Homeomorphic in the context of Fundamental group

⭐ Core Definition: Homeomorphic

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and 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.

Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a line into a point. Some homeomorphisms do not result from continuous deformations, such as the homeomorphism between a trefoil knot and a circle. Homotopy and isotopy are precise definitions for the informal concept of continuous deformation.

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👉 Homeomorphic in the context of Fundamental group

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space $X$ is denoted by $\pi _{1}(X)$ .

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⭐ Core Definition: Homeomorphic
👉 Homeomorphic in the context of Fundamental group
Homeomorphic in the context of Manifold
Homeomorphic in the context of Topological property
Homeomorphic in the context of Invariance of dimension
Homeomorphic in the context of Solid torus

Homeomorphic in the context of Manifold

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.

One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

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Homeomorphic in the context of 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 property is a proper class of topological spaces which is closed under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.

A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are not homeomorphic, it is sufficient to find a topological property which is not shared by them.

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Homeomorphic in the context of Invariance of dimension

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space $\mathbb {R} ^{n}$ . It states:

The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem.

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Homeomorphic in the context of Solid torus

In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product $S^{1}\times D^{2}$ of the disk and the circle, endowed with the product topology.

A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.

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Homeomorphic in the context of Fundamental group

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⭐ Core Definition: Homeomorphic

👉 Homeomorphic in the context of Fundamental group

Homeomorphic in the context of Manifold

Homeomorphic in the context of Topological property

Homeomorphic in the context of Invariance of dimension

Homeomorphic in the context of Solid torus