Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
Euclidean planeIn mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane.
Line (geometry)In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ).
Line segmentIn geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
Cover (topology)In mathematics, and more particularly in set theory, a cover (or covering) of a set is a family of subsets of whose union is all of . More formally, if is an indexed family of subsets (indexed by the set ), then is a cover of if . Thus the collection is a cover of if each element of belongs to at least one of the subsets . A subcover of a cover of a set is a subset of the cover that also covers the set. A cover is called an open cover if each of its elements is an open set. Covers are commonly used in the context of topology.
Isolated pointIn mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S.
GeometryGeometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.
Accumulation pointIn mathematics, a limit point, accumulation point, or cluster point of a set in a topological space is a point that can be "approximated" by points of in the sense that every neighbourhood of with respect to the topology on also contains a point of other than itself. A limit point of a set does not itself have to be an element of There is also a closely related concept for sequences.
Hilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
Primitive notionIn mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem).
Axiomatic systemIn mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication.
Boundary (topology)In topology and mathematics in general, the boundary of a subset S of a topological space X 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 and . Some authors (for example Willard, in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds.
Locally compact spaceIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.
Scalar fieldIn mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin.
