Differential topology also deals with questions like these, which specifically pertain to the properties of differentiable mappings on R < sup > n </ sup > ( for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth ).

The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed ( with respect to the Krull topology below ) subgroups of the Galois group correspond to the intermediate fields of the field extension.

If E / F is a Galois extension, then Gal ( E / F ) can be given a topology, called the Krull topology, that makes it into a profinite group.

In mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact.

* Homotopy extension property, a property in algebraic topology.

The fundamental result of class field theory states that the group G is naturally isomorphic to the profinite completion of the idele class group C < sub > K </ sub > of K with respect to the natural topology on C < sub > K </ sub > related to the specific structure of the field K. Equivalently, for any finite Galois extension L of K, there is an isomorphism

This work revealed that S4 and S5 are models of interior algebra, a proper extension of Boolean algebra originally designed to capture the properties of the interior and closure operators of topology.

# Definition of the integral for functions in L < sup > 1 </ sup >( X, μ ) as extension by continuity ( after verifying that μ is continuous with respect to the topology of L < sup > 1 </ sup >( X, μ ))

# Closed extension topology

# Open extension topology

From about the 1920s onwards, it was realised that tensors play a basic role in algebraic topology ( for example in the Künneth theorem ).

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

An important fact about the weak * topology is the Banach – Alaoglu theorem: if X is normed, then the closed unit ball in X * is weak *- compact ( more generally, the polar in X * of a neighborhood of 0 in X is weak *- compact ).

A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using relative homotopy groups ; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation.

* Invariance of domain, a theorem in topology about homeomorphic subsets of Euclidean space

It occurs in the proofs of several theorems of crucial importance, for instance the Hahn – Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that every nonzero ring has a maximal ideal and that every field has an algebraic closure.

The Baire category theorem is an important tool in general topology and functional analysis.

In the topology of metric spaces the Heine – Borel theorem, named after Eduard Heine and Émile Borel, states:

This follows from Tychonoff's theorem and the characterisation of the topology on R < nowiki ></ nowiki > X < nowiki ></ nowiki > as a product topology.

Several texts identify Tychonoff's theorem as the single most important result in general topology Willard, p. 120 ; others allow it to share this honor with Urysohn's lemma.

The theorem crucially depends upon the precise definitions of compactness and of the product topology ; in fact, Tychonoff's 1935 paper defines the product topology for the first time.

Studying the strength of Tychonoff's theorem for various restricted classes of spaces is an active area in set-theoretic topology.

The analogue of Tychonoff's theorem in pointless topology does not require any form of the axiom of choice.

From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry ; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis.

The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).

* Jordan curve theorem in topology

* Jordan – Schönflies theorem in geometric topology

Brouwer also proved the simplicial approximation theorem in the foundations of algebraic topology, which justifies the reduction to combinatorial terms, after sufficient subdivision of simplicial complexes, of the treatment of general continuous mappings.

In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a function.

Lastly, there is the primitive causality restriction which states that any polynomial in the smeared fields can be arbitrarily accurately approximated ( i. e. is the limit of operators in the weak topology ) by polynomials over fields smeared over test functions with support in

An important theorem of topology, the classification theorem for surfaces, states that all two-dimensional compact manifolds without boundary

The Gelfand representation or Gelfand isomorphism for a commutative C *- algebra with unit is an isometric *- isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak * topology.

The free product is important in algebraic topology because of van Kampen's theorem, which states that the fundamental group of the union of two path-connected topological spaces is always an amalgamated free product of the fundamental groups of the spaces.

In differential topology, Smale's paradox states that it is possible to turn a sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process often called sphere eversion ( eversion means " to turn inside out ").

The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres.

The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number of tori and some number of real projective planes.

In functional analysis and related branches of mathematics, the Banach – Alaoglu theorem ( also known as Alaoglu's theorem ) states that the closed unit ball of the dual space of a normed vector space is compact in the weak * topology.

He is best known for his work on topology, including the metrization theorem he proved in 1926, and the Tychonoff's theorem, which states that every product of arbitrarily many compact topological spaces is again compact.

In differential topology, the Whitney immersion theorem states that for, any smooth-dimensional manifold can be immersed in Euclidean-space.

The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry.

In geometric topology, the spherical space form conjecture states that a finite group acting on the 3-sphere is conjugate to a group of isometries of the 3-sphere acting by left translation.

The first two axioms are algebraic, and state that is a representation of the semigroup (); the last is topological, and states that the map is continuous in the strong operator topology.

The Mackey – Arens theorem states that all possible dual topologies are finer than the weak topology and coarser than the Mackey topology.