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open disk – polyhedron
open disk
The interior of a circle.
 Cf. neighborhood, disk.
open function
A function from one topological space into another is called open if the image of every open set of the domain is an open set in the range.
open interval
An interval of the real number line (or any other totally ordered set) which does not include its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. closed interval.
open set
A subset U of a topological space X is open if every element x of U is contained in an open set of X that is also contained in U. In a metric space, U is open if for every x in U we may find a d greater than zero such that the d neighborhood of x is also contained in U.
order-preserving function
A function f is called order-preserving if it preserves the order of its domain elements, that is, if whenever x and y are elements of its domain such that x  y then f(x) f(y). Also called isotone or inctreasing. If f reverses the order of its domain elements, then it is called antitone or decreasing. In either case f is called monotone or monotonic. If whenever x < y we have f(x) < f(y), then f is called strictly increasing (resp. decreasing).
order topology
A topology on a totally ordered set that agrees with the order. Specifically, given a totally ordered set X with total order relation <, we define the order topology T on X to be the collection of all arbitrary unions of open intervals of X under <.
order type
See total order.
outer measure
A non-negative extended real-valued set function defined on all subsets of a space X that is zero on the empty set, monotonic, and countably subadditive (see below) is called an outer measure. An outer measure is often used together with Caratheodory's Theorem (see below) to obtain a measure. Given a set X and a collection A of subsets of X which includes the empty set and X itself, and a positive real-valued function r whose domain is A and whose value on the empty set is zero, then for any F in X define m* by
Then m* is an outer measure. A set B in X is then called m*-measurable if
for all subsets C of X. (Cc denotes the complement of C in X.) Carathéodory’s Theorem states that if m* is an outer measure on X, then the collection M of m*-measurable sets is a s-algebra of sets, and the restriction of m* to M is a complete measure.
parabola
The locus of points in the plane whose distances from a fixed point, called the focus, and a fixed line, called the directrix, are equal.
Like the ellipse and hyperbola, the parabola is a conic section. See the related article for a full exposition.
Related article: Conics
parallelogram
A quadrilateral with opposite sides (and opposite angles) equal.
parallel postulate
The 5th postulate of Euclidean geometry.
partition
General: A partition of a set X is a collection of subsets of X such that every element of X is in exactly one of the subsets. Such a partition is given by (and gives rise to) an equivalence relation on X. For example, division modulo 3 partitions the set of natural numbers into three subsets, each containing all those numbers leaving remainders of 0, 1, or 2 respectively when divided by 3.
Algebra: A partition of a matrix is a division of the matrix into conformable submatrices.
Analysis: A partition of a space is a collection of pairwise disjoint regions of the space whose union is the entire space. For example, a partition of an interval [a, b] of the real line is given by a finite set of points {xi} such that a = x1 < x2 < . . . < xn = b which divide the interval into disjoint subintervals.
Number Theory: Given a positive integer n, a partition of n is a set of positive integers whose sum is n. For example, 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 are the four possible partitions of the number 4.
perfect set
A closed set X is called perfect if every point of X is an accumulation point of X. The following are equivalent characterizations:- X is closed and containes no isolated points.
- X is closed and dense in itself.
- X is equal to its derived set.
perpendicular
Two lines are perpendicular if they intersect in a right angle. In particular, two lines in the Cartesian plane are perpendicular if their respective slopes m1 and m2 satisfy m 1 × m2 = –1.
Two curves intersect perpendicularly if their tangent lines at the point of intersection are perpendicular.
Perpendicularity of planes and surfaces is defined analogously. Cf. normal.
perpendicular distance
Geometry: The shortest distance between a point and a line, or between a point and a plane. Sometimes called directed distance.
Pi (p)
The real number that is equal to the ratio of the circumference of any circle to its diameter, approximately equal to 3.1415926. It is known to be a transcendental number.
plane figure
A figure, i.e., a graphical (visual) representation of points, lines, curves, or regions in the geometric plane. A plane figure, such as a polygon, that bounds a finite region is called closed.
Platonic solid
ARTICLE
A solid having congruent, regular polygonal faces. There are five Platonic solids: the icosahedron, tetrahedron, octahedron, dodecahedron, and cube.
pointwise bounded
See bounded.
polygon
A closed plane figure having straight sides. A polygon that has equal sides and equal angles is called a regular polygon. Examples: square, hexagon, nonagon, etc.
polyhedron
A closed, solid figure consisting of four or more planar faces, pairs of which meet at edges. The faces form polygons, and the points where three or more edges meet are called vertices. The number of faces, edges, and vertices of a polyhedron are related by the Euler polyhedron formula.
A polyhedron with four faces is called a tetrahedron, a pentahedron has five faces, a hexadron six, a heptahedron seven, an octahedron eight, a dodecahedron twelve, and an icosahedron twenty. If all the faces of a polyhedron are congruent, it is called a regular solid or Platonic solid.
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