(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.

(topology) A theorem which states that for any open covering of some given set in Euclidean n-space there is an open subcovering (of that same set) which is countable.

(number theory) A theorem that states precisely which quadratic imaginary number fields admit unique factorization in their ring of integers.

(topology) The theorem that states that a simple closed curve (Jordan curve) divides the plane into precisely two distinct areas.

(quantum mechanics) A no-go theorem that places certain constraints on the permissible types of hidden-variable theories that try to explain the apparent randomness of quantum mechanics as a deterministic model featuring hidden states. It demonstrates the impossibility of quantum-mechanical observables representing "elements of physical reality".

(number theory) The theorem that the Diophantine equation aⁿ+bⁿ=cⁿ has no solutions for positive integers a,b,c,n, where n>2.

(number theory) The theorem that, for any prime number p and integer a, aᵖ-a is an integer multiple of p.

(mathematics) In real algebraic geometry, a theorem that describes the possible numbers of connected components that an algebraic curve can have, in terms of the degree of the curve.

(mathematics) A theorem stating that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities.