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notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. PL, or more precisely PDIFF, sits between DIFF (the category of smooth

closely related to the stable homeomorphism conjecture (now proved) which states that every orientation-preserving homeomorphism of Euclidean space is stable

boundary, by some homeomorphism. This homeomorphism is called the monodromy of the surface bundle. It is possible to show that the homeomorphism type of the

disk. The extension has an important equivariance property: if the homeomorphism is composed on either side with a Möbius transformation preserving the

{z}}f(z)=\mu (z)\partial _{z}f(z)} for which f is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞. A similar result is true with C replaced

an open set U {\displaystyle U} containing x, a Lie algebra L and a homeomorphism ϕ : U × L → p − 1 ( U ) {\displaystyle \phi :U\times L\to p^{-1}(U)\

is homogeneous if for every two points x and y in X, there exists a homeomorphism h: X → X such that h(x) = y. A Peano continuum is a continuum that is

imposing the stronger condition that the homeomorphism also be an involution, i.e. have period 2 in the homeomorphism group of the 3-sphere, we arrive at the

space, and the binary operation is associative and commutative (up to homeomorphism). Sometimes the wedge sum is called the wedge product, but this is not

are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian. The theorem

Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk

{\displaystyle \simeq } the relation of topological equivalence (existence of homeomorphism). A starting point of the proof of the Anderson–Kadec theorem is Kadec's

characteristic 0 in terms of coadjoint orbits. More precisely, it is a homeomorphism from the space of orbits g*/G of the dual g* of g (with the Zariski

X {\displaystyle X} as a smooth manifold as well. That is, given a homeomorphism ϕ : X → M {\displaystyle \phi \colon X\to M} , one can define coordinate

Seifert-fibered 3-manifolds Union along their boundary, using the trivial homeomorphism Prime decomposition Essentially surfaces and 3-manifolds. The decomposition

equivalence can be taken to be a diffeomorphism, PL-isomorphism, or homeomorphism, respectively), the s-cobordism theorem states that if the manifolds

seminormal if every morphism Y → X {\displaystyle Y\to X} which induces a homeomorphism of topological spaces, and an isomorphism on all residue fields, is

the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological

(i=0,1) are equivalent if there exists a diffeomorphism (or PL-homeomorphism or homeomorphism) g : M 0 → M 1 {\displaystyle g:M_{0}\to M_{1}} such that f

the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a triangulation of the given space. Many topological

such that on the unit circle they differ by a given quasisymmetric homeomorphism. Several proofs are known using a variety of techniques, including the

S_{R}} which maps onto s ~ {\displaystyle {\tilde {s}}} restricts to a homeomorphism onto each open cell. 2. A finite two dimensional CW complex R ( S R

up to homeomorphism and diffeomorphism of compact Riemann surfaces implies the classification of closed orientable 2-manifolds up to homeomorphism and diffeomorphism:

is called an Akbulut cork, if F {\displaystyle F} extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an

numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinite

characterization of a Jordan curve as the image of a circle under a homeomorphism of the plane. In particular a quasicircle is a Jordan curve. The interior

decomposition, which is unique for invertible matrices, shows that there is a homeomorphism between GL(n, R) and the Cartesian product of O(n) with the set of positive-definite

also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero. Let C(T)

fundamental reason why surgery theory does not classify manifolds up to homeomorphism. Instead, for any fixed manifold M it classifies pairs ( N , f ) {\displaystyle

consisting of a compact metric space X {\displaystyle \textstyle X} and a homeomorphism T : X → X {\displaystyle \textstyle T:X\rightarrow X} . The topological

Michigan under Frank Raymond with thesis Necessary conditions for the homeomorphism of Seifert manifolds. He became in 1966 an assistant professor and in

\sum _{n=0}^{\infty }{\frac {2a_{n}}{3^{n+1}}}.} This mapping gives a homeomorphism from 2ω onto the Cantor set, demonstrating that 2ω is indeed a Cantor

P, then f has P. Open. Closed. Quasi-compact and a homeomorphism onto its image. A homeomorphism. Additionally, for each of the following properties

structure, on the fibers besides a topology. Then is required that the homeomorphism is an isomorphism with respect to that structure, and the conditions

is called the (free) mapping space from X to Y. Moreover, there is a homeomorphism Map ⁡ ( X × Y , Z ) ≃ Map ⁡ ( X , Map ⁡ ( Y , Z ) ) {\displaystyle \operatorname

Drawing a picture, there is a homeomorphism I 2 → I 2 {\displaystyle I^{2}\to I^{2}} that restricts to a homeomorphism K → I × { 0 } {\displaystyle K\to

Drawing a picture, there is a homeomorphism I 2 → I 2 {\displaystyle I^{2}\to I^{2}} that restricts to a homeomorphism K → I × { 0 } {\displaystyle K\to

short) consisting of a compact metric space X {\displaystyle X} and a homeomorphism T : X → X {\displaystyle T:X\rightarrow X} . A set E ⊂ X {\displaystyle

and z yields the required upper bound. A homeomorphism f of the unit circle can be extended to a homeomorphism F of the closed unit disk which is diffeomorphism

isometry extends to a conformal homeomorphism of the sphere at infinity (and conversely, every conformal homeomorphism on the sphere at infinity extends

about their derivatives, so, this equivalence contains the claim that a homeomorphism that is complex differentiable must actually have nonzero derivative

they are given the subspace topology inherited from the real line. A homeomorphism between Baire space and the irrationals can be constructed using continued

continuum is homogeneous, i.e. for any two of its points there is a homeomorphism taking one to the other. Yet also in 1948, Edwin Moise showed that Knaster's

Young 1988, pp. 175–176. Spivak 1999, p. 55 Bing, R. H. (1952), "A homeomorphism between the 3-sphere and the sum of two solid horned spheres", Annals

category theory: it does not uniquely characterize this space up to homeomorphism. For example, the disjoint union of a Sierpiński carpet and a circle

demonstrate that in the finite-dimensional situation, homotopy equivalence, homeomorphism, and diffeomorphism of manifolds are distinct properties. Although Sard's

submanifold of M. The flow of any continuous timelike vector field defines a homeomorphism S × ℝ → M. By considering the restriction of the inverse to another

space, extremally disconnected space T. (This space T is unique up to homeomorphism). where if in addition, Y is a vector space over the real numbers then

University. Due to the COVID-19 pandemic, not formally awarded yet. Homeomorphism groups 2018 Osvaldo Guzmán Postdoc, University of Toronto. 53rd Spring

general orientation preserving homomorphism f of S1 to itself we call a homeomorphism F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } a lift of f

{\displaystyle \mathbb {R} ^{d}\to \mathbb {R} ^{n}.} That is, there exists a homeomorphism U → R n {\displaystyle U\to \mathbb {R} ^{n}} such that the image of

of M {\displaystyle M} and f {\displaystyle f} fails to be a local homeomorphism at x} satisfies dimension ( B {\displaystyle B} f {\displaystyle f}

2-dimensional tori with one boundary component. The monodromy map is then a homeomorphism of the 2-torus, which can be represented in this case by the matrix

^{n+k})\times X\\\phi (V,v)=(V,p(v))\end{cases}}} which is clearly a homeomorphism. Hence, the result is a vector bundle of rank n. The above definition

n2 is NP-complete for arbitrarily small positive ε. The problem of homeomorphism of 2-complexes. The definability problem for first-order logic. The

known to be in NP, but it is unknown if it is NP-complete. Subgraph homeomorphism (for a fixed graph H) Graph genus Chordal graph completion Chromatic

form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent

The considered spaces are compact, so we get the same result up to homeomorphism. The cone over a point p of the real line is a line-segment in R 2 {\displaystyle

Conversely, let i: Z → X be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map i#: OX → i⋆OZ is

states that such spaces with finite dimension are characterised up to homeomorphism as the subspaces of the Euclidean spaces, with their usual topology

similar list of criteria is satisfied The map i {\displaystyle i} is a homeomorphism of Z {\displaystyle Z} onto its image The associated sheaf map O X →

ISBN 9780412990410. Hua Luogeng (1981) Starting with the unit circle, Springer Verlag, ISBN 0-387-90589-8. Homeomorphism and the Group Structure on a Circle

[0,1].} Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric, however

real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 1-manifolds without boundary

previous statement. Therefore, h cannot exist. In fact, there is no homeomorphism of R3 sending A to a set of Hausdorff dimension < 1, since the complement

speaking, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism, i.e. that closed Haken manifolds are topologically rigid. He put forward

In 1932 George Birkhoff described his "remarkable closed curve", a homeomorphism of the annulus that contained an invariant continuum. Marie Charpentier

{\displaystyle f} maps intervals to intervals, so is an open map and thus a homeomorphism. Since f {\displaystyle f} and the inverse function f − 1 : I 2 → I

flatten the hemisphere to make it into a disk. This transformation is a homeomorphism, so the Euler characteristic is still 1. However, on the left hand side

trivialization of E {\displaystyle E} , where φ {\displaystyle \varphi } is a homeomorphism from π − 1 ( U ) {\displaystyle \pi ^{-1}(U)} to U × F {\displaystyle

is Z2 when n > 1. (When n = 1 the fundamental group is Z due to the homeomorphism with S1). A generator for the fundamental group is the closed curve

{p}}\\U&{\xrightarrow {\text{id}}}&U\end{matrix}}} where φ is a Cr diffeomorphism (a homeomorphism, if r = 0) that carries F ∣ π − 1 ( U ) {\textstyle {\mathcal {F}}\mid

globally hyperbolic spacetime has a Cauchy surface S, and that the homeomorphism (as a C0-diffeomorphism) to S × R can be selected so that every surface

adds the topological data of a "marking", which can be seen as a fixed homeomorphism to the torus). To obtain the analytic moduli space (forgetting the marking)

and the inversion operation on this set is continuous (and hence is a homeomorphism), so that it forms a topological group under multiplication. If a Banach

has B((I + g)(x), r(1 − k)) ⊂ Ω′; I + g : Ω → Ω′ is a bi-Lipschitz homeomorphism; precisely, (I + g)−1 is still of the form I + h : Ω → Ω′ with h a Lipschitz

privileged component: the component of the identity. A specific choice of homeomorphism between B and GL(V) is equivalent to a choice of a privileged basis

\operatorname {Out} (F_{n})} , induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component. If φ ∈ Out ⁡ ( F

the logistically evolving variable as y n {\displaystyle y_{n}} , the homeomorphism is x n = 2 π sin − 1 ⁡ ( y n 1 / 2 ) . {\displaystyle x_{n}={\tfrac

null-homotopy follows. Since the Whitehead link is symmetric, that is, a homeomorphism of the 3-sphere switches components, it is also true that the meridian

the invariant. In symbols, if f : M → N {\displaystyle f:M\to N} is a homeomorphism of manifolds that pulls back ω N {\displaystyle \omega _{N}} to ω M

(extended) mapping class group of the torus, meaning that every self-homeomorphism of the torus is isotopic to a map of this form. The algebraic properties

the affine line k equipped with its Zariski topology. Because of this homeomorphism, some authors use the term affine line for the spectrum of k[t]. If

roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed)

neighbourhood of the image; and similarly for vertices. Thus P is locally a homeomorphism and so takes open sets to open sets. The image P(Σ), i.e. the union

Saito–Kurokawa lift of modular forms Rotation number uses a lift of a homeomorphism of the circle to the real line. Arithmetic geometry: Andrew Wiles (1995)

of M has precisely two preimages under π. In fact, π is even a local homeomorphism, because the preimages of the open sets U mentioned above are homeomorphic

won the Mary Ellen Rudin Award for "her deep and extensive work on homeomorphism groups of manifolds". The Association for Women in Mathematics gave

of K by hypothesis and barycentric subdivision does not change the homeomorphism type of the realization. If X is a topological space with open cover

That same year, she married Novikov and published three papers: On the Homeomorphism of Canonical Elements of the 3rd Class; On Simple Functions of Class

in 2019. LaPaugh, Andrea S.; Rivest, Ronald L. (1980), "The subgraph homeomorphism problem", Journal of Computer and System Sciences, 20 (2): 133–149,

the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive

MR 0003191 Dijkstra, Jan J.; van Mill, Jan (2010), "Erdős space and homeomorphism groups of manifolds" (PDF), Memoirs of the American Mathematical Society

S 1 → R n {\displaystyle f:S^{1}\rightarrow \mathbb {R} ^{n}} be a homeomorphism and Γ = f ( S 1 ) {\displaystyle \Gamma =f(S^{1})} . Then f ∈ C 1 ,

T : X → X {\displaystyle \textstyle T:X\rightarrow X} be the shift homeomorphism ( … , x − 2 , x − 1 , x 0 , x 1 , x 2 , … ) → ( … , x − 1 , x 0 , x

that every quasiconformal homeomorphism of the disk is Hölder continuous and therefore extends continuously to a homeomorphism of the unit circle (it is

{\displaystyle S'} is simply connected, φ {\displaystyle \varphi } is a homeomorphism, and hence, a (global) isometry. Therefore, H {\displaystyle H} and

during the Lwów Period, he introduced the concept of so-called quasi homeomorphism that opened up a new field in topological studies. Kuratowski's research

S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane. Unlike Lebesgue's and Brouwer's generalization

{\displaystyle \iota ^{-1}} is also continuous, so ι {\displaystyle \iota } is a homeomorphism. Therefore the topology on a topologically finitely generated profinite

is within every prime, to show the spectrum of the quotient map is a homeomorphism, this reduces to the irreducibility of the spectrum of an integral domain

X ∗ , {\displaystyle X^{*},} which is continuous, and thus yields a homeomorphism of top Stiefel manifolds V n ( X ) → ∼ V n ( X ∗ ) . {\displaystyle

representing homology classes by real algebraic subvarieties up to homeomorphism, Compositio Mathematica, 53, (1984), 143–151. S. Akbulut and H.C. King

Alper, T. M. (1987). "A classification of all order-preserving homeomorphism groups of the reals that satisfy finite uniqueness". Journal of Mathematical

{\displaystyle \exp :\operatorname {Lie} (G)\to G} defines a local homeomorphism from a neighborhood of the zero vector to the neighborhood of the identity

(n-i-1)} -dimensional stratified space L, and a filtration-preserving homeomorphism U ≅ R i × C L {\displaystyle U\cong \mathbb {R} ^{i}\times CL} . Here

exponential map A ↦ e i A {\displaystyle A\mapsto e^{iA}} is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of

exponential map A ↦ e i A {\displaystyle A\mapsto e^{iA}} is a local homeomorphism from the space of self-adjoint complex matrices to U(n). The space of

)).} It is a map over R n {\displaystyle \mathbb {R} ^{n}} and is a homeomorphism since G ( x ) = ( φ 1 ( − t 1 , ⋯ , φ n ( − t n , x ) ⋯ ) , t ) , t

(n-i-1)} -dimensional stratified space L, and a filtration-preserving homeomorphism U ≅ R i × C L {\displaystyle U\cong \mathbb {R} ^{i}\times CL} . Here

a recursive construction using CW-complexes. Recall that there is a homeomorphism C P 1 ≅ S 2 {\displaystyle \mathbf {CP} ^{1}\cong S^{2}} to the 2-sphere

{C}})=\exp({\overline {C}})\cdot G.}} Moreover, the map (g,x) ↦ g exp x is a homeomorphism. In fact if X is in C, it is diagonalizable with real eigenvalues. So

the change in functions Uniform isomorphism – Uniformly continuous homeomorphism Uniform property – Object of study in the category of uniform topological

distance between all points Uniform isomorphism – Uniformly continuous homeomorphism Rusnock & Kerr-Lawson 2005. Bourbaki, Nicolas. General Topology: Chapters

\varinjlim X_{\alpha }} to X , {\displaystyle X,} which is in fact a homeomorphism. Here lim → ⁡ X α {\displaystyle \varinjlim X_{\alpha }} is the direct

1)-tuple as its barycentric coefficient. This correspondence is an affine homeomorphism. Aitchinson geometry is a natural way to construct an inner product

open neighbourhood U {\displaystyle U} of p {\displaystyle p} and a homeomorphism φ : U → V {\displaystyle \varphi :U\to V} to an open subset V ⊆ C n

many-to-one map. Note that it is not a covering map – while it is a local homeomorphism near the origin, it is not a covering map at rotations by 180 degrees

homeomorphic. The function from X to itself which swaps a and b is a homeomorphism. A topological space homeomorphic to one of these is called a Sierpiński

graph dual) represents the same manifold. In Poincaré duality, this homeomorphism is reflected in an isomorphism of the kth homology group and the (n − k)th

this fact; see [2]. A morphism between algebraic varieties that is a homeomorphism between the underlying topological spaces need not be an isomorphism

quasi-isometry between two hyperbolic spaces X , Y {\displaystyle X,Y} induces a homeomorphism between the boundaries. In particular the group of isometries of X {\displaystyle

"Gowers' Ramsey theorem with multiple operations and dynamics of the homeomorphism group of the Lelek fan". Journal of Combinatorial Theory, Series A.

Providence, R.I., 1980. T. Kobayashi, A construction of 3-manifolds whose homeomorphism classes of Heegaard splittings have polynomial growth, Osaka J. Math

Poincaré–Birkhoff theorem: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions

homeomorphic to a circle. Thus it is metrizable corresponding (for a given homeomorphism) to the ordinary metric on this circle (either measured straight or

φ([x1, ..., xn], [y1, ..., yn]) = ([x1, ..., xn, y1, ..., yn]) is a homeomorphism for compact X and Y. But this implies that either π1(SP(CH)) ≅ H1(CH)

curvatures all equal to − 1 {\displaystyle -1} ) and f {\displaystyle f} a homeomorphism, up to a natural equivalence relation. The Teichmüller space can be

and some ω, then ω ∈ Ap. For K > 1, a K-quasiconformal mapping is a homeomorphism  f  : Rn →Rn such that f ∈ W l o c 1 , 2 ( R n ) ,  and  ‖ D f ( x )

Hopcroft, John; Wyllie, James (1980-02-01). "The directed subgraph homeomorphism problem". Theoretical Computer Science. 10 (2): 111–121. doi:10

{\displaystyle e:S^{d}\hookrightarrow S^{d+1}} such that there exists a homeomorphism h : S d + 1 → S d + 1 {\displaystyle h:S^{d+1}\rightarrow S^{d+1}} so

{\displaystyle e:S^{d}\hookrightarrow S^{d+1}} such that there exists a homeomorphism h : S d + 1 → S d + 1 {\displaystyle h:S^{d+1}\rightarrow S^{d+1}} so

f:X\to Z} , Z {\displaystyle Z} some topological space, is a local homeomorphism that is injective on A {\displaystyle A} , then f {\displaystyle f}

uncurrying; in many special cases, they are also continuous and form a homeomorphism. Quantification. If ϕ Y {\displaystyle \phi _{Y}} is a unary predicate

[L]=\deg _{L}(c_{\mathbb {P} H}|_{L})=1} . Since T {\displaystyle T} is a homeomorphism, T ∗ c P K {\displaystyle T^{*}c_{\mathbb {P} K}} also generates H 2

Since the base space is contractible, this can be extended to a global homeomorphism.[clarification needed] Because the restricted Lorentz group SO+(1, 3)

× W T → G {\displaystyle G/T\times _{W}T\to G} and the latter is a homeomorphism on regular points and so has degree one. Hence, the degree of q {\displaystyle

⁡ S {\displaystyle G(\varepsilon )\to \operatorname {Proj} S} is a homeomorphism of the underlying topological space of X with its image. For every quasi-coherent

convergence, in short the product topology. It is the ability of self homeomorphism as well as the ability to embed every space into such a space, denoted

uniformly continuous extension of f {\displaystyle f} (which is also a homeomorphism). Complete subsets Every complete subset of a TVS is sequentially complete

Equivalently, f is a closed immersion if, and only if, it induces a homeomorphism from the underlying topological space of Y to a closed subset of the

correspondence as follows: Theorem (Moduli space version): There is a homeomorphism of the moduli space of polystable holomorphic vector bundles over (

i ) {\displaystyle \varphi _{i}:U_{i}\to \varphi _{i}(U_{i})} is a homeomorphism and φ j ∘ φ i − 1 : φ i ( U i ∩ U j ) → φ j ( U i ∩ U j ) {\displaystyle

collapsing stable and unstable laminations of the monodromy pseudo-Anosov homeomorphism of S for this fibration of M. In particular, this description implies

{Z} ^{+}} . Choose λ = 1 {\displaystyle \lambda =1} and define the homeomorphism h : ω Z + → [ 0 , 1 ] {\displaystyle h:\omega \mathbb {Z} ^{+}\rightarrow

H_{1}\rightrightarrows H_{0}} preserves their transverse geometry, i.e. it induces: a homeomorphism between the orbit spaces G 0 / G 1 {\displaystyle G_{0}/G_{1}} and H

whereas the complex plane nor the unit dis are not, so there is not even homeomorphism for (1) onto (2) or (3). A conformal equivalence of (2) onto (3) would

{\mathbb {Z} }}} is a G R {\displaystyle G_{\mathbb {R} }} -equivariant homeomorphism. So we get G Z ∖ G R → ∼ G Q ∖ G A / G Z ^ {\displaystyle G_{\mathbb

the taxonomic implications of microstructure to disentangle cases of homeomorphism". Historical Biology. 34 (12): 2304–2314. Bibcode:2022HBio...34.2304R

defines a second isomorphism. By composing both isomorphisms, a linear homeomorphism which transfers the two topologies into each other is obtained. More