Connected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjointnon-emptyopen subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X{\displaystyle X} is a connected set if it is a connected space when viewed as a subspace of X{\displaystyle X}. Some related but stronger conditions are path connected, simply connected, and n{\displaystyle n}...
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. There are several equivalent definitions of a topology, the most commonly used of which is the defini...
Connectivity (graph theory)In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphsIn an undirected graphG, two verticesu and v are called connected if G contains a path from u...
Graph theoryIn mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links, or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in dis...
Graph (discrete mathematics)In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or u...
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Mathematics typically does not define precisely what constitutes a "set" or "collection", because such a definition would have to be in terms of something else previously defined. Instead, sets serve as foundational objects whose behavior is described by ax...
Boundary (topology)In topology and mathematics in general, the boundary of a subset S of a topological spaceX is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include bd(S),fr(S),{\displaystyle \operatorname {bd} (S),\operatorname {fr} (S),} and ∂S{\displaystyle \partial S}. TerminologySome authors (for example, Wil...
Open setIn general topology and mathematical analysis, an open set is a generalization of an open interval in the real line. In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P). More generally, an open set is a member of a given collection of subsets of a given set, a collection that has the pr...ManifoldIn mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n{\displaystyle n}-dimensional manifold, or n{\displaystyle n}-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n{\displaystyle n}-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure-eight. Two-dimensional manifolds are also ca...
Disjoint setsIn set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two or more sets is called disjoint if any two distinct sets of the collection are disjoint. GeneralizationsThis definition of disjoint sets can be extended to families of sets and to indexed familie...Lie groupGroup that is also a differentiable manifold with group operations that are smooth .mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+.mw-empty-elt+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}Not to be confused with Group of Lie type or Ree group. .mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #3...
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. There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of op...*As an Amazon Associate I earn from qualifying purchases.