Reduced homologyIn mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If P is a single-point space, then with the usual definitions the integral homology group H0(P) is isomorphic to (an infinite cyclic group), while for i ≥ 1 we have Hi(P) = {0}.
Bounded functionIn mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that for all x in X. A function that is not bounded is said to be unbounded. If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below.
Join (topology)In topology, a field of mathematics, the join of two topological spaces and , often denoted by or , is a topological space formed by taking the disjoint union of the two spaces, and attaching line segments joining every point in to every point in . The join of a space with itself is denoted by . The join is defined in slightly different ways in different contexts If and are subsets of the Euclidean space , then:,that is, the set of all line-segments between a point in and a point in .
Knot polynomialIn the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial.
Tietze extension theoremIn topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem or Urysohn-Brouwer lemma) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary. If is a normal space and is a continuous map from a closed subset of into the real numbers carrying the standard topology, then there exists a of to that is, there exists a map continuous on all of with for all Moreover, may be chosen such that that is, if is bounded then may be chosen to be bounded (with the same bound as ).
Möbius transformationIn geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane.
Prime knotIn knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.
FibrationThe notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. A mapping satisfies the homotopy lifting property for a space if: for every homotopy and for every mapping (also called lift) lifting (i.e. ) there exists a (not necessarily unique) homotopy lifting (i.e.
Hyperbolic groupIn group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by . The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology (in particular the results of Max Dehn concerning the fundamental group of a hyperbolic Riemann surface, and more complex phenomena in three-dimensional topology), and combinatorial group theory.
Analytic manifoldIn mathematics, an analytic manifold, also known as a manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted. For , the space of analytic functions, , consists of infinitely differentiable functions , such that the Taylor series converges to in a neighborhood of , for all .
