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topology – universal quantifier
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 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 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.
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
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 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.
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.
uncountable
A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”).
Related MiniText: Infinity -- You Can't Get There From Here...
uncountably infinite
See uncountable.
union
The union of two sets A and B is the set consisting of those elements which are in A or in B or in both, and is denoted by
If the union is taken over a family of sets {Ai}i = 1, 2, ..., n, then it is the set consisting of those elements that are in at least one of the sets in the collection.
Sometimes the word “union” is used to indicate the sumset of a set A, and is then loosely described as “union A.” In this latter case the union is understood to be the union over the elements of A, and is denoted by
When considering an algebra of sets, the union of two or more sets is sometimes called the join of the sets.
Cf. intersection.
unit circle
A circle with radius 1.
unit interval
The interval on the real number line from 0 to 1, inclusive.
unit square
The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary.
universal quantifier
See predicate calculus.
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