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arithmetic – Borel measure
arithmetic
The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).
Aronszajn tree
Set Theory: For a a regular cardinal, an a-Aronszajn tree is a tree T with |T| = a, such that every chain in T is of cardinality less than a. Every Suslin tree is an Aronszajn tree. There are no Aronszajn trees of height w.
Ascoli-Arzelŕ Theorem
If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (fn} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets.
asin
Abbreviation of arc-sine.
associative
An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.
associative property
A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.
asymmetric relation
A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.
atan
Abbreviation of arc-tangent.
automorphism
An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.
automorphism group
The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.
axiom
In formal mathematics, a formula or schema of formulas stipulated as true in the theory under discussion, i.e., assumed to be true at the outset, and so not requiring proof. Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.
axiom of choice
An axiom of set theory, particularly ZFC, which asserts that from any (infinite) family of sets a new set may be created containing exactly one element from each set in the family. This axiom has been shown to be independent of the other axioms of ZFC (ZF). It is equivalent to: 1) Zorn's Lemma; 2) the Hausdorff Maximality Theorem; 3) the well-ordering principle; and 4) the statement that the cartesian product of an infinite family of sets is non-empty. Because the axiom of choice permits non-constructive proofs, it is rejected by intuitionism, and careful mathematicians refer explicitly to its use in proofs which require it.
Baire Category Theorem
If X is a complete metric space, then the intersection of a countable collection of dense open subsets of X is also dense. Equivalently, the union of a countable collection of closed, nowhere dense subsets of X is nowhere dense.
Banach space
A complete normed space.
Banach-Tarski Paradox
Given two bounded subsets of 3-dimensional Euclidean space, provided each has interior points, then the first may be decomposed into finitely many pieces and translated by rigid motions into a subset congruent to the second subset. For instance, a ball of unit radius can be decomposed into finitely many pieces, and then reassembled into two balls of unit radius. This shocking result is a consequence of the axiom of choice.
base
Number systems: In a place-notational number system, the number of symbols used. For example, in base two the two symbols 0 and 1 are used, and in base seven the seven symbols 0, 1, 2, 3, 4, 5, and 6 are used.
Exponential expressions: The number or expression which is “being raised to” the power of the exponent.
Topology: Given a topological space X, a base is a class B of sets such that, for every x in X and every neighborhood U of x, there is a set b in B such that x is contained in b and b is contained in U.
bijection
A bijective function, i.e., a function that is both an injection and a surjection.
bijective
A function is bijective if it is both injective and surjective, i.e., both “one-to-one” and “onto.”
binary operation
A binary operation on a set X is a function whose domain is the set of ordered pairs of elements from X and whose range is X.
Examples: addition on the set of integers, function composition.
Bolzano-Weierstrass property
A topological space X is said to have the Bolzano-Weierstrass property if every infinite sequence in X has a convergent subsequence. It is a theorem that every compact space has the Bolzano-Weierstrass property.
Borel measure
A measure m on a metric space X defined on a s-algebra of sets which contains the Borel sets is called a Borel measure. See also: Lebesgue measure.
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