Hausdorff space least upper bound axiom
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heawood Map Coloring Theorem
For every positive integer n, the chromatic number of the orientable surface of genus n (sphere with n handles) is given by
For n = 0, this statement is exactly the Four Color Theorem, which was proven later, and does not bear Heawood's name.
Every closed and bounded subset of the real numbers is compact.
A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous.
A function that maps each domain element to itself. Also called the identity map.
See identity function.
Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f.
Abbreviation of infimum.
The infimum of any subset of a linear order (linearly ordered set) is the greatest lower bound of the subset. In particular, the infimum of any set of numbers is the largest number in the set which is less than or equal to every other number in the set. In a complete linear order the infimum of any bounded set always exists.
Cf. supremum, least upper bound axiom.
An injective function, i.e., a function that is “one-to-one.” Equivalently, a function that maps exactly one element of its domain to each element of its range.
Cf. surjection, bijection.
A function is injective, also called “one-to-one,” if to each element of the range at most one element of the domain is mapped by the function.
Cf. surjective, bijective.
An antiderivative of a function. That is, if f(x) is a real-valued function, an antiderivative F(x) of f(x) has the property that the derivative of F with respect to x is f.
The definite integral (called the Riemann integral) of a real-valued function f(x) from x = a to x = b is the limit of the Riemann sum:
assuming this limit exists, where ci is in the i th subinterval of the partition of (a, b), and where a is the lower limit, and b is the upper limit, of the integral.
Cf. fundamental theorem of calculus, Lebesgue integral.
The interior of a subset E of a topological space is the largest open set contained in E. It may also be expressed as the intersection of E with the closure of its complement. If E is open, then it is equal to its interior. If E contains no open sets we say that it has empty interior, and this equivalent to saying that E is nowhere dense.
The set of all real numbers lying between two given real numbers a and b. If both a and b are included in the set it is called a closed interval. If neither a nor b is included it is called an open interval. If only one of a or b is included the interval is called half-open (equivalently, half-closed). Intervals may be denoted using either interval notation or set notation, as follows:
Although intervals are most commonly intervals on the real line, the definition carries over without modification to any totally ordered set.
An intersection graph of a finite family of intervals on the real line.
Given a function f and a subset Y of the range of f, the inverse image (under f) of Y is the set of all x such that f(x) is in Y.
A morphism that is both injective and surjective, that is, both “one-to-one” and “onto.” An isomorphism from a structure to itself is called an automorphism.
A function that is order preserving and increasing.
Jordan Decomposition Theorem
If m is a signed measure, there exist unique positive measures m+ and m– such that m = m+ – m–, with m+ and m– mutually singular. This is called the Jordan decomposition of m, and the measures m+ and m– are called the positive and negative variations of m. The total variation of m is defined to be the sum of the positive and negative variations of m.
least upper bound
An upper bound which is less than or equal to every other upper bound.
Cf. greatest lower bound.
least upper bound axiom
“Any subset of the real numbers which has an upper bound has a least upper bound.” This axiom, together with the field axioms, completely characterizes the set of real numbers.