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Given F_1(x,y,z,u,v,w) = 0 (1) F_2(x,y,z,u,v,w) = 0 (2) F_3(x,y,z,u,v,w) = 0, (3) if the determinantof the Jacobian |JF(u,v,w)|=|(partial(F_1,F_2,F_3))/(partial(u,v,w))|!=0, ...

A member of a collection of sets is said to be maximal if it cannot be expanded to another member by addition of any element. Maximal sets are important in graph theory since ...

A subset E of a topological space S is said to be nonmeager if E is of second category in S, i.e., if E cannot be written as the countable union of subsets which are nowhere ...

A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.

A subset A subset= X of a topological space X is said to be disconnected if it is not connected.

Three sets of three lines such that each line is incident with two from both other sets.

If F is a sigma-algebra and A is a subset of X, then A is called measurable if A is a member of F. X need not have, a priori, a topological structure. Even if it does, there ...

Let (A,<=) and (B,<=) be disjoint totally ordered sets with order types alpha and beta. Then the ordinal sum is defined at set (C=A union B,<=) where, if c_1 and c_2 are both ...

A point B is said to lie between points A and C (where A, B, and C are distinct collinear points) if AB+BC=AC. A number of Euclid's proofs depend on the idea of betweenness ...

If S is any nonempty partially ordered set in which every chain has an upper bound, then S has a maximal element. This statement is equivalent to the axiom of choice. Renteln ...