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closed set – continuum
closed set
Topology: A subset E of a topological space X is closed if X - E (set difference) is open. In a metric space, E is closed if every convergent sequence in E converges in E; equivalently, if every accumulation point of E is in E.
 Set Theory: If a is a limit ordinal, then a set C contained in a is called closed if and only if for every limit ordinal b less than a, if C  b is unbounded in b, then b  C. C is called c.u.b. (“cub set” or “club set”) if and only if C is closed and unbounded in a.
Cf. stationary set.
closed set system
If X is a set (or proper class) and F is a family of subsets of X, then F is called a closed set system provided- X is a member of F, and
- F is closed under arbitrary intersections.
Cf. filter.
closure
Topology: The closure of a subset E of a topological space is the smallest closed set containing E. It may also be expressed as the union of E with its accumulation points. If E is closed, then it is equal to its closure.
Algebra: An algebraic closure of a field F is a field G containing F such that every polynomial with coefficients from F has a root in G.
closure operator
If X is a set, then a function C from P(X) into P(X) (i.e., a function on the power set of X) is called a closure operator provided- Y is contained in C(Y) for every subset Y of X,
- C(C(Y)) = C(Y) for every subset Y of X, and
- If Y and Z are both subsets of X, with Y a subset of Z, then C(Y) is a subset of C(Z).
Closure operators induce closed set systems.
closure property
A property of subsets of a set X is a closure property if X has the property and the intersection of any subsets of X having the property also has the property.
club set
Closed, unbounded set. See closed set.
cofinality
A function f which maps an ordinal a into an ordinal b is said to map a cofinally if the range of f is not bounded in b. (I.e., for every b in b there is an a in a such that f(a)  b.) The cofinality of an ordinal b is the least ordinal a such that there is a cofinal map of a into b.
compact
Topology: In a topological space, a set E is compact if every open covering of E has a finite subcover, i.e., a finite subcollection which also covers E. A space X is compact if and only if every collection of closed sets with the finite intersection property has a non-empty intersection. E is called s-compact if there exists a sequence of compact sets {Ci} such that E is contained in their union.
Cf. locally compact, Bolzano-Weierstrass property, Heine-Borel property.
Set Theory: A cardinal k is called weakly compact if it is uncountable and
Equivalently, k is weakly compact if it is strongly inaccessible and there are no k-Aronszajn trees.
Lattices: an element a of a lattice L is called compact if whenever a is dominated by the join of a subset X of L then a is dominated by the join of a finite subset of X. Symbolically:
complement
The complement of a set A is the set of all elements that are not elements of A.
Graph Theory: The complement of a simple graph G with vertex set V is the simple graph Gc, which also has vertex set V, and in which two vertices are adjacent if and only if they are not adjacent in G.
complete
Analysis: A metric space X is complete if every Cauchy sequence in X converges in X.
Logic: a system of axioms for a mathematical theory is complete if every theorem in the theory is deducible from the axioms. Gödel's incompleteness theorem states that any axiom system which includes or allows the operations of arithmetic is necessarily incomplete.
Set Theory: If F is a filter on a set X and k is a regular, uncountable cardinal, then we say that F is k-complete (k-closed) if AF for every A F with |A| < k. Every filter is w-complete. If k is the first uncountable cardinal ( 1), then F is called countably complete.
Related article: Gödel's Theorems
complete lattice
A lattice X is complete if every subset of X has a least upper bound and a greatest lower bound in X.
complete measure
See measure.
complete ordered field
A linearly ordered field in which every subset with an upper bound has a least upper bound. Every complete ordered field is isomorphic to the set of real numbers.
concave
A region of space is concave if there are two points of the region such that a line joining the two points is not entirely contained within the region. In particular, a polygon is concave if any of its interior angles is greater than 180°.
Cf. convex.
concave function
A function is concave if the chord connecting any two points of its graph lies entirely below the graph.
Cf. convex function.
conditional statement
A statement of the form “if A then B,” or “A implies B.” A conditional is equivalent to its contrapositive, “not B implies not A.” See also: inverse statement and converse statement.
congruence relation
Given a structure X (i.e., X is a space, algebra, etc.), then an equivalence relation “ ~ ” on X is called a congruence relation if it is preserved by every defined operation on X. That is, for any x, y in X, if x ~ y and f is defined on X, then f(x) ~ f(y).
connected
A topological space X is connected if there are no two open sets of X whose union is X and whose intersection is empty.
constant function
A constant function f is one whose value is the same at all points of its domain.
continuous
Analysis: A function f is continuous at a point x of its domain if, whenever we are given a number e greater than 0, we may find a d greater than 0 so that whenever y is within a d-neighborhood of x, then f(y) is within an e-neighborhood of f(x).
Topology: A transformation of one topological space into another is continuous if the inverse image of every open set is open, or equivalently if the inverse image of every closed set is closed.
Cf. uniformly continuous, equicontinuous, absolutely continuous.
continuum
In mathematics, the real numbers or real number line.
Related article: Gödel's Theorems
Related MiniText: Infinity -- You Can't Get There From Here...
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