# Category: Theorems in topology

Andreotti–Vesentini theorem
In mathematics, the Andreotti–Vesentini separation theorem, introduced by Aldo Andreotti and Edoardo Vesentini states that certain cohomology groups of coherent sheaves are separated.
Denjoy–Riesz theorem
In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-inters
Fáry–Milnor theorem
In the mathematical theory of knots, the Fáry–Milnor theorem, named after István Fáry and John Milnor, states that three-dimensional smooth curves with small total curvature must be unknotted. The the
Invariance of domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space . It states: If is an open subset of and is an injective continuous map, then is open in and is a homeomorph
Urysohn's lemma
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 continuous function. Urysohn's lemma is comm
Atiyah–Segal completion theorem
The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that th
Baire category theorem
The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be
Hsiang–Lawson's conjecture
In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere S3. The conjecture was featured by the Australian Mathematical Society Gazette a
Excision theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space and subspaces and such tha
Netto's theorem
In mathematical analysis, Netto's theorem states that continuous bijections of smooth manifolds preserve dimension. That is, there does not exist a continuous bijection between two smooth manifolds of
Lebesgue's number lemma
In topology, Lebesgue's number lemma, named after Henri Lebesgue, is a useful tool in the study of compact metric spaces. It states: If the metric space is compact and an open cover of is given, then
Kuiper's theorem
In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H. It states that the space GL(H) of invertible boun
Loop theorem
In mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos in 1956, along with Dehn's lemma a
Janiszewski's theorem
In mathematics, Janiszewski's theorem, named after the Polish mathematician Zygmunt Janiszewski, is a result concerning the topology of the plane or extended plane. It states that if A and B are close
Bagpipe theorem
In mathematics, the bagpipe theorem of Peter Nyikos describes the structure of the connected (but possibly non-paracompact) ω-bounded surfaces by showing that they are "bagpipes": the connected sum of
In mathematics, in the areas of topology and functional analysis, the Anderson–Kadec theorem states that any two infinite-dimensional, separable Banach spaces, or, more generally, Fréchet spaces, are
Katětov–Tong insertion theorem
The Katětov–Tong insertion theorem is a theorem of point-set topology proved independently by Miroslav Katětov and Hing Tong in the 1950s. The theorem states the following: Let be a normal topological
Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is th
Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended
Vector fields on spheres
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division alg
Reeb sphere theorem
In mathematics, Reeb sphere theorem, named after Georges Reeb, states that A closed oriented connected manifold M n that admits a having only centers is homeomorphic to the sphere Sn and the foliation
Nielsen–Thurston classification
In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen. Given a homeom
Cyclic surgery theorem
In three-dimensional topology, a branch of mathematics, the cyclic surgery theorem states that, for a compact, connected, orientable, irreducible three-manifold M whose boundary is a torus T, if M is
Lickorish–Wallace theorem
In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3
Quillen's theorems A and B
In topology, a branch of mathematics, Quillen's Theorem A gives a sufficient condition for the classifying spaces of two categories to be homotopy equivalent. Quillen's Theorem B gives a sufficient co
Jordan curve theorem
In topology, the Jordan curve theorem asserts that every Jordan curve (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an "exterior" region containing
De Rham's theorem
No description available.
Kline sphere characterization
In mathematics, a Kline sphere characterization, named after John Robert Kline, is a topological characterization of a two-dimensional sphere in terms of what sort of subset separates it. Its proof wa
Sphere theorem
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The prec
Metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metr
Gordon–Luecke theorem
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between
Denjoy's theorem on rotation number
In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. D
Side-approximation theorem
In geometric topology, the side-approximation theorem was proved by . It implies that a 2-sphere in R3 can be approximated by polyhedral 2-spheres.
Ham sandwich theorem
In mathematical measure theory, for every positive integer n the ham sandwich theorem states that given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide each one of th
Twisted Poincaré duality
In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local
Smith conjecture
In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith showed that a non-tr
Ehrenpreis conjecture
In mathematics, the Ehrenpreis conjecture of Leon Ehrenpreis states that for any K greater than 1, any two closed Riemann surfaces of genus at least 2 have finite-degree covers which are K-quasiconfor
Mostow–Palais theorem
In mathematics, the Mostow–Palais theorem is an equivariant version of the Whitney embedding theorem. It states that if a manifold is acted on by a compact Lie group with finitely many orbit types, th
Kakutani fixed-point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact s
Lusin's separation theorem
In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A
Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class vanishes), t
Scott core theorem
In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott,. The precise statement is as follows: Given a 3-manifold
Ellis–Numakura lemma
In mathematics, the Ellis–Numakura lemma states that if S is a non-empty semigroup with a topology such that S is compact and the product is semi-continuous, then S has an idempotent element p, (that
Kuratowski–Ulam theorem
In mathematics, the Kuratowski–Ulam theorem, introduced by Kazimierz Kuratowski and Stanislaw Ulam, called also the Fubini theorem for category, is an analog of Fubini's theorem for arbitrary second c
Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. Th
Fiber bundle construction theorem
In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives con
Borsuk–Ulam theorem
In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere
Blumberg theorem
In mathematics, the Blumberg theorem states that for any real function there is a dense subset of such that the restriction of to is continuous. For instance, the restriction of the Dirichlet function
Phragmen–Brouwer theorem
In topology, the Phragmén–Brouwer theorem, introduced by Lars Edvard Phragmén and Luitzen Egbertus Jan Brouwer, states that if X is a normal connected locally connected topological space, then the fol
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a compact convex set to itself there is a
Donaldson's theorem
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalis
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayev
Double suspension theorem
In geometric topology, the double suspension theorem of James W. Cannon and Robert D. Edwards states that the double suspension S2X of a homology sphere X is a topological sphere. If X is a piecewise-
Bing metrization theorem
In topology, the Bing metrization theorem, named after R. H. Bing, characterizes when a topological space is metrizable.
Bing's recognition theorem
In topology, a branch of mathematics, Bing's recognition theorem, named for R. H. Bing, asserts that a necessary and sufficient condition for a 3-manifold M to be homeomorphic to the 3-sphere is that
Pasting lemma
In topology, the pasting or gluing lemma, and sometimes the gluing rule, is an important result which says that two continuous functions can be "glued together" to create another continuous function.
Nagata–Smirnov metrization theorem
The Nagata–Smirnov metrization theorem in topology characterizes when a topological space is metrizable. The theorem states that a topological space is metrizable if and only if it is regular, Hausdor
Nielsen realization problem
The Nielsen realization problem is a question asked by Jakob Nielsen about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by Steven Kerckhoff .
Federer–Morse theorem
In mathematics, the Federer–Morse theorem, introduced by Federer and Morse, states that if f is a surjective continuous map from a compact metric space X to a compact metric space Y, then there is a B
Novikov's compact leaf theorem
In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have
Smale conjecture
The Smale conjecture, named after Stephen Smale, is the statement that the diffeomorphism group of the 3-sphere has the homotopy-type of its isometry group, the orthogonal group O(4). It was proved in
Schoenflies problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often refe
Annulus theorem
In mathematics, the annulus theorem (formerly called the annulus conjecture) states roughly that the region between two well-behaved spheres is an annulus. It is closely related to the stable homeomor
Sphere theorem (3-manifolds)
In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos gives conditions for elements of the second homotopy group of a 3-manifold to be represented by embedde
Esenin-Volpin's theorem
In mathematics, Esenin-Volpin's theorem states that weight of an infinite compact dyadic space is the supremum of the weights of its points.It was introduced by Alexander Esenin-Volpin. It was general