convergent sequence finite measure
Naively, a region of space is convex if the line segement joining any two points of the region lies wholly within it. Thus, a polygon is convex if every line segment joining any two points on its sides lies entirely within the polygon. (This is equivalent to the condition that all its interior angles be less than 180°.)
More generally, a region in a real vector space is convex if whenever two points x and y are in the region then so is any point tx + (1 - t)y, where t lies in the interval [0, 1]. See the immediately following entries for additional uses of the descriptor “convex.”
A function is convex if the chord joining any two points of its graph lies entirely above the graph.
Cf. concave function.
A set is countable if it is finite, or if it is infinite and bijective to the set of natural numbers (finite ordinals), i.e., if there exists a complete one-to-one mapping of the set in question onto the set N. Sets that are both countable and infinite are sometimes called denumerable. Georg Cantor proved that sets may be uncountably infinite, for example the set of real numbers.
Topology: a collection of sets which contains a given set. If the sets in the covering collection are open sets, the cover is called an open cover.
Partially ordered sets: If x and y are elements of a partially ordered set such that x y, and such that there is no z such that x z y, then we say that x covers y.
Graph Theory: An edge or vertex of a graph is said to cover (verb) those vertices or edges, respectively, that it is incident on. A set of edges or vertices is said to cover any vertex or edge covered by any element of that set. A set of edges or vertices that covers all the vertices or edges, respectively, of the graph is called a cover (noun), usually with a specification of whether it consists of edges or vertices. See edge cover, vertex cover.
Given a space X and a subset A of X, we say that A is dense in X if the intersection of every open set of X with A is non-empty. E.g., the rational numbers are dense in the real numbers.
A set is denumerable if it is infinite and countable.
Given a set X, the derived set of X is the set of accumulation points of X. The second derived set is the derived set of the derived set, and so on.
Cf. Cantor-Bendixson Theorem.
Geometry: A diameter of a circle (or sphere) is a line containing the center and with endpoints on the perimeter (resp. surface).
Analysis: Given a set X in a metric space, the diameter of X is the supremum of the distances between all pairs of points of X.
Graph Theory: The diameter of a given graph G is the maximum, over all pairs of vertices u, v of G that are in the same connected component of G, of the distance between u and v. In other words, it is the greatest distance between two vertices on the graph.
General: So-called “Discrete Mathematics” consists of those branches of mathematics which are concerned with the relations among fixed rather than continuously varying quantities, e.g., combinatorics and probability.
Topology: A topology on a set X is discrete if every subset of X is open, or equivalently if every one-point set of X is open.
A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.
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.