Fundamental groupIn the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group.
HomotopyIn topology, two continuous functions from one topological space to another are called homotopic (from Ancient Greek: ὁμόςhomós'same, similar' and τόποςtópos'place') if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
N-sphereGeneralized sphere of dimension n (mathematics)2-sphere wireframe as an orthogonal projectionJust as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue), and hypermeridians (green).
Topological spaceIn mathematics, a topological space is, roughly speaking, a 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.
Group (mathematics)In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set. The following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.
Equivalence classIn mathematics, when the elements of some setS{\displaystyle S} have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S{\displaystyle S} into equivalence classes. These equivalence classes are constructed so that elements a{\displaystyle a} and b{\displaystyle b} belong to the same equivalence classif, and only if, they are equivalent.
Pointed spaceIn mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is simply one particular point, picked out from the space, and given a name, such as x0,{\displaystyle x_{0},} that remains unchanged during subsequent discussion, and is kept track of during all operations.
Loop (topology)In mathematics, a loop in a topological spaceX is a continuous functionf from the unit intervalI = [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. A loop may also be seen as a continuous map f from the pointedunit circleS into X, because S may be regarded as a quotient of I under the identification of 0 with 1.
Algebraic topologyAlgebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.
Space (mathematics)In mathematics, a space is a set (sometimes known as a universe) endowed with a structure defining the relationships among the elements of the set. A subspace is a subset of the parent space which retains the same mathematical structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself.
HoleA hole is an opening in or through a particular medium, usually a solidbody. Holes occur through natural and artificial processes, and may be useful for various purposes, or may represent a problem needing to be addressed in many fields of engineering. Depending on the material and the placement, a hole may be an indentation in a surface (such as a hole in the ground), or may pass completely through that surface (such as a hole created by a hole puncher in a piece of paper).
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.*As an Amazon Associate I earn from qualifying purchases.