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topology – Turing machine
topology
Generally, topology is the study of those properties of a space which are invariant under continuous deformations, i.e., deformations which do not create tears or holes. More specifically, given a set X, a topology on X is a collection of subsets of X, called the open sets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersections and arbitrary (i.e., not necessarily finite or countable) unions. A set X with a topology defined upon it is called a topological space.
 Cf. homeomorphism.
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.
transcendental function
A function which is not an algebraic function, i.e., a function whose action on its argument(s) cannot be represented by the arithmetic and algebraic operations: addition and subtraction, multiplication and division, raising to a power, or extraction of roots. The exponential function, the logarithmic function, and the trigonometric functions are examples of transcendental functions.
transcendental number
A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well.
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.
triangle
Geometry: A closed plane figure with three straight sides meeting at three vertices. If one side of a triangle is chosen as the base, then the height of the triangle is the perpendicular distance to the base from the vertex opposite the base.
Triangles are classified by their angle measures:- Acute all angles less than 90°.
- Obtuse one angle greater than 90°.
- Right one angle exactly 90°.
- Scalene all angles and sides unequal.
- Isosceles two angles equal (equivalently, two sides equal).
- Equilateral all angles equal (equivalently, all sides equal).
On a right triangle, the sides adjacent to the right angle are called the legs, and the side opposite is called the hypotenuse.
Cf. Pythagorean Theorem.
Graph Theory: A cycle with three vertices.
trigonometric function
The trig functions are transcendental functions defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions. See the related article for a complete description.
Related article: Trig Functions and Identities
trigonometric identity
ARTICLE
An established rule for manipulating trigonometric expressions. See the article for a comprehensive listing.
trigonometry
The mathematical subject concerned with trigonometric functions, which are defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions.
Related article: Trig Functions and Identities
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.
trivial group
The group with only one element (necessarily the identity element).
trivial topology
For a given space X, the trivial topology is the topology whose only open sets are the empty set and X itself.
tuple
An ordered list of elements from a set, usually represented as (a1, a2, a3, ... ,an). (Sometimes angle brackets are used in place of parentheses.) A tuple with only two elements is called an ordered pair, and tuples with 3, 4, and 5 elements are called ordered triples, quadruples, and quintuples, respectively. In general, a tuple with n elements is called an n-tuple. A relation on a family of sets is represented by a set of tuples.
Cf. flat pair.
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.
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