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Invariance of domain

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space . It states: If is an open subset of and is an injective continuous map, then is open in and is a homeomorph

Urysohn's lemma

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Urysohn's lemma is comm

Michael selection theorem

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following: Let X be a paracompact

Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local h

Continuous wavelet transform

In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of

Proper map

In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.

Discontinuities of monotone functions

In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) functio

Banach–Mazur theorem

In functional analysis, a field of mathematics, the Banach–Mazur theorem is a theorem roughly stating that most well-behaved normed spaces are subspaces of the space of continuous paths. It is named a

Maximum theorem

The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959

Runge's phenomenon

In the mathematical field of numerical analysis, Runge's phenomenon (German: [ˈʁʊŋə]) is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polyn

Homotopy

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμός homós "same, similar" and τόπος tópos "place") if on

Homeomorphism

In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous invers

Loop theorem

In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma a

Tietze extension theorem

In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This m

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between and at some point w

Space-filling curve

In mathematical analysis, a space-filling curve is a curve whose range contains the entire 2-dimensional unit square (or more generally an n-dimensional unit hypercube). Because Giuseppe Peano (1858–1

Banach–Stone theorem

In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, t

Minkowski's question-mark function

In mathematics, the Minkowski question-mark function, denoted ?(x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rationa

Direction-preserving function

In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacen

Quotient space (topology)

In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the or

Lefschetz fixed-point theorem

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space to itself by means of traces of the induced mapping

Bounded operator

In functional analysis and operator theory, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and

Perfect map

In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve so

Direct image with compact support

In mathematics, the direct image with compact (or proper) support is an image functor for sheaves that extends the compactly supported global sections functor to the relative setting. It is one of Gro

Local boundedness

In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that p

Stone–Weierstrass theorem

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polyn

Local diffeomorphism

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition o

Metric map

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).These maps are the morphisms in

Darboux's theorem (analysis)

In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the interm

Classification of discontinuities

Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says

Continuous linear operator

In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator

Graph continuous function

In mathematics, and in particular the study of game theory, a function is graph continuous if it exhibits the following properties. The concept was originally defined by Partha Dasgupta and Eric Maski

Quasi-continuous function

In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not tru

Blancmange curve

In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–L

Leray spectral sequence

In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectra

Quillen adjunction

In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the

Uniform continuity

In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as clo

Open and closed maps

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 is open if for any open set in the image i

Peano curve

In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. Peano's curve is a surjective, continuous function from the unit interval onto t

Symmetrically continuous function

In mathematics, a function is symmetrically continuous at a point x if The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function is symme

Bundle map

In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bun

Normal family

In mathematics, with special application to complex analysis, a normal family is a pre-compact subset of the space of continuous functions. Informally, this means that the functions in the family are

Continuous functions on a compact Hausdorff space

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers.

Borsuk–Ulam theorem

In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere

Semi-continuity

In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, l

Equicontinuity

In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In

Heine–Cantor theorem

In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if is a continuous function between two metric spaces and , and is compact, then is uniformly continuou

Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a

Direct image functor

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic

Simplicial approximation theorem

In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that a

Pasting lemma

In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function.

Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations o

Weierstrass function

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its di

Degree of a continuous mapping

In topology, the degree of a continuous mapping between two compact oriented manifolds of the same dimension is a number that represents the number of times that the domain manifold wraps around the r

Arzelà–Ascoli theorem

The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous funct

Hemicontinuity

In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets A and B. The dual concepts of upper hemicontinuity a

Continuous functional calculus

In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal element

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