mathematics and its terminology. Exoteric jokes are intelligible to the outsiders, and most of them compare mathematicians with representatives of other disciplines or with common ...
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Trefoil Knot
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory. The trefoil knot is named after the three-leaf clover (or trefoil) plant. Descriptions The trefoil knot can be defined as the curve obtained from the following parametric equations: :x = \sin t + 2 \sin 2t :y=\cos t - 2 \cos 2t :z=-\sin 3t The (2,3)-torus knot is also a trefoil knot. The following parametric equations give a (2,3)-torus knot lying on torus (r-2)^2+z^2 = 1: :x = (2+\cos 3t)\cos 2t :y=(2+\cos 3t )\sin 2t :z=\sin 3t Form of trefoil knot without visual three-fold symmetry Any continuous deformation of the curve above is also considered a trefoil knot. Specifically, any curve isotopic to a trefoil knot is also considered to be a trefoil. In addition, th ...
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is a related shape, a toroid. Real-world objects that approximate a torus of revolution include swim rings and inner tubes. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a ''solid torus'', which is formed by rotating a disk, rather than a circle, around an axis. A solid toru ...
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