In classicalEuclidean geometry, a point is aprimitive notion, defined as "that which has no part". Points and other primitive notions are not defined in terms of other concepts, but only by certain formal properties, calledaxioms, that they must satisfy; for example,"there is exactly onestraight line that passes through two distinct points". As physical diagrams,geometric figures are made with tools such as acompass,scriber, or pen, whose pointed tip can mark a small dot or prick a small hole representing a point, or can be drawn across a surface to represent a curve.
A point can also be determined by theintersection of two curves or three surfaces, called avertex orcorner.Since the advent ofanalytic geometry, points are often defined or represented in terms of numericalcoordinates. In modern mathematics, a space of points is typically treated as aset, apoint set.
Anisolated point is an element of some subset of points which has someneighborhood containing no other points of the subset.
Points, considered within the framework ofEuclidean geometry, are one of the most fundamental objects.Euclid originally defined the point as "that which has no part".[2] In the two-dimensionalEuclidean plane, a point is represented by anordered pair (x, y) of numbers, where the first numberconventionally represents thehorizontal and is often denoted byx, and the second number conventionally represents thevertical and is often denoted byy. This idea is easily generalized to three-dimensionalEuclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted byz. Further generalizations are represented by an orderedtuplet ofn terms,(a1, a2, … , an) wheren is thedimension of the space in which the point is located.[3]
Many constructs within Euclidean geometry consist of aninfinite collection of points that conform to certain axioms. This is usually represented by aset of points; As an example, aline is an infinite set of points of the formwherec1 throughcn andd are constants andn is the dimension of the space. Similar constructions exist that define theplane,line segment, and other related concepts.[4] A line segment consisting of only a single point is called adegenerate line segment.[citation needed]
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points, that any two points can be connected by a straight line.[5] This is easily confirmed under modern extensions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts known at the time. However, Euclid's postulation of points was neither complete nor definitive, and he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.[6]
The dimension of a vector space is the maximum size of alinearly independent subset. In a vector space consisting of a single point (which must be the zero vector0), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non-trivial linear combination making it zero:.
The topological dimension of a topological space is defined to be the minimum value ofn, such that every finiteopen cover of admits a finite open cover of whichrefines in which no point is included in more thann+1 elements. If no such minimaln exists, the space is said to be of infinite covering dimension.
A point iszero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set.
LetX be ametric space. IfS ⊂X andd ∈ [0, ∞), thed-dimensionalHausdorff content ofS is theinfimum of the set of numbersδ ≥ 0 such that there is some (indexed) collection ofballs coveringS withri > 0 for eachi ∈I that satisfies
TheHausdorff dimension ofX is defined by
A point has Hausdorff dimension 0 because it can be covered by a single ball of arbitrarily small radius.
Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgot it, e.g.noncommutative geometry andpointless topology. A "pointless" or "pointfree" space is defined not as aset, but via some structure (algebraic orlogical respectively) which looks like a well-known function space on the set: an algebra ofcontinuous functions or analgebra of sets respectively. More precisely, such structures generalize well-known spaces offunctions in a way that the operation "take a value at this point" may not be defined.[7] A further tradition starts from some books ofA. N. Whitehead in which the notion ofregion is assumed as a primitive together with the one ofinclusion orconnection.[8]
Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common inclassical electromagnetism, where electrons are idealized as points with non-zero charge). TheDirac delta function, orδ function, is (informally) ageneralized function on the real number line that is zero everywhere except at zero, with anintegral of one over the entire real line.[9] The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealizedpoint mass orpoint charge.[10] It was introduced by theoretical physicistPaul Dirac. In the context ofsignal processing it is often referred to as theunit impulse symbol (or function).[11] Its discrete analog is theKronecker delta function which is usually defined on a finite domain and takes values 0 and 1.
Gerla, G (1995)."Pointless Geometries"(PDF). In Buekenhout, F.; Kantor, W (eds.).Handbook of Incidence Geometry: Buildings and Foundations. North-Holland. pp. 1015–1031. Archived fromthe original(PDF) on 2011-07-17. Retrieved2017-12-22.
Whitehead, A. N. (1920).The Concept of Nature. Cambridge: University Press.{{cite book}}: CS1 maint: publisher location (link). 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered atTrinity College.
