(set theory) A theorem which (in a simpler formulation) states that a closed uncountable set in Euclidean n-space is equal to the disjoint union of a perfect set and a countable set.

(calculus) A theorem that states for each value between the least upper bound and greatest lower bound of the image of a continuous function there is a corresponding point in its domain that the function maps to that value.

(mathematics) Any of various theorems that saliently concern mean values.

(mathematical analysis) A theorem which states that any real-valued Lebesgue integrable function can be approached arbitrarily closely from below by an upper semicontinuous function and also from above by a lower semicontinuous function.

(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.