(mathematical analysis) A theorem which states that for any subset S of an n-dimensional Euclidean space, S is compact if and only if it is both closed and bounded.

(rare) Synonym of Lindemann-Weierstrass theorem

(mathematics) A theorem stating that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence.

(mathematics) A theorem stating that for each positive integer d there is a finite set Y_d of prime numbers, such that if p is any prime not in Y_d then every homogeneous polynomial of degree d over the p-adic numbers in at least d²+1 variables has a nontrivial zero.

A theorem dealing with deterministic ordinal electoral systems that choose a single winner, stating that for every voting rule, one of the following three things must hold: (i) the rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or (ii) the rule limits the possible outcomes to two alternatives only; or (iii) the rule is susceptible to tactical voting: in certain conditions some voter's sincere ballot may not defend their opinion best.

(mathematics) An important statement about surfaces in differential geometry, connecting their geometry (in the sense of curvature) to their topology (in the sense of the Euler characteristic).

(mathematics) A theorem showing that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups; one of the foundational results of anabelian geometry.

(mathematics) A result about injectivity and surjectivity of polynomials, often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. The full theorem generalizes to any algebraic variety over an algebraically closed field.

(probability) A theorem relating the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions.

(computing theory) A fundamental theorem about the complexity of computable functions, stating that for any complexity measure there are computable functions that are not optimal with respect to that measure.