** Gromov's compactness theorem ( geometry ) in Riemannian geometry

** The Banach – Alaoglu theorem about compactness of sets of functionals.

** Well-ordering theorem: Every set can be well-ordered.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.

** The Nielsen – Schreier theorem, that every subgroup of a free group is free.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

** The theorem that every Hilbert space has an orthonormal basis.

** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.

** Hilbert's basis theorem

** Bayes ' theorem

** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.

** Lyapunov's central limit theorem

** Superposition theorem, in electronics

** " Kelvin's vorticity theorem for incompressible or barotropic flow ".

** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem

** Various proofs of the four colour theorem.

** In the product topology, the closure of a product of subsets is equal to the product of the closures.

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

** The set of infinite strings is the inverse limit of the set of finite strings, and is thus endowed with the limit topology.

** the cap, cup and slant product in algebraic topology.

** Category ( topology ) in the context of Baire spaces

** Section ( fiber bundle ), in topology

** Join ( topology ), an operation combining two topological spaces

** Frobenius theorem ( differential topology )

** Cover ( topology ), a system of ( usually, open or closed ) sets whose union is a given topological space

** The topology of F ( M ) depends only on the topology of M, not on the metric d.

** Hausdorff topologies are better-behaved than those in arbitrary general topology.

** Differential topology, in multivariable calculus, the differential of a smooth map between Euclidean spaces or differentiable manifolds is the approximating linear map between the tangent spaces, called pushforward ( differential )

** Krull topology

** In geographic information systems and their data structures, the terms " topology " and " planar enforcement " are used to indicate that the border line between two neighboring areas ( and the border point between two connecting lines ) is stored only once.

** Also in cartography, a topological map is a much simplified map that preserves the mathematical topology while sacrificing scale and shape

** triangulation ( topology ), generalizations to topological spaces other than R < sup > d </ sup >

** Arbitrated loop, for second alternate FC topology

** Fibre Channel point-to-point, for alternate FC topology

** Continuity ( topology ), a generalization to functions between topological spaces

** End ( topology )