# Category: Homeomorphisms

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
Lantern relation
In geometric topology, a branch of mathematics, the lantern relation is a relation that appears between certain Dehn twists in the mapping class group of a surface. The most general version of the rel
Homeotopy
In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.
Local homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If is a local h
Uniform isomorphism
In the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces that respects uniform properties. Uniform spaces with uniform maps
Toronto space
In mathematics, in the realm of point-set topology, a Toronto space is a topological space that is homeomorphic to every proper subspace of the same cardinality. There are five homeomorphism classes o
Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous invers
Carathéodory's theorem (conformal mapping)
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that
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
Local property
In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some sufficiently small or arbitra
Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multi
Homeomorphism (graph theory)
In graph theory, two graphs and are homeomorphic if there is a graph isomorphism from some of to some subdivision of . If the edges of a graph are thought of as lines drawn from one vertex to another
Topological conjugacy
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flow
Y-homeomorphism
In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces. It can be constructed by sliding a Möbius band included on the surface aroun
Mapping torus
In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed in
Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition o
Quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded
Topological property
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological
Universal homeomorphism
In algebraic geometry, a universal homeomorphism is a morphism of schemes such that, for each morphism , the base change is a homeomorphism of topological spaces. A morphism of schemes is a universal
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 .
Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its de
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
Borel conjecture
In mathematics, specifically geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. I
Alexander's trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.
Self-similarity
In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as
Mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group