A morphism f from X to Y is called an epimorphism when it is surjective, that is, when to each element y of Y there corresponds an x in X such that f(x) = y.
Let C[a,b] denote the space of all continuous functions on the real interval [a,b]. A subset S of C[0,1] is called equicontinuous at x in [a,b] if for any e greater than zero there is some d greater than zero such that for every function f in S we have:
Two sets are equipotent if there exists a function between them that is bijective, that is, which is “one-to-one” and “onto.”
A space satisfying the axioms of Euclidean geometry. Specifically, a space in which the parallel postulate holds; equivalently, a space in which the distance between points is given by a straight line.
extended real numbers
The set of real numbers together with a “point(s) at infinity.” If both a positive and a negative infinity are added, then the resulting space is order equivalent and topologically equivalent (i.e., homeomorphic) to the unit interval [0, 1]. If only a single point at infinity is added, the resulting space is topologically equivalent to the unit circle.
If X is a set (or class) and F is a family of subsets of X, then F is called a filter provided
If X and F are as above, and if in addition for every subset Y of X either Y or its complement is in F, then F is called an ultrafilter (or maximal filter). Equivalently, an ultrafilter is a filter that is not the proper subset of any filter. A free ultrafilter is an ultrafilter which contains the complement of every finite set. If A is any subset of X, then the collection of supersets of A together with all finite intersections of supersets of A is called the filter generated by A. More generally, if U is any family of subsets of X that is closed unter finite intersections, then the family F of subsets which includes precisely U and all of the intersections of supersets of members of U is the filter generated by U, and U is called its filter base. If a filter F is generated by a singleton set, then it is called a principal filter.
- F is closed under intersections, i.e., for any sets A, B in F their intersection is also in F, and
- F is closed under supersets (upwardly closed), i.e., if A is any set in F and B is any set in X containing A, then B is in F.
finite intersection property
A collection of sets has the finite intersection property if every finite subcollection has non-empty intersection.
Topology: Given a topological surface S, the genus of S is the ordered pair (h, c) in which h is the number of handles on the surface and c is the number of cross-caps.
Graph Theory: The genus of a graph is the minimum genus of the topological surface on which the graph can be drawn without crossovers. Intuitively, the minumum number of handles one must attach to a sphere to draw the graph on it.
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.