The **Alexander horned sphere** is a pathological object in topology discovered by J. W.Alexander ( 1924 ).

The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:^{ [1] }

- Remove a radial slice of the torus.
- Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.
- Repeat steps 1–2 on the two tori just added
*ad infinitum*.

By considering only the points of the tori that are not removed at some stage, an embedding results in the sphere with a Cantor set removed. This embedding extends to the whole sphere, since points approaching two different points of the Cantor set will be at least a fixed distance apart in the construction.

The horned sphere, together with its inside, is a topological 3-ball, the **Alexander horned ball**, and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. The exterior is *not* simply connected, unlike the exterior of the usual round sphere; a loop linking a torus in the above construction cannot be shrunk to a point without touching the horned sphere. This shows that the Jordan–Schönflies theorem does not hold in three dimensions, as Alexander had originally thought. Alexander also proved that the theorem *does* hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between the categories of topological manifolds, differentiable manifolds, and piecewise linear manifolds became apparent.

Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space **R**^{3}. The closure of the non-simply connected domain is called the **solid Alexander horned sphere**. Although the solid horned sphere is not a manifold, R. H. Bing showed that its double (which is the 3-manifold obtained by gluing two copies of the horned sphere together along the corresponding points of their boundaries) is in fact the 3-sphere.^{ [2] } One can consider other gluings of the solid horned sphere to a copy of itself, arising from different homeomorphisms of the boundary sphere to itself. This has also been shown to be the 3-sphere. The solid Alexander horned sphere is an example of a crumpled cube; i.e., a closed complementary domain of the embedding of a 2-sphere into the 3-sphere.

One can generalize Alexander's construction to generate other horned spheres by increasing the number of horns at each stage of Alexander's construction or considering the analogous construction in higher dimensions.

Other substantially different constructions exist for constructing such "wild" spheres. Another example, also found by Alexander, is Antoine's horned sphere, which is based on Antoine's necklace, a pathological embedding of the Cantor set into the 3-sphere.

- Cantor tree surface
- List of topologies
- Platonic solid
- Wild arc, specifically the
**Fox–Artin arc**

In the mathematical field of topology, a **homeomorphism**, **topological isomorphism**, or **bicontinuous function** is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In the part of mathematics referred to as topology, a **surface** is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

In mathematics, **topology** is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

**Algebraic topology** is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In mathematical analysis, a **space-filling curve** is a curve whose range contains the entire 2-dimensional unit square. Because Giuseppe Peano (1858–1932) was the first to discover one, space-filling curves in the 2-dimensional plane are sometimes called *Peano curves*, but that phrase also refers to the Peano curve, the specific example of a space-filling curve found by Peano.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In the mathematical field of geometric topology, a **handlebody** is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds. Handles are used to particularly study 3-manifolds.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, the **Whitehead manifold** is an open 3-manifold that is contractible, but not homeomorphic to . J. H. C. Whitehead (1935) discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead where he incorrectly claimed that no such manifold exists.

In mathematics, a **4-manifold** is a 4-dimensional topological manifold. A **smooth 4-manifold** is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique.

In mathematics, topology generalizes the notion of triangulation in a natural way as follows:

In mathematics, a **manifold** is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or *n-manifold* for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space.

In topology, a branch of mathematics, a **topological manifold** is a topological space which locally resembles real *n*-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

The **Hauptvermutung** of geometric topology is the question of whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern.

In mathematics, a **solenoid** is a compact connected topological space that may be obtained as the inverse limit of an inverse system of topological groups and continuous homomorphisms

In the mathematical area of topology, the **generalized Poincaré conjecture** is a statement that a manifold which is a homotopy sphere *is* a sphere. More precisely, one fixes a category of manifolds: topological (**Top**), piecewise linear (**PL**), or differentiable (**Diff**). Then the statement is

In mathematics **Antoine's necklace** is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by Louis Antoine (1921).

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 referred to as the **Jordan–Schoenflies theorem.**

- ↑ Hocking & 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 of Mathematics*, Second Series,**56**: 354–362, doi:10.2307/1969804, ISSN 0003-486X, JSTOR 1969804, MR 0049549

- Alexander, J. W. (1924), "An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected",
*Proceedings of the National Academy of Sciences of the United States of America*, National Academy of Sciences,**10**(1): 8–10, Bibcode:1924PNAS...10....8A, doi:10.1073/pnas.10.1.8, ISSN 0027-8424, JSTOR 84202, PMC 1085500 , PMID 16576780 - Fuchs, Dmitry; Tabachnikov, Serge (2007),
*Mathematical Omnibus. 30 Lectures on Classical Mathematics*, Providence, RI: American Mathematical Society, doi:10.1090/mbk/046, ISBN 978-0-8218-4316-1, MR 2350979 - Hatcher, Allen,
*Algebraic Topology,*http://pi.math.cornell.edu/~hatcher/AT/ATpage.html - Hocking, John Gilbert; Young, Gail Sellers (1988) [1961].
*Topology*. Dover. ISBN 0-486-65676-4. - Spivak, Michael (1999).
*A comprehensive introduction to differential geometry (Volume 1)*. Publish or Perish. ISBN 0-914098-70-5.

- Weisstein, Eric W. "Alexander's Horned Sphere".
*MathWorld*. - Zbigniew Fiedorowicz. Math 655 – Introduction to Topology. – Lecture notes
- Construction of the Alexander sphere
- rotating animation
- PC OpenGL demo rendering and expanding the cusp

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.