(informal) The proposition that a monkey hitting typewriter keys at random is bound to produce meaningful text (such as a work by William Shakespeare), given an infinite amount of time.

(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.

(mathematics) The theorem that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients, and that each component of the link can be assumed to be unknotted.

(mathematics) A theorem that describes the properties of the support of the convolution of two functions.

A theorem of representation theory and quantum mechanics, stating that matrix elements of spherical tensor operators on the basis of angular momentum eigenstates can be expressed as the product of two factors, one of which is independent of angular momentum orientation, and the other a Clebsch-Gordan coefficient.

(mathematical logic) A theorem stating that, if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ. The result implies that first-order theories are unable to control the cardinality of their infinite models, and that no first-order theory with an infinite model can have a unique model up to isomorphism.

(mathematics) A theorem stating that, if L is a complete lattice and f : L → L is an order-preserving function, then the set of fixed points of f in L is also a complete lattice. It has important applications in formal semantics of programming languages and abstract interpretation.

(combinatorial game theory) A theorem stating that every impartial game under the normal play convention is equivalent to a nimber.

(mathematics) A theorem in operator theory, stating that, if T is a contraction on a Hilbert space H, and U is its minimal unitary dilation acting on some Hilbert space K, and R is an operator on H commuting with T, then there is an operator S on K commuting with U such that RTⁿ=P_HSUⁿ|_H;∀n≥0, and ‖S‖=‖R‖. In other words, an operator from the commutant of T can be "lifted" to an operator in the commutant of the unitary dilation of T.

(mathematics) A basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices q > 0 are automatically zero.