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supremum – Tychonoff’s Theorem
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.
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.
Suslin tree
Set Theory: For a an infinite cardinal, an a-Suslin tree is a tree T such that |T| = a, and every chain and every antichain of T has cardinality less than a.
Cf. Aronszajn tree.
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.
symmetric relation
A relation “ ~ ” on a set X is symmetric if for any two elements x and y in X we have x ~ y if and only if y ~ x. The relation “ ~ ” is called asymmetric if for any two elements x and y we have that x ~ y implies it is not true that y ~ x, and it is called antisymmetric if whenever x ~ y and y ~ x then x = y. Note that a relation may be neither symmetric, asymmetric, nor antisymmetric.
Cf. reflexive relation, transitive relation.
Tarski Truth Theorem
Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let T j be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’) Tj.
In other words, “truth” in a theory T is not definable in T.
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.
totally bounded
Given a metric space X, a subset E of X is called totally bounded if for every e greater than zero there is a finite covering of E by open spheres in X whose radius is less than e.
Cf. bounded.
totally ordered set
A set with a total order defined on it.
total order
An order relation “<” on a set S is a total order exactly if, for any two elements x and y of S, either x < y, x = y, or y < x, but no two of the three. Sometimes also called a linear order. Every totally ordered set is order-isomorphic to an ordinal, and this ordinal is called its order type (or sometimes just “type.”)
Cf. partial order
total variation
(For total variation of a signed measure, see the Jordan Decomposition Theorem.) If f is a function on the real numbers with range in the complex numbers, then the function Tf given by
is called the total variation function of f, where here the supremum is being taken over all finite partitions of the real line up to x. If the limit of this function as x goes to infinity is finite, then f is said to be of bounded variation. The space of all such functions is usually denoted by BV. Functions which are increasing and bounded are in BV, and differentiable functions whose derivative is bounded are in BV.
transitive closure
If x is any set, the transitive closure of x is the set containing all of the elements of x, and all of the elements of the elements of x, and all of the elements of the elements of the elements of x, and so on. Formally, the transitive closure Tc(x) of x is defined by the recursive formulas
where w is the set of finite ordinals.
transitive relation
A relation “ ~ ” on a set X is called transitive if it is the case that for every x, y, and z in X, if x ~ y and y ~ z , then x ~ z. For example, the relation “ < ” (less-than) on the set of natural numbers is transitive.
Cf. reflexive relation, symmetric relation.
transitive set
A set is called transitive if it is equal to its transitive closure. That is, x is a transitive set if whenever y is an element of x and z is an element of y, then z is an element of x.
tree
Graph Theory: A connected graph that does not contain any cycles. So named because they vaguely resemble trees in appearance.
Cf. forest.
Set Theory: A partial order T in which, for each x in T, the set {y T: y < x} is well-ordered. The height of x, denoted ht(x, T ), is the order type of the set {y T: y < x}. For each ordinal a, the a-th level of T, denoted Leva(T ), is the set {x T: ht(x, T ) = a}. The height ht(T ) of T is the least a such that Leva(T ) =  . A sub-tree of T is a subset S  T with the induced order such that whenever x S and y < x, then y S.
trivial
The smallest, simplest, and usually least interesting example of any object or construction. Every field has a specific definition of what is considered the trivial object of study in that field. The following entries provide examples.
Logic: A conclusion is trivial if it is so obvious that no proof or demonstration is required to establish its truth.
trivial automorphism
The identity function is a trivial automorphism of any set.
Turing machine
An abstract machine consisting of a collection of ordered quadruples, corresponding to i) the contents of the current memory location; ii) the current state of the machine; iii) the action to be performed (erase or write to the current memory location and move to a new memory location); and iv) the new state of the machine. All modern computers are universal Turing machines.
Tychonoff’s Theorem
If {Xa}, a in some index set A, is any family of compact topological spaces, then the cartesian product of the Xa, with the product topology, is compact.
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