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sphere – topological space
sphere
A closed surface, all points of which are equidistant from a given point, called the center.
In 3-dimensional Euclidean space, the equation of a sphere of radius r and center (h, j, k) is
The term sphere may also refer to the solid bounded by this surface, and the interior is then called the open sphere of radius r.
 More generally, a sphere may be defined as the set of points in n-dimensional space (or any metric space) equidistant from a given point. The unit sphere in n-dimensional space is typically denoted S n - 1. Thus, the unit sphere in ordinary 3-space is denoted S2, and the unit circle in the plane is denoted S1.
square
A regular polygon having four equal sides and four right angles.
Stone-Weierstrass Theorem
If X is a compact space and C(X) denotes the space of all continuous functions on X, and A is an algebra of functions in C(X) which separates the points of C(X) and which contains a constant function f not identically zero, then A is dense in C(X).
story problem
ARTICLE
A mathematical problem presented as a real-world situation. See the article for problem solving techniques.
subbase
In a topological space, a subbase is a collection of sets, the collection of all finite intersections of which constitutes a base.
subset
A set A is a subset of a set B if every element of A is also an element of B. If in addition B is a subset of A, then A = B, but if not then A may be said to be a proper subset of B.
Cf. superset.
subtract
To subtract a number m from a number n is to calculate the difference of m and n. If m is less than n we take the positive difference, otherwise we take the negative of the difference. This is tantamount to adding the negative of m to n.
successor
In a structure with an order relation defined upon it, the successor of an element a is the least element greater than a, if such exists.
Cf. predecessor.
sup
Abbreviation of supremum.
supplemental angles
Two angles are supplemental if they add up to 180 degrees (p radians).
Cf. complementary angles.
supremum
The supremum of any subset of a linearly ordered set is the least upper bound of the subset. In particular, the supremum of any set of numbers is the smallest number in the set which is greater than or equal to every number in the set. In a complete linear order the supremum of any bounded set always exists.
Cf. infimum, least upper bound axiom.
surd
(rare) An irrational root of a number, e.g., the square root of two.
Related article: Irrationality of the Square Root of 2
surjection
A surjective function, i.e., a function that maps at least one element of its domain to each element of its range.
Cf. injection, bijection.
surjective
A function f from a set X to a set Y is surjective, also called “onto,” if to each element y of Y there is an element x of X such that f maps x to y, i.e., f (x)=y. Compare: injective, bijective.
symbolic logic
Logic reduced to syntax, i.e., which works only with uninterpreted symbols. The two most often used kinds of symbolic logic are the propositional calculus and the predicate calculus.
symmetric difference
The symmetric difference of two sets A and B is the set of those elements that are in either A or B but not both.
Cf. intersection, union.
Taylor series
Given a function having derivatives of all orders, the Taylor series of the function is given by
where f (k)(a) is the kth derivative of f at a. A function is equal to its Taylor series if and only if its error term Rn can be made arbitrarily small, where
It can be shown that
for some c between a and x.
tesselation
A tiling of the plane, i.e. the use of plane figures to completely cover the plane without overlaps or gaps. A regular tesselation uses only a finite number of distinct shapes. Most regular tesselations are periodic, but some are aperiodic.
Cf. polygon, Penrose tiles.
Related MiniText: Mathematical Art of M.C. Escher
tetrahedron
A polyhedron having four faces.
The faces of a regular tetrahedron are congruent, equilateral triangles.
Cf. Platonic solid.
tiling
See tesselation.
topological space
A set with a topology defined upon it.
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