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}.
SubmanifoldIn mathematics, a submanifold of a manifoldM{\displaystyle M} is a subsetS{\displaystyle S} which itself has the structure of a manifold, and for which the inclusion mapS→M{\displaystyle S ightarrow M} satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions.
Topological pairIn mathematics, more specifically algebraic topology, a pair (X,A){\displaystyle (X,A)} is shorthand for an inclusion of topological spacesi:A↪X{\displaystyle i\colon A\hookrightarrow X}. Sometimes i{\displaystyle i} is assumed to be a cofibration.
Image (mathematics)Set of the values of a functionFor the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}.
TopologyTopology (from the Greek words , 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuousdeformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
Half-space (geometry)In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space. If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional space is called a half-line or ray.More generally, a half-space is either of the two parts into which a hyperplane divides an n-dimensional space.
Category (mathematics)In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
HomeomorphismIn mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and 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.
CrumplingIn geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is of interest to mathematicians studying topology.
Differentiable manifoldManifold upon which it is possible to perform calculusA nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner.
Mechanical engineeringMechanical engineering is the study of physical machines and mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and maintain mechanical systems. It is one of the oldest and broadest of the engineering branches.*As an Amazon Associate I earn from qualifying purchases.